Charge density topological analysis in CRYSTAL

Ab Initio
Modelling in
Solid State
Chemistry
Turin (Italy)
1-5 September 2013
Charge Density Topological Analysis
in CRYSTAL
Carlo Gatti
CNR-ISTM - Milano
Outline
Deformation electron densities, : shortcomings
Topology of the total density  and the elements of
chemical structure (QTAIM). How it is implemented in
CRYSTAL. Examples in solid state
The Laplacian of the electron density,2 : atomic shell
structure, electron pairs and Lewis model, classification of chemical
interactions, chemical reactivity, H-bonds mechanisms
Topology of 2. How it is implemented in CRYSTAL
Examples in solid state
Atomic properties and applications. How are they
determined in CRYSTAL. Examples in solid state
Plotting scalar and vectorial fields of interest to
QTAIM in CRYSTAL. Examples in solid state
1
QTAIM basic bibliography
Bader, R. F. W., Atoms in Molecules: A Quantum Theory. International
Series of of Monographs on Chemistry, 22, (1990) Oxford Science
Publications
The Quantum Theory of Atoms in Molecules: from Solid State to DNA
and Drug Design, C. Matta and R. Boyd (Eds.), Wiley-VCH, 2007;
chapters 7 (Gatti) and 8 (Martin Pendas et al.) largely dedicated to QTAIM in
the solid state
Atoms in Molecules: an introduction, P. Popelier, Prentice Hall 2000
QTAIM applied to crystalline materials
Koritsanszky, T. S. , Coppens, P., Chemical Applications of X-ray
Charge-Density Analysis, ( 2001) Chem. Rev. 101, 1583-1627.
Gatti, C. , Chemical bonding in crystals: new directions, (2005) Z.
Kristallogr. 220, 399-457
Several chapters in Modern Charge Density Analysis, C. Gatti and P.
Macchi Editors (2012) Springer
Several chapters in Electron Density and Chemical Bonding I and II D.
Stalke Editor , Structure and Bonding volumes 146-147 (2012) Springer
2
Chemical bonding studies through deformation densities
(r ; X) = (r ; X) -R(r ; X)
 standard deformation electron density
 total electron density, either th or exp,MM,static
R reference density
promolecular density
procrystal density
R= atoms a(r-Ra) where a is the ground state
spherically averaged atomic electron density
C
O
H’’
N
H’
C
N
H’
H’
O
Total electron density
, urea crystal
0.01 au contour intervals
Dovesi et al., JCP 92, 7402 (1990)
Deformation electron density
 (bulk minus atoms)
0.01 au cont. int.
Interaction electron density
 (bulk minus molecular superposition)
0.002 au cont. int.
But they have serious
drawbacks……
3
R(r ; X) is a non physical quantity : the antisymmetry requirement
is only fulfilled separately by each a
R(r ; X) is not unique
In the case of an atom with spatially degenerate or nearly degenerate
GS (e.g. atoms with an open valence shell, like B, C, O or F) many
alternative a may be chosen.

Orientational freedom exists for
degenerate atomic GSs
F2
H
O
The spherically averaged density might
not be the best choice to give insight
on the way it is bonded to other atoms
H2O2
O
Some covalent bonds do not show the
expected accumulation of the electron
density along the internuclear axis
Available reflections are finite
Fhkl 2 very weak at high angle
 Phase is unknown
 Need of a model to deconvolute
thermal motion and get stat
ED deformation calculated between two set of Fhkl (observed –
calculated) do not suffer from the termination problems. The
calculated model provides the reference Fhkl and phases
The best way to obtain ED from experiment is to refine a model using
the scattered Bragg X-ray reflection intensities for optimizing the
parameters of the model
Hansen and Coppens model
i (r)  Pi ,corei ,core (r)  Pi ,valence i3 i ,valence (r) 
 3

 i Rl ( lr )  Plm ylm (r / r )
l 0 ,lmax 
m0 ,l


4
Since 1990’ experimental and modelling improvements have shifted the
interest from  to 
Decisive step:
Increasing popularity of the various topological approaches to the
study of bonding [s(r) via s(r)]
Quantum Theory of Atoms in Molecules
(QTAIM) Bader RFW
Richard F.W.
Bader
All chemistry is already hidden in . No need to invoke any
arbitrary reference density
The ED  exhibits maxima only at
nuclear positions
Topological study of 
H
H
S (3,-1)

