61
Introduction to the X-ray Charge (Electron) Density Modelling Dr. Anatoliy Volkov April 21, 2016 Department of Chemistry & Computational Science Ph.D. Program
Introduction to the X-ray Charge
(Electron) Density Modelling
Dr. Anatoliy Volkov
April 21, 2016
Department of Chemistry &
Computational Science Ph.D. Program
X-ray diffraction
intensities
Molecular / Crystal
wavefunctions
Electron density
(ED)
•accurate geometries•anisotropic displacement parameters
•simple analysis of intermolecular interactionsetc.
•energies•excited states• reaction pathsetc
•bonding analysis
•atomic/molecular/crystal properties
• intermolecular interactions
Electron density
(ED)
atomic/molecular charges,
electrostatic moments
electrostatic
interactions
between
molecular
units
intramolecular
energies ????
electrostatic potential
electric field
electric field gradient
empirical analysis of
orbital occupancies
electron static
polarizabilities
non-linear optical
susceptabilities
intermolecular
interaction
energies
chemical bonding analysis:
deformation densities
interaction densities
Bader’s topological analysis
4
• Electrons are treated as spreading the entire molecule
Molecular orbital (MO) theory
– every electron contributes to the strength of every bond
• Molecular orbital (MO), (r) – one-electron
wavefunction for an electron that spreads throughout
the entire molecule
MO’s spread over the entire molecule, not just the adjacent
atoms of the bond
each i-th MO, i(r), is a linear combination of all AO’s
(LCAO) in the molecule
)()(AO
1
rr j
N
j
iji c
j(r) – j-th AO
NAO – total number of AO’s
cij – MO expansion coefficients
r
opposite signs of the wavefunctions are represented by
different colors (our convention: positive, negative)
5
• Once MO expansion coefficients are known, MO’s can be
visualized using stylized shapes to represent the basis set, and
then scale their size to indicate the value of the coefficient in
LCAO
E1 -26.3 Eh E2 -1.6 Eh E3 -0.8 Eh E4/5 -0.7 Eh
(degenerate)
z
x
y
Example: H-F molecule (Rexp=0.917 Å)
Hartree-Fock (HF)/cc-pV6Z
only occupied MOs (2 electrons per MO) are shown here
6
We can also build up a representation of the electron
density (r) (ED) in the molecule
i
iin 2)()( rr
ni - occupation number (1 or 2) of the i-th MO i
only occupied orbitals contribute to ED
electron density is non-negative everywhere
the outcome is commonly represented by an isodensity
surface – surface of constant ED
• Isodensity surface in H-F (contour is 0.1 electron Å-3)
elec.
3)( Nd rr i
inNelec.
7
Ok, great! But where is bonding in ED ?
atoms
)()( atom spherical freeepromoleculN
i
i rr
promolecule – superposition of free (non-interacting)
spherically-averaged atomic densities
Solution: Compare molecular density with that of promolecule
promolecule electron density must be calculated using the
same formalism as the molecular density
the comparison is expected to reveal redistribution of ED as
atoms form bonds when combined into a molecule
also called the Independent Atom Model (IAM)
elec.
3epromolecul )( Nd rr
8
Example: H-F molecule
ED calculated at the Hartree-Fock/cc-pV6Z level
FH
HF
9
Example: H-F molecule
H F H F z
Isocontour 0.1 e Å-3
H F H Fz
Contour levels are 0.008, 0.02, 0.04, 0.08, 0.2,
0.4, 0.8, 2, 4, 8, 20, 40, 80 … e Å-3
)(epromoleculr
)(rmolecular
10
Deformation electron density, (r)
molecular ED minus promolecule ED
)()()( epromoleculrrr
promolecule ED must be calculated using the same formalism
as the molecular density
deformation density (r) is expected to reveal redistribution
of ED as atoms form bonds when combined into a molecule
perhaps, we can say that (r) reveals bonding density
non-bonding density (i.e. lone pairs) is also revealed
positive (r) – accumulation of charge (electrons) in
molecule as compared to a promolecule
negative (r) – reduction of charge (electrons) …
0)( 3 rr d
11
Deformation electron density (r) in the H-F molecule
F
contour levels are at 0.1 e Å-3
positive density - accumulation of charge
negative density – reduction of charge
dotted black line ( ) → zero contour
positive density isocontour 0.1 e Å-3
negative density isocontour -0.1 e Å-3
FH
z
z
H
)()()( epromoleculrrr
F
H
12
Core electron density is polarized
E1 -26.3 Eh
contour levels are at 0.1 e Å-3
positive density - accumulation of charge
negative density – reduction of charge
core MO (AO) of fluorine
positive density
isocontour 0.1 e Å-3
negative density
isocontour -0.1 e Å-3
)(core r
)(corer
)(core r
H F
Core polarization is small
13
Core electron density polarization is also predicted by
the Density-Functional calculations
Core electron polarization effects are significantly smaller than
the deformation density
frozen-core approximation: unperturbed spherical atom
electron core
)(core r
Electron density at each point in space r, (r), is
described by a sum of contributing nuclei-centered
pseudoatoms pseudoatom(r):
atoms
1
pseudoatom)()(
N
i
i rr x
y
z
(r1) ?
