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Shant Shahbazian

Faculty of chemistry, Department of pure chemistry, Shahid Beheshti

University

The Multi-Component Quantum Theory of Atoms in Molecules

(MC-QTAIM): New developments and novel opportunities

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Introduction

The conflicting views of quantum mechanics and the Structural

theory of chemistry on physical “reality” in micro-world

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Pattern recognition: Basic “forms” as the intuitive basis of science

Inhomogeneity of matter Recognition of forms in

real 3D space System and environment

and their boundary System is composed of

subsystems Classification of systems …Then comes the

abstraction in science

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The Structural theory of chemistry: “The” theory of chemical “forms”

The concept of molecular/chemical structures

Chemical structures as “forms”: real space entities

Modern conformational analysis: Handful laws and infinite diversity

Recognition of structural subsystems

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Atoms in molecules: Teaming the chemical diversity

Atoms in molecules as “well-defined” subsystems Atoms in molecules as quasi-rigid entities: Stability of

“underlying” subsystems Atoms in molecules as “transferable” subsystems:

Chemical anatomy based on functional groups The simplest model of molecules composed of atoms in

molecules: Molecular mechanics

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Experimental observation of “atoms in molecules”

The scanning tunneling microscope (STM) is an example of a modern probe to “see” atoms in molecules

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The Copenhagen interpretation of quantum mechanics and its anti-

realist attitude QM is the theory of the

measurements on outcomes: Probabilities

No “genuine” mechanical picture of system emerges from QM: Transition from Bohr to Heisenberg atom

Description of “physical reality” is done in abstract mathematical spaces

Wavefunction acts as a ghost in an alien world: Multi-dimensional Hilbert spaces…

Since 1976 Guy Woolley has forcefully stressed on the fact that “molecular structures” does notemerge directly from basic principles of QM

The orthodox response to the reconciliation of QM and the Structural theory

From the dawn of QM the Born-Oppenheimer approximation (BOA) was utilized to incorporate something akin of molecular structure into QM

The BOA justifies the clamped nucleus model which treats molecules as a decoupled systems of electrons, as quantum waves, and nuclei, as clamped points charges

From this viewpoint the concept of potential energy hypersurface (PES) and electronic wavefunctions emerges

In all of these entities nuclear coordinates acts as “parameters” rather than “variables” that may be varied independent from electronic coordinates

The structures are usually assumed to the geometry of clamped nuclei at stationary points of the PES while chemical reactions are paths on the PES connecting to stationary stable points through a saddle point (transition state) 8

Why the orthodox response is not adequate The clamped nuclei are not representatives of “atoms in

molecules” The whole procedure as well as the concept of PES is only

justifiable within the BOA Then, when the BOA does not work how structures must

be described? The BOA is just an approximation then is it reasonable to

attach the whole foundations of Structural chemistry to an approximate outlook of molecules?

Some even cast doubt that molecular structures, even in principle, are derivable from QM without imposing some sort of approximation, e.g. the BOA, into the basic equations of QM…

So, is chemistry based on shaky grounds? 9

Why bothering to reconcile the pictures emerging from QM and the Structural theory of molecules?

Why not assume that molecules have structures as an axiom? Why trying to extract the concept of “molecular structure” from QM?

In the reductionist approach to the science one always assumes that the properties of a system is somehow emerges from the its constituents and the nature of their interactions

Examples are “temperature” and “consciousness ” A vessel containing a gas is composed of atoms of the gas each governed by the

laws of dynamics while temperature emerges somehow from the collective effect of all atoms of the gas; a dynamical model of atoms must make a connection between dynamical variables and temperature and this is indeed done in kinetic theory of gases

A brain is composed of the neurons governed by biochemical and biophysical laws while consciousness emerges from the collective effect of all neurons of the brain; a network model of neurons must make a connection between electrical and chemical properties of neurons and the consciousness and this has yet to be archived…

Then, can we pursue the same reductionist approach making a connection between basics principles and equations of the QM and the concept of molecular structure? Woolley’s dilemma… 10

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The QTAIM

Building a “bridge” between quantum mechanics and the

Structural theory of chemistry: The Quantum Theory of

Atoms In Molecules

The basic ideas behind the QTAIM “Projection” from Hilbert/configuration space to the real space (3D) is the

necessary step:

Each function in real space carries some of the “information” encoded in the wavefunction though one needs to concrete “axioms” delineating the exact nature of the projection

Each function is a “property density” in real space; the energy density determines the distribution of energy in 3D space while the particle density determines the distribution of particles in 3D space, etc.