H
C
leads to an exhaustive partitioning of R3 into a
set of non overlapping mono-nuclear domains 
 is bounded by a surface, S(, rs)
C
C C
C
C
C
H
C

•n(rs) = 0 , rs S(, rs)
Recovers the atoms and the atomic group
of chemistry and their experimentally
established transferable properties
The topological definition of  provides
the boundary condition for  being a
quantum open system

h
v
Cyclopropane, D3h
5
BH3
Cyclop.
Diethyl ether.
NH3
H2C=O
C2H6
CH4
Molecular
structure
C2H2
B2H6
C
C C
H
S (3,-1)

C2H4
B5H9
CH3F
H
C
Bond paths
Bcp, =0
H
C
C
H
C

Bicyclo
butane
H2O2
(rc) = 0
H
rc = Critical Point 1) Find
H12= 2/(xy)
2) Classify
H eigenvalues i and eigenvectors
(3,-3)

3  2  1
(3,-1)
NUCLEI
BONDED ATOMS (“chemical bonds”)
(3,+1)
RING
(3,+3)
CAGES
Non degenerate CPs (rank 3)
Classify by (RANK, SIGN) the CP of  in R3
2D display of the structure diagram for an
A-B-C system
RFW Bader, Atoms in Molecules, Oxford Press 1990
RFW Bader et al., JACS 105 5061 (1983)
6
 and total 
in strained
hydrocarbons
Jackson J, Allen, LC
C-C, 1.87 Å
C-C, 1.54 Å
Bicyclo [1.1.1] pentane
C
C
JACS, 106, 591 (1984)
1.1.1 propellane
C
C
(3,+3) cage;
cage = 0.098
au
C
C
Wiberg KB et al.,
JACS,109, 985 (1987)
(3,-1) saddle;
b= 0.203 au 4/5
of a normal CC
bond (in this plane
is a local maximum)
Working with Carla Roetti and Vic Saunders
TOPOND
An electron density topological program for
system periodic in N (N=0-3) dimensions
C. Roetti
First Implementation of QTAIM (Bader’s theory) for periodic wfs
Interfaced to CRYSTAL96/98 (periodic M/LCAO ; M= HF, various
kind of DFT; gaussian basis sets) (Turin, Italy and Daresbury, UK).
Now (2010-2012) with CRYSTAL-09
Performs fully authomated topological analyses of scalar fields
derived from the electron density by working in the crystal space
rather than on a molecule or cluster extracted from the crystal
Atomic properties within QTAIM basins, molecular graphs, 
plots, etc.
N=0 molecules; N=1 polymers; N=2 slab; N=3 crystals
C. Gatti, C. Roetti and Vic Saunders JCP 101 , 10686-10696 (1994)
7
Keyword TOPO in CRYSTAL-13
Properties
Silvia Casassa
Section
Task
TRHO
Topological analysis of  (r)
TLAP
Topological analysis of -2 (r)
ATBP
Atomic basin properties
PL2D
2D plots
PL3D
3D grids
CP search TRHO (TLAP)
f(rcp) = 0
(3,-3)
(3,-1)
(3,+1)
(3,+3)
f   or 2
Local maxima
1D saddle
2D saddle
Local minima
Nuclei (NNA)
bonds
rings
cages
f
 Search algorithms (NR, EF)
 Search strategies (selection of starting points)
8
CP search: Newton Raphson Method
(x0+h) = (x0) + g+h + 1/2 h+ h+ ……
h= -  -1g
NR step x0  CP
 -1= ii-1 vi vi+; h= -ii-1 vi vi+g = -ii-1 viFi
with Fi = vi+g projection of the gradient along the local eigenmode vi
i > 0
h
i < 0
h
g
Fi
vi
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
CP search: Newton Raphson Method
x0 
No hope in general
BCP 
, 3,+1
, 3,-1
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.
9
CP search: Eigenvector Following Method (*)
A modified NR algorithm with a suitable and locally
defined shift s for the NR step
hNR= -ii-1 viFi
Sp
hEF= -i(i – s)-1 viFi
for eigenmodes along which the
function is to be maximized
s
Sn
for 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)
CP search: Eigenvector Following Method
hNR= -ii-1 viFi
hEF= -i(i – s)-1 viFi
Example: search of a (3,-1) CP
1
0
F1
h1
0
2
F2
h2
F1
F2
0
1
3
F3
F3
0
h3
1
h1
h2
= Sp
Sp = highest eigenvalue
1
= Sn
h3
1
Sn = lowest eigenvalue
10
CP search: Eigenvector Following Method
hNR= -ii-1 viFi
hEF= -i(i – s)-1 viFi
Example: search of a (3,+1) CP
1
F1
h1
F1
0
= Sp
h1
1
Sp = highest eigenvalue
1
2
0
F2
h2
0
3
F3
h3
F2
F3
0
1
h2
= Sn
Sn = lowest eigenvalue
h3
1
CP search: Eigenvector Following Method
hNR= -ii-1 viFi
hEF= -i(i – s)-1 viFi
Example: search of a (3,+3) CP
1 0
0 2
0
0
F1
F2
h1
h2
0
3 F3
h3
0
F1 F2 F3
0
1
h1
h2
= Sn
h3
Sn = lowest eigenvalue
1
EF method locates reliably all kind of CPs, regardless of
the  structure at the starting point.
Separate searches for (3,-3), (3,-1), (3,+1) and (3,+3) CPs
are implemented
11
Search strategies (TRHO)