H(r1) F(r1)
r1
(r1) =F(r1) + H(r1)
Pseudoatom model of the electron density
H
F
Each pseudoatom is expanded as :
spherical core
density
spherical
valence density
aspherical
deformation density
pseudoatom(r) = CORE(r) + VALENCE(r) + DEFORMATION(r)
nucleus
of the i-th
pseudoatom
The Hansen-Coppens pseudoatom model of the
electron density
atoms
1
pseudoatom)()(
N
i
i rr
b) superposition of deformation
pseudoatomsa) individual deformation
pseudoatoms in the molecule
Example: superposition of the deformation density of
pseudoatoms in the plane of the water molecule
Contour levels are 0.1 eÅ-3; positive - red, negative – blue
atoms
1
pseudoatom)(
N
i
i r
Pseudoatom parameters
max
0
33pseudoatom ),()'(')()()(l
l
l
lm
lmlmlvvcc dPrRrPrP r
aspherical deformation part is refined: populations Plm account
for the shift of density between regions of opposite sign of
angular functions dlm(,)
parameters describe the expansion-contraction of radial
density functions Rl( r)
spherical core is fixed at that from a free-atom ab initio
calculation at Hartree-Fock/DFT level ( electron population Pc
is not refined, i.e. the “frozen-core approximation” )
spherical valence ED is refined: electron population Pv
accounts for charge transfer, and parameter allows
contraction-expansion of the spherical valence shell
CORE(r) VALENCE(r) DEFORMATION(r)
Bunge, Barrientos & Bunge (BBB, 1993)
Core and Valence: STO-based atomic wavefunctions
Clementi & Roetti (1974)
n l
orbital
exponents
rn
i
n
in
ii
i
ier
nrR
1
2/1
2/1
)!2(
2)( ),()(),( 0
0 YrRrninii
(1s)
expansion
coefficients
(2s)
expansion
coefficients
description of orbitals of s symmetry
description of orbitals of
p symmetry
n lorbital
exponents
(2p)
expansion
coefficients
),( i
i
m
lY
wavefunction-
normalized spherical
harmonics
radial
function
total i-th Slater function
angular function
8
9
10
11
12
1
2
3
4
5
6
7
for a spherical
atom, l = m = 0
2
1),(0
0 Y
19
1s
2s
2p
1s radial wavefunction :
1s total wavefunction (spherical) :
Spherical core density
Spherical core electron density :
)(),,( 2
1 rPrP ccsc
Pc = 2 electrons
CORE(r)
2020
1s
2s
2p
2s total spherical wavefunction:
Unperturbed spherical valence electron density :
2
22
2
22 )()( rPrP ppss
P2s = 2 electrons, P2p = 5 electrons
Spherical valence density
2p total spherical wavefunction :
VALENCE(r)
21
Valence density parameters
Unperturbed spherical valence electron density of F-atom :
P2s = 2 electrons, P2p = 5 electrons
Refined spherical valence electron density of F-atom:
)()()( 2
22
2
2
22
233 rPP
Pr
PP
PPrP p
ps
p
s
ps
svvalencev
)()()()( 2222
2
22
2
22 rPrPrPrP ppssppss VALENCE(r)
electron population parameter Pv accounts for charge transfer
parameter accounts for contraction-expansion of the
spherical valence shell relative to that of a free atom
VALENCE(r)
max
0
33pseudoatom ),()'(')()()(l
l
l
lm
lmlmlvvcc dPrRrPrP r
VALENCE(r)
22
Valence density parameters: kappa ()
parameter accounts for contraction-expansion of the
spherical valence shell relative to that of a free atom
= 1: free spherical atom valence density
> 1: contraction relative to free atom
< 1: expansion relative to free atom
Note: all three densities
integrate to 7 electrons
0
23 )(4 drrkrP valencev
max
0
33pseudoatom ),()'(')()()(l
l
l
lm
lmlmlvvcc dPrRrPrP r
max
0
33pseudoatom ),()'(')()()(l
l
l
lm
lmlmlvvcc dPrRrPrP r
23
