Quantum Theory of Atoms in
Molecules
by
Cina Foroutan-Nejad
1CINA FOROUTAN-NEJAD
Are There “Atoms” in Molecules
• A Physicist Response:
NO! There is no operator in QM describing such a things!
• A Chemist Response:
Yes! I am classifying many reactions based on “functional
groups” there must be something!
• My Recommendation:
• An Skeptic Response:
Maybe!
2CINA FOROUTAN-NEJAD
How To Define an Atom in a
Molecule?
• There are various approaches based on orbital
view point but the lecturer’s recommendation is
electron density-based approaches!
• Electron density is a 3-dimensional function in
contrast to wavefunction which is 4N-dimensional
perN electrons.
• There are various approaches for definition of
atoms in molecules based on electron density,
here we focus on the topological approach for
defining an atom in a molecule; the QTAIM
approach.
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The Free Atom in the Context of
QTAIM
• The electron density distribution
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The Free Atom in the Context of
QTAIM
• The Laplacian of electron density.
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Atoms in a Molecule
An atomic boundary
should satisfies this
equation:
𝛻𝜌. 𝑛 = 0
Where n is the
normal vector to the
interatomic surface
6CINA FOROUTAN-NEJAD
What is a Critical Points in QTAIM?
A critical point, CP, is a point at which
𝛻𝜌 = 0
Four different types of CP could be
found in a molecular space of a
molecule. Each CP can be introduced by
a rank and a signature, (r, s).
Rank represents the number of non-zero
diagonal eigenvalues of the Hessian
matrix of the Laplacian of electron
density. Signature denotes local
curvatures of the CP with respect to the
x, y and z directions.
7CINA FOROUTAN-NEJAD
Critical Points of QTAIM
• A (Non)-Nuclear Attractor or (3,-3) CP is a point in which
electron density is maximum with respect to all directions (x,
y, z).
• A Bond CP or (3,-1) CP is a point in which electron density is
minimum with respect to the maximum density path,
connecting two (3,-3) points but maximum respecting to the
rest of directions
• A Ring or (3,+1) CP is a point on a surface which is minimum
in electron density on that surface but maximum with respect
to the path perpendicular to the surface
• A Cage or (3,+3) CP is a point which is minimum with respect
to all other points around it.
8CINA FOROUTAN-NEJAD
Critical Points of QTAIM; a Summary
Full Name Acronym λ1 λ2 λ3 (r, s)
(Non)-Nuclear Attractor (N)NA – – – (3, –3)
Bond (Line) Critical Point B(L)CP – – + (3, –1)
Ring Critical Point RCP – + + (3, +1)
Cage Critical Point CCP + + + (3, +3)
9
Poincare-Hopf rule:
(N)NA – B(L)CP + RCP – CCP = 1
CINA FOROUTAN-NEJAD
Critical Points and Bond Paths of
QTAIM
10CINA FOROUTAN-NEJAD
Bonding in the Context of QTAIM
Presence of BP or AIL between two atoms was first
introduced as the necessary and sufficient conditions for
presence of chemical bond. In QTAIM terminology each
bond could be classified either as Shared or Closed Shell.
If in a (3,-1) CP, 𝛻2 𝜌 < 0, then the corresponding bond
could be classified as Shared.
In contrary, if in a (3,-1) CP 𝛻2
𝜌 > 0, then the
corresponding bond is classified as a Closed Shell
interaction.
Shared interactions usually are found between covalently
bonded atoms. Closed Shell interactions usually
correspond to ionic and van der Waals bonds.
11CINA FOROUTAN-NEJAD
Bonding in the Context of QTAIM
Properties Description & Application
ρ(r) Electron Density; in homologous series a higher ρ(r) means stronger
bond. ρ(r) could be used for defining the bond order.