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.)
within a finite region of space which
encloses a finite cluster built-up around a
specified “seed point” Size and origin of
the cluster are defined by input.
Fully-automated
searches can be
performed
within a finite region of space which is
defined by building-up a “supercluster”
composed by 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.
12
Seed point







IAUTO=-2 (nnb =3)


Each star may have a different atom’s
multiplicity (here 3,4,3)


A
B
3 stars of symmetry related atoms
C
IAUTO= -1 CPs are searched for in the
cluster AB C (A,B,C being the unique
atoms in the unit cell)
IAUTO= 0 (3,-1) CPs are searched for among
the unique atom pairs i-j, in the cluster AB
C, with Rij threshold
Fully-automated and chain-like CP searches
(3,-3)
(3,-1)
(3,+1)
(3,+3)
Sequence of the chain-like search
strategy for locating CPs
At each search stage, an EF
step specific for the kind of CP
searched for is adopted.
13
Fully-automated and chain-like CP searches
1. Search of (3,-3) associated to nuclear maxima, starting from the
nuclear position of each of the unique atoms of the unit cell
2. Search of all (3,-1) unique bcps associated to the unique bonded atom
pairs within the cluster. Search started from internuclear axis midpoint
3. Non-nuclear (3,-3) attractors, if any, are recovered at this stage
by determining the nature of the termini of the atomic lines (bond
paths) associated to the unique (3,-1) CPs found
4. Search of unique (3,+1) rcps by considering all unique nuclear triplets
having at least 2 of their 3 atoms bonded to each other and CM
(Center of Mass) not too differently distant from each of the 3 nuclei.
CP search started from CM (mass =1 assigned to each nucleus)
5. Search of unique (3,+3) cage CPs between all pairs of ring CPs
Brute force approach: Search on a grid
A grid-search for CPs in a given portion of the cell