Aspherical deformation density parameters
Basic quantum mechanics:
Example: hydrogenic wavefunctions
),()(),,( ,,,, ll mllnmln YrRr n, l, ml –
quantum numbers
),()'('max
0
3
l
l
l
lm
lmlml dPrR Pseudoatom model:
populations Plm account for the shift of density between
regions of opposite sign of angular functions dlm(,)
parameters describe the expansion-contraction of radial
density functions Rl( r)
Atomic orbitals can be written as a product of radial and angular
functions
DEFORMATION(r)
24
Deformation density:
Each function in the expansion has the form
),( )'( ' 3 lmllm drRP
- radial function)'( rRl - angular function
),()'('max
0
3
l
l
l
lm
lmlml dPrR
),( lmd
population Plm (refined) accounts for the shift of density
between regions of opposite sign of angular function dlm(,)
parameter (refined) describes the expansion-contraction of
radial density function Rl( r)
usually, the same parameter is applied to all Rl( r) of
a given atom
parameter Plm is refined separately for each dlm(,)
25
),( )'( ' 3 lmllm drRP
- radial function)'( rRl - angular function),( lmd
Radial function, - nodeless density-normalized Slater-
type function
)'exp('
!2')'(
33 rr
nrR l
n
l
n
ll
l
l
l – energy-optimized orbital exponents of single-Slater
representations of the electron subshells of isolated atoms
Clementi & Raimondi (1963)
multiplied by 2 (density functions, not orbitals)
nl – coefficients that allow density functions reproduce orbital
products
rule: nl l
first-row atoms: n1=n2=2, n3=3, n4=4
1)'(0
2
rrRl
)'( rRl
26
nl for first-row atoms: n1=n2=2, n3=3, n4=4
2s = 2.5638 (Bohr-1), P2s = 2 electrons
2p = 2.5500 (Bohr-1), P2p = 5 electrons
values (Clementi & Raimondi,1963):
l for density functions (assumed to be the same for all l) is
the weighted average
1-1- Å 6525.9Bohr 1079.527
55500.225638.2
l
)'exp('
!2')'(
33 rr
nrR l
n
l
n
ll
l
l
1)'(0
2
rrRl
Radial function, - nodeless density-normalized Slater-
type function
)'( rRl
Example: Fluorine atom
density
function
27
l = 9.6525 Å for all l
Here, functions
for l=1 and l=2
are the same
)'exp('
!2')'(
33 rr
nrR l
n
l
n
ll
l
l
1)'(0
2
rrRl
Radial function, - nodeless density-normalized Slater-
type function
)'( rRl
nl for first-row atoms: n1=n2=2, n3=3, n4=4
Example: Fluorine atom
28
parameter describes the expansion-contraction of the
radial density function Rl( r)
Same n and ,
but different
)'exp('
!2')'(
33 rr
nrR l
n
l
n
ll
l
l
1)'(0
2
rrRl
Radial function, - nodeless density-normalized Slater-
type function
)'( rRl
Example: Fluorine atom
29
Angular function, - real density-normalized Cartesian
spherical harmonics
),( )'( ' 3 lmllm drRP
),( lmd
spherical harmonics, ),( m
lY
angular part of the solutions for the Laplace equation
0sin
1sin
sin
11),,(
2
2
222
2
2
2
f
r
f
rr
fr
rrrf
normalization (“wavefunction-based”)
mmll
m
l
m
l ddYY
2
0 0
* sin),(),(
- complex-conjugate (most functions are complex)
ji
jiij
if 1
if 0 ij - Kronecker delta:
rule: l = 0…., m = -l,...,0,…,+l
),( m
lY
30
spherical harmonics, ),( m
lY
Angular function, - real density-normalized Cartesian
spherical harmonics
),( )'( ' 3 lmllm drRP
),( lmd
imm
l
m
l ePml
mllY )(cos
)!(
)!(
4
)12(),(
2/1
general form:
- associated Legendre polynomial of x
solutions to the associated Legendre differential equation
)(xPm
l
0)(1
)1()(
2)(
)1(2
2
2
22
xf
x
mll
dx
xdfx
dx
xfdx
definition: m
l
mmmm
ldx
xPdxxP
