X-ray Charge Density Analysis and the XD programming package

Buffalo NY,USA13-17 May 2003

Search of critical points of various scalar fields in crystals

Carlo Gatti

CNR-ISTM, Milano, Italy

TOPXD Sections

TRHO Topological analysis of (r)

TLAP Topological analysis of 2 (r)

ATBP Atomic basin properties

PL2D 2D plots

PL3D 3D plots

VZ3D 3D visualization of atomic basins

CP search

Search algorithms (NR, EF)

Search strategies (selection of starting points)

f(rcp) = 0 f or 2

cagesLocal minima(3,+3)

rings2D saddle(3,+1)

bonds1D saddle(3,-1)

NNA (nuclei *)Local maxima(3,-3)

* No CP if Slater bfs are used as in TOPXD

f

I. CP search: Newton Raphson Method

(x0+h) = (x0) + g+h + 1/2 h+ h+ ……

NR step x0 CP h= - -1g

-1= ii-1 vi vi

+; h= -ii-1 vi vi

+g = -ii-1 viFi

with Fi = vi+g projection of the gradient along the local eigenmode vi

viFi

gi > 0 h

i < 0 h

NR step

Minimizes the scalar (, -2 ) along modes with positive Hessian eigenvalues

Maximizes the scalar (goes in the same direction of g) along modes with negative Hessian eigenvalues

II. CP search: Newton Raphson Method

NR method is suitable for the location of a CP only when one moves in a region where has the same structure (i.e. the

same number of positive and negative eigenvalues) as the CP that is searched for.

This is a problem, in particular with 2 field, since 2 generally varies quite rapidly in R3.

No hope in generalx0

BCP , 3,+1 , 3,-1

III. CP search: Eigenvector Following Method (*)

(*) A. Banerjee, et. al., J. Phys. Chem. 89, 52 (1985)

(*) P.L.A. Popelier, Chem. Phys. Lett. 228, 160 (1994)

A modified NR algorithm with a suitable and locally

defined shift s for the NR step

hNR= -ii-1 viFi hEF= -i(i – s)-1 viFi

s

Sp for eigenmodes along which the

function is to be maximized

Sn for eigenmodes along which the

function is to be minimized

IV. CP search: Eigenvector Following Method

0F2F1

F220

F101

1

h2

h1

= Sp

1

h2

h1

Sp = highest eigenvalue

= Sn 0F3

F33

1

h3

1

h3

Sn = lowest eigenvalue

hNR= -ii-1 viFi hEF= -i(i – s)-1 viFi

Example: search of a (3,-1) CP

V. CP search: Eigenvector Following Method

0F3F2

F330

F202

1

h3

h2

= Sn

1

h3

h2

Sn = lowest eigenvalue

= Sp 0F1

F11

1

h1

1

h1

Sp = highest eigenvalue

hNR= -ii-1 viFi hEF= -i(i – s)-1 viFi

Example: search of a (3,+1) CP

VI. CP search: Eigenvector Following Method (*)

hNR= -ii-1 viFi hEF= -i(i – s)-1 viFi

Example: search of a (3,+3) CP

0F3F2F1

F3300

F2020

F1 001

1

h3

h2

h1

= Sn Sn = lowest eigenvalue

1

h3

h2

h1

EF method locates reliably all kind of CPs. The method seeks for the CPs of a given kind (e.g. 3,+3) regardless of the structure at

the starting point. Separate searches for (3,-3), (3,-1), (3,+1) and

(3,+3) CPs are implemented

I. Search strategies

Fully automated and chain-like search for some (or all) kinds of CPs

Exhaustive grid search in the asymmetric unit

More standard searches (along a line, from a given set of starting points, etc.)

II. Search strategies

TRHO (*)seed (*)all (*)ail (*)debug nstep nstep nnb nnb rmax rmax th th

[ (*)fra | (*)car ] x y z

. . . . . . . . . . . . . . . . . . .

Fully-automated and chain like search strategy for all kinds of critical points (*) using at each stage the EF step specific for the kind of CP searched for.

The search is performed within a finite region of space, which encloses a finite molecular cluster built up around a specified “seed point” A. Size and origin of the cluster are defined by input.