L(r) -1/4 𝜵 𝟐 𝝆(r); definition of charge concentration/depletion and
discriminating bonds as Shared or Closed Shell
K(r) −
𝟏
𝟒
𝑵 𝒅𝝉 𝝍∗
𝛁 𝟐
𝝍 + 𝝍 𝛁 𝟐
𝝍∗
, Hamiltonian Kinetic Energy
G(r) −
𝟏
𝟒
𝑵 𝒅𝝉𝛁 𝝍∗. 𝛁 𝝍 , Lagrangian Kinetic Energy; G(r) = K(r) – L(r)
G(r)/ρ(r) Defines the local kinetic energy per electron; a value bigger than
unity denotes presence of ionic-type of bonding, less than unity
denotes presence of covalent-type bonds
V(r) If the Virial ratio in a molecule is exactly 2 then V(r) = - K(r) = H(r),
the Energy Density. Negative values represent covalency and vice
versa
ε(r) 𝝀 𝟏
𝝀 𝟐
− 𝟏, Bond Ellipticity; defines the preferred directional distribution
of electron density in (3, -1) CP
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Bond Path or Line Path?
It was shown that B(L)Ps may appear in unexpected positions! A
good example is the bond critical point between two hydrogen
atoms of a planar biphenyl.
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Unexpected BPs! H-H Bonding
Controversy…
14CINA FOROUTAN-NEJAD
The Quantum Atom
• The Gauss Divergence Theorem:
if Ω is a region in space, bounded by a surface S, V is a vector field
defined in Ω and n is the normal vector to S, then:
𝑑𝑉𝛻. 𝑉 = 𝑑𝑆 𝑉. 𝑛
𝑆Ω
This theorem is valid since integration and differentiation are
inverse operators.
• A Quantum Atom is a region in space in which
𝑑𝑉𝛻2
𝜌 = 0Ω
since
𝑑𝑉𝛻2 𝜌 =
Ω
𝑑𝑉𝛻. 𝛻𝜌 = 𝑑𝑆 𝛻𝜌. 𝑛
𝑆Ω
• The 𝑑𝑆 𝛻𝜌. 𝑛𝑆
is simply integration over the zero-flux surface of a
topological atom.
15CINA FOROUTAN-NEJAD
Atomic Properties
• Atomic Energy, E(Ω)
• Atomic Charge, q(Ω)
• Atomic Volume, Vol(Ω)
• Atomic Multiple Electric Moments, μ(Ω),
Q(Ω)
• Atomic Magnetizability, χ(Ω)
• Localization and Delocalization Indices, λ(Ω),
δ(Ω|Ω’)
16CINA FOROUTAN-NEJAD
Non-Nuclear Attractors and United
Atoms; Beyond Classical Chemistry!
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Two Problematic Types of MG
18
I. Unexpected Bond Paths[1-3] II. Missing Bond Paths[4-6]
1. C. F. Matta, J. Hernandez-Trujillo, T.-H. Tang, R. F. W. Bader, Chem. Eur. J., 2003, 9, 1940.
2. J. Poater, M. Sola, M. Bickelhaupt, Chem. Eur. J., 2006, 12, 2889.
3. R. F. W. Bader, Chem. Eur. J., 2006, 12, 2896.
4. J. Farrugia, C. Evans, M. Tegel, J. Phys. Chem. A 2006, 110, 7952.
5. M. Mousavi, G. Frenking, J. Organomet. Chem. 2013, 748, 2.
6. M. Mousavi, G. Frenking, Organometallics 2013, 32, 1743.CINA FOROUTAN-NEJAD
Dynamic vs. Static Molecular Graphs
• Distribution of electron density depends on the nuclear
geometry so any change in the nuclear geometry is followed
by sudden changes in the topology of electron density, i.e.
molecular graph.
• In general, it is believed that finite geometry changes due to
molecular vibrations/rotations do not affect molecular graph.
Though, some exotic cases are known that break this general
rule.[7]
• The role of geometry in the distribution of the electron density
for problematic cases of type II was investigated.
19
7. C. Foroutan-Nejad, G. H. Shafiee, A. Sadjadi, S. Shahbazian, Can. J. Chem. 2006, 84,
771.
CINA FOROUTAN-NEJAD
Missing BP in TMM Complexes of
TMs[6]
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Missing BP in TMM Complexes of
TMs[6]
Geometry changes in low-force constant normal
modes of TMM complexes were investigated.