)()1()1()( 2/2
- Legendre polynomials:)(xPl
n
n
n
nn xdx
d
nxP )1(
!2
1)( 2
0)()1()(
2)(
)1(2
22 xfll
dx
xdfx
dx
xfdxLegendre differential equation
31
real spherical harmonics,
Angular function, - real density-normalized Cartesian
spherical harmonics
),( )'( ' 3 lmllm drRP
),( lmd
),( m
ly
linear combinations of complex spherical harmonics,
0 if ),()1(),(2
1
0 if ),(
0 if ),()1(),(2
),( 0
||
mY-Y
mY
mY-Yi
y
m
l
mm
l
l
m
l
mm
l
m
l
),( m
lY
normalization
(“wavefunction-based”)
mmll
m
l
m
l ddyy
2
0 0
sin),(),( alternatively,
)cos()(cos|)!|(
|)!|(
2
)12( ),(
)(cos4
)12(),(
)sin()(cos|)!|(
|)!|(
2
)12(),(
||
2/1
||
0
2/1
0
||
2/1
||
mPml
mlly
Pl
y
mPml
mlly
m
l
m
l
ll
m
l
m
l
definitions in terms of the
associated Legendre
polynomials
32
Angular function, - real density-normalized Cartesian
spherical harmonics
),( )'( ' 3 lmllm drRP
),( lmd
real density-normalized spherical harmonics, ),( lmd
real spherical harmonics, , renormalized as:),( m
ly
0 for 2sin),(
2
0 0
ldddlm
0 for 1sin),(
2
0 0
ldddlm
rarely used (have spherical core and valence densities)
population of 1 electron of the spherically-symmetric ),(00 d
aspherical functions (l>0) represent a shift of density
between regions (lobes) of opposite sign
normalization allows the shift of 1 electron from the
negative to the positive region
33
Angular function, - real density-normalized Cartesian
spherical harmonics
),( )'( ' 3 lmllm drRP
),( lmd
real density-normalized Cartesian spherical harmonics, ),,( zyxdlm
switching from the spherical to Cartesian representation
),,(),( zyxdd lmlm
yx
rz
zyxr
/arctan
/arccos
222
expressing spherical coordinates
(,) in terms of Cartesian
coordinates (x,y,z)
34
Angular function, - real density-normalized Cartesian
spherical harmonics
),( )'( ' 3 lmllm drRP
),( lmd
Aspherical functions are named as follows:
l = 0 : monopole → ml = 0
l = 1 : dipole(s) → ml = -1, 0, 1
l = 2 : quadrupole(s) → ml = -2, -1, 0, 1, 2
l = 3 : octupole(s) → ml = -3, -2, -1, 0, 1, 2, 3
l = 4 : haxadecapole(s) → ml = -4, -3, -2, -1, 0, 1, 2, 3, 4
Altogether, these functions are called “multipoles” or
“multipolar” functions
35
Angular function, - real density-normalized Cartesian
spherical harmonics
),( )'( ' 3 lmllm drRP
),( lmd
Example: function (quadrupole)),,(1,2 zyxd
check
check
this is “your very own” dxz function!
Angular functions in the pseudoatom model are represented by
real density-normalized Cartesian spherical harmonics
dipolequadrupole quadrupole
octupole octupole hexadecapole
Spherical harmonic functions for each atom are defined
in the atomic local coordinate systems
Convenient :
some Plm are zero due to local symmetry
chemical-equivalency constraints can be easily applied
(atoms share the same set of pseudoatom parameters)
L, R – left- and right-handed coordinate systems
chemically
equivalent
Independent Atom Model (IAM)
max
0
33pseudoatom ),()'(')()()(l
l
l
lm
lmlmlvvcc dPrRrPrP r
aspherical deformation part = 0
unperturbed core of a free (non-interacting) spherical atom electron population Pc = number of core electrons in a free
atom
unperturbed valence density of a free spherical atom electron population Pv = number of valence electrons in a free
atom
expansion-contraction parameter = 1
CORE(r) VALENCE(r) DEFORMATION(r)
IAM is used in a “standard”/”conventional” refinement of
crystal structures (i.e., SHELX etc.)