* Search of(3,-3) CPs associated to nuclear maxima is skipped in TOPXD. NNAs recovered either as termini of a BP or in the grid search for CPs.

III. Search strategies

search of (3,-3) associated to nuclear maxima, starting from the nuclear position of each of the unique atoms of the unit cell

Not performed in TOPXD

search of all (3,-1) unique bcps associated to the unique bonded atom pairs within the cluster. Search started from internuclear axis midpoint. Non-nuclear (3,-3) attractors, if any, are recovered at this stage by determining the nature of the termini of the atomic interaction lines associated to the unique (3,-1) CPs found.

Fully automated and chain-like CP search

search of unique (3,+1) rcps by considering all unique nuclear triplets having at least two associated atoms bonded to each other and center of mass (with mass 1 given to any nucleus) not too differently distant from each of the three nuclei. CP search started from the center of mass.

search of unique (3,+3) CPs between all pairs of ring CPs.

IV. Search strategies

(*) ail if activated atomic AIL lengths and termini are evaluated numerically for each unique (3,-1)CP. This is costly, but the only safe way to know to which nuclei a bcp is linked to. The associated ordinary differential equations are solved using a 5th order Runge-Kutta method with monitoring of local truncation error and an adaptive stepsize control. Atomic interaction lines are generally determined with less than 80-130 integration steps and 500-1200 and evaluations.

(*) debug if activated debug printing during the CP search

TRHO (*)seed (*)all (*)ail (*)debug nstep nstep nnb nnb rmax rmax th th

[ (*)fra | (*)car ] x y z

. . . . . . . . . . . . . . . . . . .

(*) all if activated all kinds of CPs are searched for, otherwise chain-like search is stopped after (3,-1) CPs search

V. Search strategies

TRHO (*)seed (*)all (*)ail (*)debug nstep nstep nnb nnb rmax rmax th th

[ (*)fra | (*)car ] x y z

. . . . . . . . . . . . . . . . . . .

nstep nstep maximum number of EF steps along each CP search

nnb nnb maximum number of symmetry related stars of atoms to be included in the cluster generated around the “seed point”.

nnb =3

3 stars of symmetry related atoms Each star may have a different atom’s multiplicity (here 3,4,3)

Seed point

VI. Search strategies

TRHO (*)seed (*)all (*)ail (*)debug nstep nstep nnb nnb rmax rmax th th

[ (*)fra | (*)car ] x y z

. . . . . . . . . . . . . . . . . . .

rmax rmax maximum radius of the cluster (Å). Each cluster includes all atoms within a sphere of radius rmax, centered on the “seed-point”. rmax may reduce the actual value of nnb.

th th if th0., (3, 1) CP search is performed among all the unique atom pairs whose internuclear distance falls below th (Å), otherwise (th=0.) the default value is used (5Å)

Red sphere has radius rmax. If nnb was settled as to include all the yellow nuclei, rmax excludes some of them

Seed point

VII. Search strategies

TRHO (*)cluster (*)all (*)ail (*)debug nstep nstep nnb nnb rmax rmax th th

Fully-automated and chain like search strategy for all kinds of critical points (*) using at each stage the EF step specific for the kind of CP searched for.

The search is performed within a finite region of space, which is defined by building-up a “supercluster” defined as the union of the separate clusters build around each of the unique atoms in the unit cell. Size of the atomic clusters is given in input.

* Search of(3,-3) CPs associated to nuclear maxima is skipped in TOPXD. NNAs recovered either as termini of a BP or in the grid search for CPs.