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Missing BP in TMM Complexes of
TMs[6]
22CINA FOROUTAN-NEJAD
Missing BP in TMM Complexes of
TMs[6]
23CINA FOROUTAN-NEJAD
24
Another Example[8,9]
8. J. Henn, D. Leusser, D. Stalke, J. Comput. Chem.
2007, 28, 2317.
9. H. Jacobsen, J. Comput. Chem. 2009, 30, 1093.
CINA FOROUTAN-NEJAD
25
Open Question:
Ionic Interaction[9]
[9] N. W. Mitzel, K. Vojinovic´, R. Frçhlich, T.
Foerster, H. E. Robertson, K. B.
Borisenko, D. W. H. Rankin, J. Am. Chem. Soc. 2005,
127, 13705.
CINA FOROUTAN-NEJAD
Take Home Messages
• (3, -1) CPs are not chemical bonds. A bond cannot
appear and disappear by vibrations.
• In analogy with Ring and Cage Critical Points, (3, -1)
CPs are suggested to be called Line Critical Points
(LCP) to avoid misunderstanding.
• Similarly, Bond Paths are suggested to be called Line
Paths (LP).
• Electron Delocalization (Delocalization Index) is the
best descriptor of covalency. It does not depend on
presence or absence of LCPs.
26CINA FOROUTAN-NEJAD
Take Home Messages
• A new terminology was suggested to classify atoms
in a molecule in three groups:
-Non-neighbors, which are not connected by LCP/LP
because have no common interatomic surface
-Neighbors that are connected by an LCP
-Passionate neighbors! These atoms are separated
because a tiny atomic basin penetrates between them
but they can form LCP in a non-equilibrium but lowenergy
structure.
27CINA FOROUTAN-NEJAD
28CINA FOROUTAN-NEJAD
How to Identify a Chemical Bond?
• Do not rely on one descriptor
• Inspect both molecular orbitals and electron
density
• Always compare your system with well-known
systems
• See the next slides…
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Are These Uranium Atoms Bonded?
30
U-U Distance: 3.905 Å
U2@C80
CINA FOROUTAN-NEJAD
MO Analysis
31
𝑯𝑶𝑴𝑶 𝑯𝑶𝑴𝑶 − 𝟏 𝑯𝑶𝑴𝑶 − 𝟐
𝑯𝑶𝑴𝑶 − 𝟑 𝑯𝑶𝑴𝑶 − 𝟒 𝑯𝑶𝑴𝑶 − 𝟓
CINA FOROUTAN-NEJAD
Electron Density Analysis
32
𝝆 𝜶 − 𝝆 𝜷 𝑬𝒏𝒆𝒓𝒈𝒚 𝑫𝒆𝒏𝒔𝒊𝒕𝒚, 𝑯(𝒓) 𝛁 𝟐
𝝆
𝛁 𝟐 𝝆 𝜷 𝛁 𝟐 𝝆 𝜶CINA FOROUTAN-NEJAD
Don’t Forget Comparison & DI
• Selected model system free U2 molecule
• U2 is known to have a quintuple bond
• DI (U2): 5.02
• DI U2@C80: 1.01
• DI is consistent with a single bond or two
single-electron bonds also confirms MO
analysis results
U2@C80 are Bonded
33CINA FOROUTAN-NEJAD
Background: Chemical Bond
Unlike ionic bonds, which are formed by attraction between
ions, formation of covalent bonds does not have an electrostatic
driving force.[1]
[1] J. C. Slater, J. Chem. Phys.,
1933, 1, 687.
34CINA FOROUTAN-NEJAD
Different Views On Covalency
1. Morokuma-Zeigler-Rauk
2. Bader
3. Ruedenberg
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Bond Order from IUPAC Definition to
QTAIM
• What is a covalent bond order?
• According to the IUPAC definition the bond order is defined as
the total orbital overlap population between a pair of atoms.
In QTAIM, this quantity is called delocalization index, δ(A,B),
and can be calculated between any pair of atoms like A and B
in a molecule or a supramolecular complex.