39
In a three-dimensional periodic system (crystal), the
generalized structure factor F(hkl) is given by the Fourier
transform of the electron density distribution function (xfyfzf)
Vc – volume of unit cell
{xf,yf,zf} - fractional coordinates defined w.r.t. the lengths of
unit cell axes a, b and c
Structure factor
1
0
1
0
1
0
fff
)(2
ffffff)()( dzdydxezyxVhklF
lzkyhxi
c
cz
by
ax
z
y
x
/
/
/
f
f
f
fx
l
k
h
S
1
0
f
32
ff)()( xxS
xSdeVF
i
c
Fourier transform
SSx
xS 32
ff)(
1)( deF
V
i
c
inverse Fourier transform
(back-transform)
max
min
max
min
max
min
fff )(2
f )(1
)(h
hh
k
kk
l
ll
lzkyhxi
c
ehklFV
xFourier synthesis
40
“The Procedure”
Quantum Mechanics Pseudoatom model
rrr
rrr
3*
3*
)()(
)(ˆ)(
d
dHE
...)()()( 2211 rrr fcfc
0
ic
E
Variational principle
trial wavefunction
energy
minimize E w.r.t. linear
coefficients ci
exponents in primitive
functions fi(r) are fixed
Least-squares fit to observed
thermally-corrected X-ray
structure factors,
i
ii
i
iii
hklFhklw
hklFhklFhklwS
2
2calcobs
)()(
)()()(
)(obs hklFi
i = 1..Nhkl
minimize error function S
wi(hkl) – weight of i-th observation
)(calc hklFi – Fourier transform of
the model (pseudoatom) density
adjust parameters Pv, , Plm, and
for each pseudoatom
41
The accuracy of a determined structure is given by the “R-
factor”, RF :
RF is often reported in %
RF < 5% → very accurate structure
5% < RF < 10% → relatively accurate structure (not bad)
for macromolecules RF can be as large as 20% or even
higher
Crystallographic R-factor
For conventional structures only,
i
i
i
i
i
i
i
ii
FhklF
hklF
hklF
hklFhklF
R)(
)(
)(
)()(
obsobs
calcobs
42
Example: Pseudoatom model fit to the theoretical
(quantum mechanical) density of the H-F molecule
H-F molecule (Rexp=0.917 Å)
theory: Hartree-Fock (HF) / cc-pV6Z
molecule enclosed in a tetragonal cell
c = 5 Å
a=
4 Å
H
F
15,531 structure factors
generated up to
-1
maxÅ 8.1sin
“static” structure factors –
no thermal motion (signal
is only due to the electron
density)
unit weights
43
Theoretical wavefunction calculated using Gaussian-type
functions via the cc-pV6Z basis set (Wilson, Mourik &
Dunning, 1999; Peterson, Woon, Dunning, Unpublished)
Pseudoatom model uses Slater-type functions (Bunge,
Barrientos & Bunge,1993)
The Good:
The theoretical model, and the pseudoatom core and
valence densities are based on the Hartree-Fock
calculations
Theoretical model vs Pseudoatom model
The Bad:
Is that a problem ?
44
Gaussian-type functions (GTF, GF)
GTFs are widely used in quantum chemistry (Gaussian9x/0x,
GAMESS, Molpro, NWChem etc.)
2
),,()( rcba ezyxNzyxg
r
2/1
4/)3222()(4/3
!)!12(!)!12(!)!12(
22
cbaN
cbacba
a, b, and c - integer numbers
l = a + b + c : orbital angular momentum quantum number
- orbital exponent
N - normalization factor
45
Gaussian-type functions (GF, GTF) are less accurate than Slater-
type orbitals (STO), i.e. wrong cusp and wrong radial decay, but
integrals are much easier to evaluate
many “bad” Gaussians one “good” Slater
STO-nG basis sets - 'n' primitive Gaussian-type function are
fitted to a single Slater-type orbital (STO)
46
Hydrogen atom (Hartree-Fock calculations)
Total atomic energy (Eh). Exact wavefunction: -0.5 Eh
STO-1G -0.42441 STO-6G -0.49986
STO-2G -0.48315 6-311++g(2df,2pd) -0.49982
STO-3G -0.49574 cc-pV6Z -0.4999992
Energy is well-
recovered when
using extended
GTF basis sets
Electron density is well-recovered when using extended GTF basis
sets, …. … except at r 0 (nuclear cusp)
47
Fluorine atom (Hartree-Fock calculations)
Total atomic energy (Eh)
STO/BBB -99.40935
GTF/6-311++g(2df,2pd) -99.40173
GTF/cc-pV6Z -99.41627
Energy is well-recovered when
using extended GTF basis sets
Electron density is well-recovered when using extended GTF basis
sets, ….