One of these functions must be used to determine the “shape” of AIM (atomic basins), delineating the boundaries of AIM, while the integration of property densities within an atomic basin yields the properties of the AIM 12

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The basic principles of the QTAIM The shape of AIM emerges from the electron density:

Each property density emerges from following integral where M stands for the operator of the desired property :

The “zero-flux equation” of the “one-electron density” determines the boundaries of AIM and atomic basins

Through the integration of property densities in each atomic basin the atomic properties emerge:

Ne ddNRr ..., 2

0 nr

RrMdM ee

,

MddNRrM Neˆ...Re, 2

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The X-ray crystallography: Evidence of atoms in molecules in electron density maps

The X-ray diffraction patterns reveals the structure of electron densities of crystals

The resulting electron densities reveal the fact that there areatoms in molecules

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Some features of the electron densities Monotonicity: Electron

density of atoms at their ground state decrease monotonically away from nucleus

Robustness: In contrast to wavefunction, electron densities of atoms are not affected much after formation of a molecule

These two features guarantee the stability of underlying atomic forms and their relative transferability

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“Topological analysis” of electron densities: Unraveling details

To the electron density as a scalar field one may attribute an associated vector gradient of field:

This vector field reveals the details of “internal” structure of the electron density through identifying the critical points (CPs)

A gradient path is a succession of infinitesimally small gradient vectors

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Typical gradient vector fields of the electron density: Molecular graphs

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A 3D view to the “topological” structure of an inter-atomic boundary

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AIM: The basic “chemical subsystems”

The AIM of formaldehyde serve as an example where the yellow surfaces are the inter-atomic surfaces or the boundary between AIM

An outer boundary may be conceived to encompass the whole system that are the white, black and red surfaces

The morphology of AIM depends on their chemical environment…

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The subsystem hypervirial theorem and atomic properties

The hypervirial theorem for quantum systems is easily derived from the Heisenberg equations of motion for stationary states:

This theorem has been extended within the context of the QTAIM to “open” subsystems and is termed the subsystem hypervirial theorem that contains an “extra” surface term:

Each atomic property is derived from this theorem using a hermitian“generator”:

The surface term makes the AIM properties “non-trivial” since they contribute to atomic properties but “null” for molecule (system)

0],ˆ[,],ˆ[ GHGHd

nrJdSGHi G

Re],ˆ[,Re

GGdmiNrJ

RG

3

2

MGHi ˆ],ˆ[

The subsystem virial theorem and basin energies at the equilibrium geometry

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Concluding by a metaphor The QTAIM formalism acts like a “machine”

that its “input” is the molecular electronic wavefunction while its “output” is the AIM morphology and their properties; the axioms of the QTAIM are just the gearwheels of this machine!

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MC-QTAIM

Extending QTAIM beyond the Born-Oppenheimer paradigm

and to the exotic species

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Why extending the QTAIM? The wavefunctions used for the QTAIM analysis are always “electronic

wavefunctions” derived from the electronic Schrodinger equation that assumes nuclei as clamped point charges; electrons are solely described as quantum waves

For wavefunctions derived beyond the BOA, assuming nuclei as quantum waves instead of clamped nuclei, the QTAIM methodology is not applicable

There are molecular species that are not made solely from electrons and nuclei that the positronic and muonic molecules are just examples; positron and muon are fundamental particles interacting with molecules with electric Columbic forces making “bound” states that are novel molecular species

The QTAIM methodology is not also applicable to the wavefunctions derived for these exotic species since positron (anti-electron) and the positive muon, similar to electrons, must be treated as quantum waves instead of clamped particles; both are light particles (positon’s mass is equal to the electron’s mass and muon’s mass is ~206.8 of the electron’s mass) and none can be modeled as clamped particles

All these are examples of “multi-component” systems that include quantum systems with more than a single type of quantum particles

The orthodox QTAIM is intrinsically a “single-component” theory thus unable to use the “multi-component wavefunctions” as input

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The basic principles of the MC-QTAIM The form of AIM emerges from the zero-flux equation of the

Gamma density:

The Gamma density is a mass-scaled combined density of all one-particle densities of the quantum particles of the multi-component system; “P” is called the “cardinal number” and is the total number of particle types while type “1” is the set of lightest quantum particles of system

It is feasible to demonstrate analytically that when the masses of quantum particles approach infinity the Gamma density reduces to the familiar one-electron density:

The same combination strategy is also used to construct the property densities assuming that they originate from all types of quantum particles:

,2

11 rmmrr n

P

nn

P

0 nrp

rr emn

1lim

rMrM n

P

n

1

~

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The extended subsystem hypervirialtheorem and atomic properties