xmin
ymin
zmin
xmax xinc
ymax yinc
zmax zinc
all in fractionary units
Constraint
Constraint type
1
X  ay
2
X  (a+y)/2
3
Y ax
4
Y  min a - x, (a + x)/2 
5
Y  min (x, a - x)
6
Y  min (2x, a - x)
7
Ya–x
8
z  ay
9
za+y
10
z  min (y, a - x)
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
recommended here, because the starting point of the CP search is
moved slightly and smoothly during the search on a grid. On top of this,
the general interest is in locating all CPs in the asymmetric unit, rather
than one peculiar type of CP
Space group constraints among x,y,z fractional coordinates may be
exploited
14
Brute force approach: Search on a grid
Why?
For a closed domain
Morse Relation
n-b+r-c=0
n
(3,-3) nuclei or NNA
b
(3,-1) bcps
r
(3,+1) rcps
c
(3,+3) ccps
Grid search in the
asymmetric unit may be
the only way to fulfill
Morse’s relationship
Fulfilment of the relationship implies a compatible CP set, not necessarily
the complete set of CPs within the cell !
Fulfillment of Morse relationship in urea crystal
w
m
c
2
(3,-3)
C
c
2
(3,-3)
O
e
4
(3,-3)
N
e
4
(3,-3)
H’
e
4
(3,-3)
H’’
c
2
(3,+1) ring
w, Wyckoff positions
e
4
(3,+1) ring
m, multiplicity
e
4
(3,+1) ring
f
8
(3,+1) ring
f
8
(3,+1) ring
c
2
(3,-1)
C-O
e
4
(3,-1)
C-N
e
4
(3,+3) cage
a
2
(3,+3) cage
b
2
(3,+3) cage
e
4
(3,-1)
N-H’
e
4
(3,-1)
N-H’’
e
4
(3,-1)
O…H’
e
4
(3,-1)
O…H’’
d
4
(3,-1) N…N, 4.3 Å
f
8
(3,-1) N…N, 3.4 Å
n - b + r - c = 0
16 – 26 + 18 – 8 = 0
Gatti et al. JCP, 101,10686 (1994)
16 – 34 + 26 – 8 = 0
15
The complete bond network in urea crystal
N-H…O
1.99 Å
2.06 Å
N…N, 3.4 Å
N…N, 4.3 Å
JCP 101 , 10686-10696 (1994)
 How important are packing effects on
the properties of intramolecular bonds?
 Does the packing have different impact on the different
atoms/chemical groups present in the molecule?
 How large is the enhancement of the molecular dipole on
crystallization?
 How can each oxygen atom in the urea crystal be involved in
four OH…O hydrogen bonds (HBs)?
 How does the global molecular volume contraction observed
in the solid result from the individual atomic volume change
on crystallization?
16
Interaction density and changes in bcp properties of urea
X-Y

2 
3

C-O
0.374
0.384
0.384
0.299
-0.550
-0.666
-0.666
0.118
1.409
1.239
1.229
0.974
0.038
0.125
0.125
0.038
C-N
0.342
0.334
0.334
0.262
-0.952
-0.825
-0.825
-0.099
0.721
0.817
0.817
0.627
0.143
0.097
0.097
0.044
N-H
0.347
0.346
0.346
0.234
-2.003
-1.735
-1.735
-0.271
0.874
0.945
0.945
1.240
0.052
0.070
0.070
0.006
O…H’N
dO…H =
1.992
0.023
0.021
0.030
0.081
0.092
0.088
0.143
0.147
0.164
0.024
0.003
0.059
O…H’’N
dO…H =
2.058
0.019
0.019
0.026
0.080
0.086
0.083
0.130
0.136
0.145
0.034
0.006
0.045
= ( 1/  2)-1
all in au
CRYSTAL
MOLECULES
MOLECULE
ATOMS
(procrystal)
The dicothomous classification based on the sign of 2
Bader, R.F.W.; Essén, H., J. Chem. Phys. 80 (1984) 1943
Local expression of quantum virial theorem
¼ 2 (r) = 2 G(r) + V(r)
2b
G(r) > 0
V(r) < 0
Chemical Interactions
 2b < 0
V(rb) in local excess
respect to the average
2G() = -V()
 2 b > 0
G(rb) in local excess
respect to the average
2G() = -V()
17
The dicothomous classification based on the sign of 2
Property
Shared shell, 2b< 0
Covalent and polar bonds
Closed-shell, 2b> 0
Ionic, H- bonds and vdW
molecules
I
1,2 dominant ; 1,2/3 > 1
3 dominant; 1,2/3 <<1
VSCC
The VSCCs of the two
atoms form one
continuous region of CC
2 >0 over the entire
interaction region. The spatial
display of 2 is mostly atomiclike
b
Large
Small
Energy
Lowering
By accumulating  in the
interatomic region
Regions of dominant V(r) are
separately localized within the
boundaries of interacting atoms
Energy
components
2Gb<Vb; Gb/b<1; Gb<<
Gb; Hb<0
2Gb >Vb ; Gb/b>1, Gb>> Gb; Hb
any value
Electron sharing is decreasing
(ond polarity increasing)
Electron sharing (covalency) is
increasing (and polarity decreasing)
Interaction density and changes in BCP properties of urea
CG molecule
OG molecule
Bulk molecule
10%
increase/decrease
% 3 (bcp)
+0.05
increase/decrease
 (bcp)
C=O becomes more ionic; C-N more covalent
18
Interaction density and changes in BCP properties of urea. The
dimer model
Bulk -molecule
Bulk - dimer
% 3 (bcp)
%  (bcp)
Chemical bond nature vs
BCP properties in solids
b, 2b, 1-3 of closed shell
interactions are generally one or
two order of magnitude smaller
than for shared interactions
3 / 1,2  1 shared interactions
3 / 1,2  0.1 closed-shell int.
19
The Laplacian of the electron density, 2
=2= ii = (2/x2ii), i=1-3
Ar
1st shell
3rd
2nd
a positive (negative) second
derivative at x0 indicates that
the ED is on average lower
(higher) in x0 than it is in a
symmetrical neighborhood of x0.
The ED is depleted
(concentrated) at x0.
VSCC
RFW Bader, Atoms
in Molecules,
Oxford Press 1990
L(r)=-2
Net  flux
Through the
Divergence theorem
CC regions : a net  flux enters the region
CD regions : a net  flux leaves the region
V2(r) d = V(r) d
2>0 electron density is depleted at r
= S()n(rs)dS
2<0 electron density is concentrated at r
The Laplacian as a Magnification Glass for the Shell Structure of Atoms
10
(r)
-2(r)
K
argon
atom
100 eÅ-5
(r)
10 eÅ-3
L
M
0
0
(1.59 Ǻ)
W. Scherer private
communication
relief maps
contour maps
-2(r)
100
0 KL M
-100
Quantum
shell M
0
-2r(r) > 0
(1.59 Ǻ) region of charge
concentration (CC)
~ 3 au
Level of approximation:
B3LYP/6-311++G(2d,2p)
20
2 and the Lewis Electron Pair Model
r (3,+1)
s (3,-1)
Nbm (3,-3), non-bonded max.
O
non-bonded
VSCCs “lone pairs”
O
H
H
O
H
bm (3,-3), bonded max.
O
H
H
bonded VSCCs
H
s (3,-1) between two nbm
s (3,-1) between two bm
R. F. W. Bader et al. J. Am. Chem. Soc. 1984, 106, 1594.
CPs
 = 0
(2) = 0
Section
Task
TRHO
Topological analysis of  (r)
TLAP
Topological analysis of -2 (r)
ATBP
Atomic basin properties
PL2D
2D plots
PL3D
3D grids
TLAP: a topological analysis of L(r)=-2 is performed, so as
to associate positive values of L(r), with an electronic charge
concentration and negative L(r) values with charge depletion
21
Search strategies (TLAP)

