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X-ray Charge Density Analysis and the XD programming package Buffalo NY,USA 13-17 May 2003

X-ray Charge Density Analysis and the XD programming package Buffalo NY,USA 13-17 May 2003. Carlo Gatti CNR-ISTM, Milano, Italy. Search of critical points of various scalar fields in crystals. TOPXD Sections.  f( r cp ) = 0 f   or  2 . (3,-3). Local maxima. NNA (nuclei *).

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X-ray Charge Density Analysis and the XD programming package Buffalo NY,USA 13-17 May 2003

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  1. X-ray Charge Density Analysis and the XD programming package Buffalo NY,USA13-17 May 2003 Carlo Gatti CNR-ISTM, Milano, Italy Search of critical points of various scalar fields in crystals

  2. TOPXD Sections

  3. f(rcp) = 0 f   or 2 (3,-3) Local maxima NNA (nuclei *) f   (3,-1) 1D saddle bonds (3,+1) 2D saddle rings (3,+3) Local minima cages * No CP if Slater bfs are used as in TOPXD CP search • Search algorithms (NR, EF) • Search strategies (selection of starting points)

  4. g i > 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 vi Fi I. CP search: Newton Raphson Method • (x0+h) = (x0) + g+h + 1/2 h+ h+ …… • NR step x0  CPh= - -1g • -1=ii-1vivi+; h= -ii-1vivi+g = -ii-1viFi • with Fi =vi+g projection of the gradient along the local eigenmode vi

  5. x0  No hope in general BCP  , 3,+1 , 3,-1 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.

  6. A modified NR algorithm with a suitable and locally defined shift s for the NR step hNR= -ii-1viFi hEF= -i(i – s)-1viFi Spfor eigenmodes along which the function is to be maximized s Snfor eigenmodes along which the function is to be minimized (*) A. Banerjee, et. al., J. Phys. Chem. 89, 52 (1985) (*) P.L.A. Popelier, Chem. Phys. Lett. 228, 160 (1994) III. CP search: Eigenvector Following Method (*)

  7. h1 1 0 F1 h1 Example: search of a (3,-1) CP Sp= highest eigenvalue 0 2 F2 h2 h2 = Sp hNR= -ii-1viFi hEF= -i(i – s)-1viFi F1 F2 0 1 1 h3 h3 3 F3 = Sn Sn= lowest eigenvalue F3 0 1 1 IV. CP search: Eigenvector Following Method

  8. h2 2 0 F2 h2 Example: search of a (3,+1) CP Sn= lowest eigenvalue 0 3 F3 h3 h3 = Sn hNR= -ii-1viFi hEF= -i(i – s)-1viFi F2 F3 0 1 1 h1 h1 1 F1 = Sp Sp= highest eigenvalue F1 0 1 1 V. CP search: Eigenvector Following Method

  9. Example: search of a (3,+3) CP 1 0 0 F1 h1 h1 hNR= -ii-1viFi hEF= -i(i – s)-1viFi 0 2 0 F2 h2 h2 = Sn Sn= lowest eigenvalue 0 0 3 F3 h3 h3 F1 F2 F3 0 1 1 VI. CP search: Eigenvector Following Method (*) 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

  10. 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.)

  11. II. Search strategies  TRHO (*)seed (*)all (*)ail (*)debug nstep nstep nnb nnb rmax rmax th th  [ (*)fra | (*)car ] x y z  . . . . . . . . . . . . . . . . . . . • Fully-automated and chain­like 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 built­up 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.

  12. III. Search strategies  Fully automated and chain-like CP search • 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. • 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.

  13. TRHO (*)seed (*)all (*)ail (*)debug nstep nstep nnb nnb rmax rmax th th  [ (*)fra | (*)car ] x y z  . . . . . . . . . . . . . . . . . . . (*) allif activated all kinds of CPs are searched for, otherwise chain-like search is stopped after (3,-1) CPs search 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. (*) debugif activated debug printing during the CP search

  14. Seed point     nnb =3 3 stars of symmetry related atoms Each star may have a different atom’s multiplicity (here 3,4,3)        V. Search strategies  TRHO (*)seed (*)all (*)ail (*)debug nstep nstep nnb nnb rmax rmax th th  [ (*)fra | (*)car ] x y z  . . . . . . . . . . . . . . . . . . . nstepnstep maximum number of EF steps along each CP search nnbnnb maximum number of symmetry related stars of atoms to be included in the cluster generated around the “seed point”.

  15. Red sphere has radius rmax. If nnb was settled as to include all the yellow nuclei, rmax excludes some of them Seed point VI. Search strategies  TRHO (*)seed (*)all (*)ail (*)debug nstep nstep nnb nnb rmax rmax th th  [ (*)fra | (*)car ] x y z  . . . . . . . . . . . . . . . . . . . rmaxrmax 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. thth if th0., (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)

  16. TRHO (*)cluster (*)all (*)ail (*)debug nstep nstep nnb nnb rmax rmax th th • Fully-automated and chain­like 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. (3,-1) CPs are searched for among the unique atom pairs y-z, in the cluster AB C, with Ry-z th ( A,B,C being the unique atoms in the unit cell) B A C VII. Search strategies 

  17. 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 ABC…. Y (A,B,C…Y being the unique atoms in the unit cell) • It differs fromTRHOclustersearch since: • it is not a chain-like search; • 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)] • it allows to make a search for a specific kind of CP (if EF method is selected)

  18. IX. Search strategies  TRHO (*)pairs meth (*)ail (*)debug nstep nstep nnb nnb rmax rmax th th {pc pc } methmeth 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 ifmeth =an {pc pc } only ifmeth = nr pc0 : if a CP is not found between A­B atom pair, the starting point of the NR search is displaced, along the internuclear axis, from the axis mid-point to: rstart = rA+pc *(rB rA ); r''start = rA+(1. pc)*(rB rA ), pc=0 : specifies the default value of pc (0.4).

  19. 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 CPSEARCHpoint since it allows to select the CP search algorithm and (meth = EF) the kind of CP to be searched for. 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 zCoord. of the starting point. Repeat this line n times for n starting points

  20. TRHO (*)line meth (*)ail (*)debug nstep nstep nnb nnb rmax rmax  line specification • This is CP search along a line. It differs from XDPROP CPSEARCHbond 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. 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

  21. 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

  22. CLUSTERS AROUND EACH OF THE UNIQUE ATOMS UNIQUE ATOM 1 O(1) (N. 1 in UNIT CELL), CLUSTER OF 11 ATOMS NEW  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 Ice VIII Label of unique atom 1 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.

  23. 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 searchalgorithm 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 valuefor 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)

  24. 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: • it allows to select the search algorithm (NR, EF) • 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.

  25. methmeth 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 nstepmaximum number of EF or NR steps for each search (10-15) ntheta nthetaCP 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 andnphi: intervals for polar coordinates  and  II. Search strategies -2 TLAP (*)auto meth (*)ail (*)debug nstep nstep nnb nnb rmax rmax ntheta ntheta nphi nphi

  26. 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 nmaxdifferent CPs of the required type are found (only with EF) rstar = 0 default value  0 sphere radius (Å) NNA specifications (*)nnaxxyyzznmaxnmax rstarrstar

  27. 2nd row atom VSCC max VSCD max 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]

  28. 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 V. Search strategies -2

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