… except at r 0 (nuclear cusp)
48
Core structure factors only
Parameters( sin / )max
0.7 Å-1 1.2 Å-1 1.8 Å-1
Nhkl 909 4,651 15,531
RF (%) 3.410-3 7.510-3 1.110-2
Theoretical structure factors: GTF cc-pV6Z wavefunction
Model structure factors: Independent Atom Model using STO
BBB wavefunction
contour levels are at 0.01 e Å-3
red - positive density, blue - negative density
dotted black line ( ) – zero contour
HF
residual Fourier (Fobs-Fcalc) core
density for (sin / )max = 1.8 Å-1
Core polarization is not
observed
“Frozen-core”
approximation is valid
49
Total vs Valence-only structure factors
Parameters( sin / )max
0.7 Å-1 1.2 Å-1 1.8 Å-1
Nhkl 909 4,651 15,531
RF (%) TOTAL 1.5 1.6 1.1
RF (%) VALENCE 3.8 9.5 13.7
Theoretical structure factors: GTF cc-pV6Z wavefunction
H F H F H F
contour levels are at 0.1 e Å-3; red - positive density, blue - negative density
residual Fourier (Fobs-Fcalc) valence density (Fourier deformation density)
(sin / )max = 0.7 Å-1 (sin / )max = 1.2 Å-1 (sin / )max = 1.8 Å-1
original (theoretical)
deformation density
H F
Model structure factors: Independent Atom Model using STO
BBB wavefunction
50
Valence-only structure factors. Kappa refinement
Parameters( sin / )max
0.7 Å-1 1.2 Å-1 1.8 Å-1
Nreflections 909 4,651 15,531
RF (%) IAM 3.8 9.5 13.7
RF (%) Kappa ref. 1.6 7.0 12.2
parameter Pv F / H 7.55 / 0.45 7.54 / 0.46 7.51 / 0.49
parameter F / H 0.97 / 1.55 0.97 / 1.56 0.97 / 1.52
max
0
33pseudoatom ),()'(')()()(l
l
l
lm
lmlmlvvcc dPrRrPrP r
refine parameters Pv and for each atom
CORE(r) VALENCE(r) DEFORMATION(r)
51
Valence-only structure factors. Kappa refinement
contour levels are at 0.05 e Å-3; red - positive density, blue - negative density
Residual Fourier (Fobs-Fcalc) density
(sin / )max
= 0.7 Å-1
(sin / )max
= 1.2 Å-1
(sin / )max
= 1.8 Å-1
IAM Kappa refinementRF = 3.8% RF = 1.6%
RF = 9.5% RF = 7.0%
RF = 13.7% RF = 12.2%
H F H F
52
Full aspherical pseudoatom (multipolar) refinement
Which aspherical deformation density parameters should be
refined?
Option 1: let Least Squares (LSQ) decide will likely work for the H-F example (theoretical data)
may (and often does!) yield unphysical results when refining
experimental data
Option 2: manually select functions to be refined take into account the so-called atomic “local” symmetries
max
0
33pseudoatom ),()'(')()()(l
l
l
lm
lmlmlvvcc dPrRrPrP r
CORE(r) VALENCE(r) DEFORMATION(r)
53
Deformation density functions are defined in local framesxF
zF
H
F
Flashback: population Plm accounts for the
shift of density between regions of opposite
sign of angular function dlm(,)
monopole d00 is not
needed (already have
spherical valence)
dipoles d1,-1 and d1,+1 violate local
symmetry
bond-directed dipole
d1,0 is Ok
!
quadrupoles d2,-2 , d2,-1 , d2,+1 and d2,+2 violate local symmetry
!