An automated strategy, using NR (for all kinds
of CPs) or EF (for a required kind of CPs)
 More specific searches (along a line, from a
given set of starting points, etc.)
Automated search (TLAP, IAUTO=0)
2 distribution retains an atomic-like portrait even upon
chemical combination. CP search is thus “atom’s-based “
-2
N2
-2ρ(r) eÅ-5
shared
Ar2
vdW-like
F2
slightly
shared
W. Scherer, private communication
CP’s search within the concentration (or
depletion) shells of each of the nonequivalent atom (NEA) and of the nonnuclear attractors (NNAs) given in input
RFW Bader, Atoms in Molecules, Oxford Press 1990
22
Usually the search is performed in the valence shell charge
concentration (VSCC) of each NEA
Which shell is sampled, depends however on the RSTAR value
given in input
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.
Using different values of RSTAR different
regions of CC (2<0) or CD (2>0) are
explored.
Different values of RSTAR are required when
the VSCC has undergone a substantial change (or
is even missing) due to the large CT occurring in
some solids, like the ionic crystals [Ca (2+) vs Ca
or O(2-) vs O]
2nd row atom
VSCC
max
VSCD
max
 CPs search is started from points located on the surface of a
sphere or radius RSTAR, centered on the nucleus of a given NEA
or at the Non Nuclear Attractor (NNA) location
 The number of starting points is fixed
by the intervals chosen for the polar
coordinates  and 
 NR method is suggested for a generic
exploratory CP search
2nd row atom
 EF method is very important for a function rapidly
2
varying as it is   (the CP search is often started
VSCC
max
from a region where the Hessian of 2 has a
VSCD
max
different structure from that of the CP)
Use the EF method to make a search for a specific kind of CP [e.g
the - 2 (3,-3) lone pairs of a carbonyl oxygen]
23
f
AIL (bond paths) f
3,-1
f
A
AGL (atomic graph lines) f
-2
B
3,-3
If IBPAT is activated, AIL [AGL] 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 [(3,-3) -2 CPs] a bcp [(3,-1) -2
CP] is linked to.
The associated ODE are solved using a 5th order
Runge-Kutta method with monitoring of local
truncation error and an adaptive stepsize control.
3,-3
Li
H
H
Li
AILs are generally determined with less than
80-130 integration steps and 500-1200  [2]
and  [ (-2) evaluations.
JCP 101 , 10686-10696 (1994)
 How important are packing effects on
intramolecular bonds?
 Does the packing have different impact on the different
atoms/chemical groups present in the molecule?
 How large is the enhancement of the molecular dipole on
crystallization?
 How can each oxygen atom in the urea crystal be involved in
four OH…O hydrogen bonds (HBs)?
 How does the global molecular volume contraction observed
in the solid result from the individual atomic volume change
on crystallization?
24
The Laplacian distribution and the H-bonds
HBs may be seen in terms of a generalized Lewis acid and base
interaction
(3,-3)
(3,+3)
Generally the approach of the acidic
hydrogen to the base will be such as to align
the (3,+3) minimum in the VSCC of the H with
the most suitable (3,-3) Base maximum
3D-Hydrogen Bonding network in urea: the -2 description
gas phase
C
O
1.229 Å
(3,-1)
(3,-3)
(3,+3)
(3,-3)
(3,-1)
(3,-1)
(3,+3)
crystal
C
-(2crystal - 2molecule )
O
1.261 Å
Gatti et al. , JCP 101,
10686 (1994)
2crystal
25