AB

C

(3,-1) CPs are searched for among the unique atom pairs y-z, in the cluster AB C, with Ry-z th ( A,B,C being the unique atoms in the unit cell)

VIII. Search strategies

TRHO (*)pairs meth (*)ail (*)debug nstep nstep nnb nnb rmax rmax th th {pc pc }

This is CP search among the unique atoms pairs in the cluster ABC…. Y (A,B,C…Y being the unique atoms in the unit cell)

It differs from TRHO cluster search since:

1. it is not a chain-like search;

2. it allows to select the search algorithm (NR, EF or Cioslowski’s analytical simultaneous determination of bcps and AILs [J. Cioslowski et al., CPL 219, 151 (1994)]

3. it allows to make a search for a specific kind of CP (if EF method is selected)

IX. Search strategies

TRHO (*)pairs meth (*)ail (*)debug nstep nstep nnb nnb rmax rmax th th {pc pc }

meth meth is the algorithm used for the CP search :

nr NR

ef type (one of the following keywords: ncp, bcp, rcp, ccp)

an Cioslowsky’s analytical simultaneous determination of AIL and bcp

ail don’t use if meth = an

{pc pc } only if meth = nr

pc0 : if a CP is not found between A B atom pair, the starting point of

the NR search is displaced, along the internuclear axis, from the axis

mid-point to: rstart = rA+pc *(rB rA ); r''start = rA+(1. pc)*(rB rA ),

  pc=0 : specifies the default value of pc (0.4).

X. Search strategies

nnb nnb In this case there is no “cluster(s)” construction and nnb defines

just the number of start of neighbors in the nearest neighbor

analysis around each unique recovered CP.

[fra | car ] Fractional (fra) or cartesian (car), Å, coordinates

x y z Coord. of the starting point. Repeat this line n times for n starting points

TRHO (*)points meth (*)ail (*)debug nstep nstep nnb nnb rmax rmax [fra | car ]

x y z

…..

This is CP search from a starting set of points. It differs from XDPROP CPSEARCH point since it allows to select the CP search algorithm and (meth = EF) the kind of CP to be searched for.

XI. Search strategies

Meth nr strongly recommended unless one is looking for a specific kind

of CP and does not care of the other CPs which may be

present along the line)

ef type (one of the following keywords: ncp, bcp, rcp, ccp)

Nstep nstep NR or EF steps for each search. Use a small nstep value, say no more

than 8, since the search is repeated 40 times, starting from 40 evenly

distributed points along the line

TRHO (*)line meth (*)ail (*)debug nstep nstep nnb nnb rmax rmax

line specification

This is CP search along a line. It differs from XDPROP CPSEARCH bond since it allows to select any line (not necessary an internuclear axis), the CP search algorithm and (meth = EF) the kind of CP to be searched for.

XII. Search strategies

TRHO (*)line meth (*)ail (*)debug nstep nstep nnb nnb rmax rmax

line specification

line specification given in one of the formats:

(*)atom label toneighbor i1… i(n)

CP search along the line(s) joining the unique atom A with label label and its i1..i(n) neighbor(s) (atom B), where i is the “NEW'” number in the “Clusters around each of the unique atoms” printing at the beginning of the TOPXD output.

(*)point [ car | fra ] x1 y1 z1 x2 y2 z2

CP search along the line joining two points a and b with coordinates (x1 y1 z1) and (x2 y2 z2), respectively. The coordinates are in Cartesian (Å) (car) or fractional (fra) units. The search will only be performed if keyword point is activated (i.e., *point). Repeat this line n times for n point pairs

XIII. Search strategies

line specification

(*)atom label toneighbor i1… i(n)

CP search along the line(s) joining the unique atom A with label label and its i1..i(n) neighbor(s) (atom B), where i is the “NEW'” number in the “Clusters around each of the unique atoms” printing at the beginning of the TOPXD output.

CLUSTERS AROUND EACH OF THE UNIQUE ATOMS

UNIQUE ATOM 1 O(1) (N. 1 in UNIT CELL), CLUSTER OF 11 ATOMSNEW  OLD   CELL  ATOM           COORD.(ANG)       DISTANCE (ANG)  1     9    0 0 0  H(1)        0.000   0.393   1.311          0.968  2   23    0 0 0   H(1)        0.000   1.935   1.311          0.968  3   12    0 0-1  H(1)        1.557   1.164  -0.383         1.911  4   19  -1 0-1  H(1)       -1.557   1.164  -0.383         1.911  5   13    0-1-1  H(1)        0.000  -0.393  -1.311         2.563  6  24    0 0-1  H(1)       0.000   2.721  -1.311         2.563  7    5    0-1-1   O(1)        0.000  -1.164  -0.726         2.743  8    5    0 0-1  O(1)       0.000   3.492  -0.726         2.743  9   15  -1 0 0  H(1)       -2.328    0.393   2.077         2.800 10    15   0 0 0  H(1)        2.328   0.393   2.077         2.800 11   22  -1 0 0  H(1)       -2.328   1.935   2.077         2.800