bond-directed
quadrupole d2,0 is Ok
bond-directed
octupole d3,0 is Ok
!
bond-directed
hexadecapole
d4,0 is Ok
!All other multipoles also violate
local symmetries of H and F !
yF
zH
xH
yH
54
Final refined parameters and statistics
refine parameters Pv , , Plm and (one for all Rl’s) for
each atom
Parameters( sin / )max
0.7 Å-1 1.2 Å-1 1.8 Å-1
Nreflections 909 4,651 15,531
RF (%) IAM 3.8 9.5 13.7
RF (%) Kappa ref. 1.6 7.0 12.2
RF (%) Full 0.2 1.0 2.2
Pv F / H 7.45 / 0.55 7.08 / 0.92 7.03 / 0.97
F / H 0.97 / 1.42 0.99 / 1.08 1.00 / 1.06
F / H 1.01 / 1.39 1.28 / 1.22 1.30 / 1.17
P10 F / H 0.42 / 0.04 -0.04 / 0.23 -0.02 / 0.26
P20 F / H 0.02 / -0.01 -0.10 / 0.11 -0.10 / 0.14
P30 F / H -0.07 / ‒ 0.01 / ‒ 0.01 / ‒
P40 F / H -0.01 / ‒ 0.02 / ‒ 0.02 / ‒
55
Residual Fourier (Fobs-Fcalc) densitycontour levels are at 0.05 e Å-3; red - positive density, blue - negative density
(sin / )max
= 0.7 Å-1
(sin / )max
= 1.2 Å-1
(sin / )max
= 1.8 Å-1
Kappa refinement
RF = 1.6%
RF = 7.0%
RF = 12.2%
H F
Full multipolar refinement
H F
RF = 0.2%
RF = 1.0%
RF = 2.2%
56
Fluorine multipoles in H-F from the (sin/)max = 1.2 Å-1
refinement (=1.28)
2D contour levels are at 0.02, 0.04,
0.0,8 0.2, 0.4, 0.8, 2., 4., 8.,… e Å-3;
red - positive density, blue - negative
density
F
04.010 P 10.020 P 01.030 P 02.040 P
superposition, F(r)
3D isocontours: +0.02 e Å-3 , -0.02 e Å-3
FF
F FH H H H
H
57
Hydrogen multipoles in H-F from the (sin/)max = 1.2 Å-1
refinement (=1.22)
2D contour levels are at 0.02, 0.04, 0.0,8 0.2, 0.4, 0.8, 2., 4.,
8.,… e Å-3; red - positive density, blue - negative density
23.010 P
3D isocontours: +0.02 e Å-3 , -0.02 e Å-3 superposition, H(r)
11.020 P
H H HFF F
FH
58
contour levels are at 0.1 e Å-3; red - positive density, blue - negative density
theory (original)
H
(sin / )max = 1.8 Å-1
(sin / )max = 1.2 Å-1
H
H
Low-resolution dataset is
“no good” !
High-resolution datasets
are OK
Deformation electron density (r) in the H-F molecule
(sin / )max = 0.7 Å-1
59
Deformation electron density (r) in the H-F molecule
positive density isocontour 0.1 e Å-3, negative density isocontour -0.1 e Å-3
FH
theory (original)
(sin / )max = 1.2 Å-1
High-resolution datasets
are OK
(sin / )max = 0.7 Å-1
FH
F
H FH
(sin / )max = 1.8 Å-1
Low-resolution dataset is “no
good” !
60
Dataset
Net atomic charge, q = qH = -qF
(electrons)
Molecular
dipole moment
z-component, z
(Debye)Mulliken Pseudoatom Hirshfeld QTAM
theory 0.31 — 0.23 0.78 -1.93
(sin/)max= 0.7Å-1 — 0.45 0.19 0.63 -0.77
(sin/)max= 1.2Å-1 — 0.08 0.22 0.73 -1.74
( sin/)max= 1.8Å-1 — 0.03 0.22 0.74 -1.70
Selected atomic and molecular properties
calculated from model densitiesx
y
zH
F comparing Mulliken and Pseudoatom
charges is like comparing and
Comparison of properties makes sense
only if the same definition is used
no problem if using the same definition, for example,
Hirshfeld, QTAM (Bader), ESP etc.
Questions ?
Comments ?
Suggestions ?