n
||
Atomic
properties
For a generic surface
d’ integration over the
coordinates of all
electrons but one and
summation over all spins
Any molecular property O which can be expressed in terms of a
corresponding property density in space, o(r), can be written as
a sum of atomic contributions
Atomic electron population and net charge
N () =  (r) d
q () = Z - N()

Atomic moments
Mj() = -  d (r) rj ; j = 1-3
Atomic dipole
x
origin
r
r
Qij() = -  d (r) (3rirj – r2 ij) ; i,j = 1-3
Atomic quadrupole moment
tensor (traceless)
Real symmetric matrix which
can be diagonalized
RFW Bader, Atoms in Molecules, Oxford Press 1990
26
Atomic integrated Laplacian
L () =  2(r) d =  (r) d = S() n(rs)dS
Should be  0 for an atom bounded by a ZFS (zero-flux surface)
Deviation from 0 measures the numerical accuracy of atomic integration (due to
approximate definition of the ZFS and numerical integration within the basin
Atomic kinetic energy
Many alternative formulas for the kinetic energy density
K(r) = N k  d’ [*2 + (2)* ]
k = (-h2/(162m))
In terms of the Laplacian
operator
In terms of the dot product
of the momentum operator
G(r) = N ½ k  d’ * 
K(r) - G(r) = k 2(r)
 [ K(r) - G(r)] d = K() – G() = k  2(r) d =k S() n(rs)dS  0
K()  G() = -E()
Atomic volumes
V () = d
 of all cellV () = Vcell
Generally infinite in the molecular case; always finite in the
crystalline case
Normally the atomic volume is however defined as the region of
space enclosed by the intersection of the atomic zero-flux
surface and a particular envelope of 
V0.001()V1 = *d ;
*: r where 0.001 au
V0.001 yields molecular sizes in agreement with those
determined from the analysis of the kinetic theory data for
gas-phase molecules (van der Waals volumes)
V0.002()V2 = **d ;
**: r where 0.002 au
V0.002 yields molecular sizes compatible with the closer packing
found in the solid state
27
Keyword TOPO in CRYSTAL-13 Properties
Section
Task
TRHO
Topological analysis of  (r)
TLAP
Topological analysis of 2 (r)
ATBP
Atomic basin properties
PL2D
2D plots
PL3D
3D grids
Atomic properties (ATBP)
Integration over each non equivalent atom (NEA)  is
essentially performed in two main steps:
a) ZFSs determination
Li
H
H
Li
28
Atomic properties (ATBP)
Li
Li
H
H
Li
b) integration within the volume bounded by the ZFSs
(Indirect) determination of the ZFSs
 performed
according to the algorithm
“PROMEGA” devised by T.A. Keith
T.A. Keith, Ph.D. thesis, McMaster University,
Ontario, Canada (1993)
 time determining step: 2-3 order of
magnitude > integration within ZFS
29
 determines within a given accuracy (variable ACC) the lengths of the
rays starting from the nucleus of the integrated atom and ending up at
the intersection with the ZFSs. The rays are associated to all possible
pairs (i,j) of the angular grid (spherical coordinates)
1st intersection
2nd intersection
Goal is to find R1i, R2i , etc.) for
each Integration Ray (IR)
IRi (R1i, R2i, i,i ) i=1,ith*iphi
ACC  typically 0.001 au
NOSE=1 to search for second
intersections
The computational time required by the ZFS determination is
proportional to ITH*IPHI, i.e. to the product of the number of 
and  angular points
(Indirect) determination of the ZFSs
 Based on the fact that each point within an atomic basin lies on a
 trajectory (GP) which terminates at the nucleus of the atom
dr(s)/ds = ± [ r(s); X]
r(s) = r0 ±  [ r(t); X] dt
for a given r(0) r0
GRADIENT PATHS (GPs)
 The objective is to determine
those segments of the IRs which
intersect  trajectories ending
up at the nucleus of the atom
5
4
30
(Indirect) determination of the ZFSs
 R1 for each ray is determined by
stepping outward a user specified
distance along the ray and by
launching a GP for each step
(determined via Predictor-Corrector
method of order 6 or Runge-Kutta
method)
IR
 Starting point along each IR:
a) r the radius of a sphere
centered on the nucleus of  and
called a capture or a beta sphere
(it is a subdomain of )
r O1