Label of

unique atom 1

Ice VIII

XIV. Search strategies

TRHO (*)grid meth (*)ail (*)debug nstep nstep nnb nnb rmax rmax

xmin xmin xmax xmax xstep xstep

ymin ymin ymax ymax ystep ystep

zmin zmin zmax zmax zstep zstep

This is a grid-search for CPs in a given portion of the cell. Warning: the grid search is very costly if the whole asymmetric unit is explored.

The CP search algorithm can be chosen (EF or NR). NR is strongly recommend here, unless one is looking for a specific kind of CP in the cell volume explored and does not care of other CP types that may be there present.

Use a small value for nstep provided the grid step is small enough ( 0.2-0.3 Å)

Grid intervals and steps in fractional units. Space group constraints among x,y,z fractional coordinates to be included in TOPXD (implemented in TOPOND)

I. Search strategies -2

TLAP (*)auto meth (*)ail (*)debug nstep nstep nnb nnb rmax rmax ntheta ntheta nphi nphi

< atom(s) specifications >

< NNA specifications >

. . . . . . . . . . . . . . . . . . . . .  

This is a systematic search of -2 Cps. It bears some resemblance to XDPROP CPSEARCH SHELL, but differs from it since:

1. it allows to select the search algorithm (NR, EF)

2. it allows to make a search for a specific kind of CP (if EF method is selected)

Choice of starting points takes into account that the Laplacian distribution retains an atomic-like portrait, even following chemical combination.

II. Search strategies -2

TLAP (*)auto meth (*)ail (*)debug nstep nstep nnb nnb rmax rmax ntheta ntheta nphi nphi

meth meth is the algorithm used for the -2 CP search :

nr NR

ef type (one of the following keywords: cccp, s1cp, s2cp, cdcp)

(3,-3) (3,-1) (3,+1) (3,+3)

ail if activated AGL lengths and termini are evaluated numerically for each

unique (3,-1) CP. AGL is the union of the unique pair of (-2 )

trajectories that originate at the (3,-1) -2 CP and terminate at

neighboring (3,-3) -2 CPs

nstep nstep maximum number of EF or NR steps for each search (10-15)

ntheta ntheta CP search is started from points located on the surface of a sphere

centered on the nucleus of a given unique atom or at NNA location

nphi nphi ntheta and nphi : intervals for polar coordinates and

III. Search strategies -2

TLAP (*)auto meth (*)ail (*)debug nstep nstep nnb nnb rstar rstar ntheta ntheta nphi nphi

< atom(s) specifications >

< NNA specifications >

. . . . . . . . . . . . . . . . . . . . .  

Atom(s) specifications

(*)atoms label1 ….label (n) nmax nmax rstar rstar

nmax = 0 normal search

0 the search is stopped when nmax different CPs of the required type are

found (only with EF)

rstar = 0 default value

0 sphere radius (Å)

NNA specifications

(*)nna x x y y z z nmax nmax rstar rstar

IV. Search strategies -2

rstar by default: the distance from the nucleus to the spherical surface where -2 attains its maximum value in the VSCC of the isolated atom).

By using different values of rstar, one explores different region of CC (negative laplacian) or CD (positive Laplacian) or one may take into account that the size of the VSCC has undergone a substantial change in size as due to the considerable CT which occurs in some solids, like the ionic crystals [ Ca (2+) vs Ca or O(2-) vs O]

2nd row atom

VSCC max

VSCD max

V. Search strategies -2

TLAP (*)points meth (*)ail (*)debug nstep nstep nnb nnb rmax rmax nmax nmax

[(*)car | (*)fra ] x y z

…..

If meth = ef Insert also type

cccp (3,-3)

s1cp (3,-1)

s2cp (3,+1)

cdcp (3,+3)

To stop the search when it has been already successful