The extended subsystem hypervirial theorem for a multi-component open quantum subsystem has the following form:

In this equation all types of quantum particles contribute to the surface term while “g” is the hermitian generator for the property “M”

The atomic properties are derived by integration of property densities in each atomic basin:

rJdSrMd G

~~

rrrrnnnnG ggdimNrJ

ˆˆ2

P

n

nGG rJrJ

1

~

rMdM ~~

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Some applications of the MC-QTAIM

AIM analysis beyond the BO paradigm nuclei: Since nuclei are treated as quantum waves, in contrast to the electronic wavefunctions, the non-BO wavefunctions are sensitive to the nuclear mass and a molecule with various isotope compositions has distinct non-BO wavefunction for each isotope composition

Thus, the MC-QTAIM analysis of such species yield distinct AIM that carry the “fingerprint” of their isotopes

The AIM analysis may be extended to the positronic species (molecular species containing one positron)

The AIM analysis may be extended to the muonic species (molecular species containing one or more muons)

Both positronic and muonic molecular species are now routinely produced and considered in various laboratories around the world thus their AIM analysis is chemically relevant

Exotic phenomena: Mass induced topological structural transformations and AIM analysis of the “cat” states

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The main steps for “computational” implementation of the MC-QTAIM analysis A multi-component wavefunction of a molecular species is the

“input” of the MC-QTAIM analysis A novel computer code was developed in our lab to perform ab

initio at the multi-component Hartree-Fock (HF), multi-component MP2, multi-component configuration interaction (CI) and multi-configurational SCF (MCSCF) levels for polyatomic molecules (an extension of the original NEO code implemented in the GAMESS package)

Proper basis set were designed for light nuclei, i.e. hydrogen isotopes, as quantum waves, as well as for the positive muon

The derived wavefunctions are coded in “extended” wfnprotocols and then used as input for the computer code developed to perform the MC-QTAIM analysis including the topological analysis of the Gamma density and integration to derive basin properties

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LiX species (X=H, D, T): Hydrogen nucleus has been treated as a quantum wave

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Some numerical results of the MC-QTAIM analysis: Isotope dependence of

hydrogen's electronegativity trend!

Some basic hydrocarbons: Molecular graphs assuming hydrogen nuclei as quantum waves

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Volumes of atomic basins containing proton is smaller than those contacting deuterium!

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Atomic basins and their properties in the positronic LiH (LiH,e+)

Muonic Hydrogen molecule (Mu+2e+P): The positive muon makes its own atomic basin!

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Comparison of proton and muon atomic basins in LiH and LiMu species

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Molecular graphs and atomic basins of various muonic hydride like species

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The positive muon is a lighter isotope of hydrogen!

Second row

Third row

Mass induced topological structural transformation: X-C≡N and C=N-X series

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Phase diagram!

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Schrödinger’s cat: Quantum superposition in action

Any linear superposition of quantum states is also a new quantum state

This is best exemplified in Schrodinger’s cat paradox where a cat in a superposition of “live” and “dead” states

While for macroscopic states such superpositions have not been yet observed, in 1996 Wineland and coworkers produced such “cat” states for a Beryllium ion in an electromagnetic trap (Science 272, 1131 (1996))

In such cat states an atom is in two positions at the same time!

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Toward realization of a “cat” state through double well potential

Double well potentials are simplest examples of the realization of a cat state

Such states are usually interpreted through the concept of “quantum tunneling”

However, this interpretation is only legitimate for “non-stationary” states not for “stationary” states without time evolution

At stationary states the quantum “object” is distributed in space! 46

Malonaldeyde as a double well system

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MC-QTAIM analysis of Malonaldeyde’scat state: Two “half” instead one

complete hydrogen atom!

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Conclusion and prospects The MC-QTAIM widens the AIM analysis considerably to

systems and quantum states that were considered traditionally not amenable to AIM analysis

The isotope effects are predominate in chemistry and it is possible to analyze these effects from the viewpoint of AIM analysis employing the MC-QTAIM analysis, the isotope dependence of barriers of internal rotations is a prime example

The AIM analysis of exotic species other than positronic and muonic species is also an interesting aria, alpha clustering in certain nuclei called “nuclear molecules” is a leap in AIM analysis beyond the electronic matter while metallic hydrogen is another encouraging target…

The time-dependent MC-QTAIM analysis is also a promising area for future considerations of the non-stationary states

.. And hopefully more novel applications are awaiting for future studies…

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Thanks for your attention

There must be chemistry in all these wave functions because we live in one world only

Paul Popelier