IR

H2
H3
b) using as starting point a guess
for R1 from previously determined
R1s for a convenient subset of IRs
(“feeler rays”).
(Indirect) determination of the ZFSs
 If the GP leads back to the
nucleus on which the ray is
centered, the step outward is
repeated until the gradient path
leads to a different nucleus or a
maximum allowed length of the
ray has been reached
IR
R1
 If the size of the step is >
ACC (the required precision for
the ZFS) move the launching
point forward and backward
along the ray and bisect the step
at each attractor oscillation,
until the step size is < ACC
H3
O1
H2
31
Evaluation of molecular dipole from the basin charge
and first moment
JCP 87, 1142 (1987)
 = el + nuc =  [ -r(r)d + XZ]
r = r + X ; q = Z -  dr(r) = Z - N
 =   - r(r)d + X(Z  -   (r) d
 =   M  + X  q   = AP + CT
Both sums are origin independent (if q=0)
For AP also each term in the  is origin independent
Enhancement of molecular dipole
moment of urea in the bulk
The non-interacting molecules and the
crystal periodic RHF densities look very
much alike despite the 37% ||
enhancement in the crystal
Inter. density , Highest
contour level ±2·10-3 au
Non interacting
molecules
Crystal
Contribution
OG Mol.
CG Mol
Crystal
A,(|A|%)
0.71
0.54
0.45 (-16.7)
CT,(|CT|%)
-2.52
-2.56
-3.22 (+25.8)
 , (||%)
-1.81
-2.01
-2.77 (+37.1)
|| values are
very sensitive to
the atomic
boundaries location
and to the atomic
ED distribution
changes
Gatti et al, JCP 101, 10686 (1994)
32
Clearly the result of a more
polarized molecule in the bulk
Crystal
All the heavy atoms gain
electrons at the expense
of the H’s in the bulk
Following HB formation there is
a net flux of 0.067 e- from the
amino-group hydrogen donor to
the carbonyl acceptor.

Mol (CG)
Crystal
N ()
N()
N()
C
3.512
3.545
+0.032
O
9.383
9.476
+0.092
N
8.481
8.568
+0.087
H’
0.508
0.454
-0.053
H’’
0.565
0.465
-0.100
CO
12.895
13.021
+0.126
NH2
9.554
9.487
-0.067
Gatti et al, JCP 101, 10686 (1994)
Keyword TOPO in CRYSTAL-13 Properties
Section
Task
TRHO
Topological analysis of  (r)
TLAP
Topological analysis of 2 (r)
ATBP
Atomic basin properties
PL2D
2D plots
PL3D
3D grids
33

trajectories
A window
larger than
the plot area
may be
defined so as
to include all
the
attractors
sending GPs
in the plot
area
Many thanks for your attention
34