𝛿 𝐴, 𝐵 = −2𝑐𝑜𝑣 𝑛 𝐴, 𝑛 𝐵 = −2 𝑛 𝐴 𝑛 𝐵 − 𝑛 𝐴 𝑛 𝐵
𝛿 𝐴, 𝐵 = − 𝑆𝑖,𝑗(𝐴)𝑆𝑖,𝑗(𝐵)
𝑖,𝑗
36CINA FOROUTAN-NEJAD
Towards a Physically Meaningful
Energy Decomposition Scheme
No matter how you select molecular fragments, a physically
meaningful energy decomposition scheme (EDS) must have
certain properties.
• Should recover binding energy in terms of physical terms, i.e.,
kinetic and potential energy components, with least number
of arbitrary energy components such as “polarization”,
“orbital interaction”, etc. (Occam’s Razor).
• Must enable user to recover chemically relevant data.
37CINA FOROUTAN-NEJAD
Interacting Quantum Atoms Energy
Decomposition Scheme
• IQA analysis can be performed employing overlapping (fuzzy
atoms) or non-overlapping (QTAIM) molecular subspaces.
• IQA analysis in QTAIM atomic space has unique advantages.
Non-overlapping atoms permits defining “chemically
meaningful fragments” and studying inter-fragment
interactions.
• QTAIM atoms satisfy atomic virial theorem.
38CINA FOROUTAN-NEJAD
IQA Energy Decomposition Scheme
• It has been demonstrated that QTAIM fragments need the least
promotion energy[9], also satisfying Li and Parr partitioning model,
among various partitioning methods.[10]
• For any atomic space one can define self-interaction energy as the
following:
Eself (Ω) = T(ΩA) + Ven (ΩA) + Vee (ΩA)
• Similarly interatomic interaction energy can be broken to the
following components:
Einter (ΩA,ΩB) = Vnn (ΩA,ΩB) + VXC_ee (ΩA,ΩB) + Vcoulomb_ee (ΩA,ΩB) +
Ven (ΩA,ΩB) + Vne (ΩA,ΩB)
= VEl + VXC
[9] L. Li and R. G. Parr, J. Chem. Phys. 1986, 84, 1704.
[10] A. M. Pendas, M. A. Blanco and E. Francisco, J. Comput. Chem. 2007, 28, 161. 39CINA FOROUTAN-NEJAD
IQA Energy Decomposition Scheme
• IQA recovers the binding energy components in terms of
physically meaningful terms as the following:
EBind = VEl + VXC + EPr
• VEl and VXC represent contribution of all classical and
exchange-correlation energy components, respectively.
• The promotion energy, EPr, is the sum of all factors that
change the energy of an atomic sub-space, including kinetic
energy, electron-electron repulsion, and electron-nucleus
attraction.
• In IQA analysis the kinetic energy has no interatomic
component and nuclear-nuclear repulsion clearly has no
atomic component.
40CINA FOROUTAN-NEJAD
How Electron Delocalization Lowers
Energy of a molecule?
• This old question can be quantitatively answered in the context of
IQA. The electron delocalization and exchange correlation energy
component can be linked together by a first order approximation.
𝛿 Ω 𝐴, Ω 𝐵 = −2 [𝜌2
𝐴𝐵
(𝒓 𝟏, 𝒓 𝟐)
Ω 𝐵
− 𝜌 𝐴(𝒓 𝟏)𝜌 𝐵(𝒓 𝟐)]𝑑𝒓 𝟏 𝑑𝒓 𝟐
Ω 𝐴
𝑑𝒓 𝟏 𝑑𝒓 𝟐
𝑉𝑋𝐶 Ω 𝐴, Ω 𝐵 = [𝜌2
𝐴𝐵
(𝒓 𝟏, 𝒓 𝟐)
Ω 𝐵
− 𝜌 𝐴(𝒓 𝟏)𝜌 𝐵(𝒓 𝟐)]𝑟12
−1
𝑑𝒓 𝟏 𝑑𝒓 𝟐
Ω 𝐴
𝜹(𝜴 𝑨, 𝜴 𝑩)
𝑹 𝑨𝑩
= 𝒎 𝑽 𝑿𝑪 (𝜴 𝑨, 𝜴 𝑩)
41CINA FOROUTAN-NEJAD
Thank you for your
attention!
Now it is time for
question!
42CINA FOROUTAN-NEJAD