Topological Analysis of Electron Density Louis J Farrugia Jyväskylä Summer School on Charge Density August 2007

Topological Analysis of Electron Density

Louis J Farrugia

Jyväskylä Summer School on Charge Density August 2007

http://www.westchem.ac.uk/

What is topological analysis ?Jyväskylä Summer School on Charge Density August 2007

A method of obtaining chemically significant information from A method of obtaining chemically significant information from the electron density the electron density (rho) (rho)

is a quantum-mechanical is a quantum-mechanical observableobservable, and may also be , and may also be obtainedobtainedfrom experimentfrom experiment

““It seems to me that experimental study of the scattered It seems to me that experimental study of the scattered radiation, in particular from light atoms, should get more radiation, in particular from light atoms, should get more attention, since in this way it should be possible to determine attention, since in this way it should be possible to determine the arrangement of the electrons in the atoms.”the arrangement of the electrons in the atoms.”

P. Debye, Ann. Phys. (1915) 48, 809.


Why analyse the charge density ?

P. Hohenberg, W. Kohn Phys. Rev. 1964, 136, B864.

The traditional way of approaching the theoretical basis of The traditional way of approaching the theoretical basis of chemistry is though the chemistry is though the wavefunction wavefunction and theand the molecular molecular orbitals orbitals obtained through (approximate) solutions to the obtained through (approximate) solutions to the Schrödinger wave equationSchrödinger wave equation

HH = E = EThe Hohenberg-Kohn theorem confirmed that the density, The Hohenberg-Kohn theorem confirmed that the density, ((rr), ), is the fundamental property that characterises the ground is the fundamental property that characterises the ground state of a system - once state of a system - once ((rr) is known, the energy of the ) is known, the energy of the system is uniquely defined, and from there a diverse range of system is uniquely defined, and from there a diverse range of molecular properties can, in principle, be deduced. molecular properties can, in principle, be deduced.

Thus a knowledge of Thus a knowledge of ((rr) opens the door to understanding of ) opens the door to understanding of all the key challenges of chemistry. all the key challenges of chemistry. is a quantum- is a quantum-mechanical observable.mechanical observable.9


Can we ever observe orbitals ?

Simply not ever possible - see E. Scerri (2000) J. Chem. Ed. 77, 1492.

However ?? - see J. Itani et al (2004) Nature 432, 867


R. F. W. Bader Atoms in Molecules : A quantum theory, OUP 1990

P. L. A. Popelier Atoms in Molecules : An Introduction, Pearson, 2000.R. J. Gillespie & P. L. A. Popelier Chemical Bonding and Molecular Geometry, OUP, 2001.

What is topological analysis ?


Topology is a branch of mathematics involving the study of the Topology is a branch of mathematics involving the study of the nature of space. The theorems are used extensively in subjects nature of space. The theorems are used extensively in subjects like geographylike geography

In chemical topology, we are concerned with the properties of In chemical topology, we are concerned with the properties of scalarscalarfieldsfields, like the (continuous) real function , like the (continuous) real function ((rr). These properties ). These properties are displayed in global terms in the are displayed in global terms in the gradient vector fieldgradient vector field of of ((rr). An analysis of the topology of ). An analysis of the topology of ((rr) leads directly to the ) leads directly to the chemical concepts of chemical concepts of atomsatoms, , moleculesmolecules, , structuresstructures & & bondsbonds..are cio09

What is topological analysis ?

Representation of the total charge density


A representation of the total A representation of the total charge density charge density ((rr) in the plane ) in the plane of the formamide molecule Hof the formamide molecule H22N-N-CHO.CHO.

This is a typical picture of This is a typical picture of ((rr), ), where the distribution is where the distribution is dominated by the electrostatic dominated by the electrostatic attraction of the electrons for the attraction of the electrons for the positively charged nucleipositively charged nuclei. The . The maxima occur at the nuclear maxima occur at the nuclear sites and sites and ((rr) ) decays in a decays in a nearly spherical manner away nearly spherical manner away from the nuclei.from the nuclei.

The other obvious features The other obvious features are the saddle points are the saddle points between the nuclei. between the nuclei. e cio09

The Hessian matrix and LaplacianJyväskylä Summer School on Charge Density August 2007

A quantitative way to analyse the topology of A quantitative way to analyse the topology of is to consider is to consider the first derivative (gradient) the first derivative (gradient) ((). At certain points, called ). At certain points, called critical pointscritical points, this gradient vanishes. The characteristic of , this gradient vanishes. The characteristic of these points is determined by the second derivative these points is determined by the second derivative 22((), and ), and the so-called Hessian of the so-called Hessian of . The Hessian is the (3. The Hessian is the (33) symmetric 3) symmetric matrix of partial second derivativesmatrix of partial second derivatives

22//xx22 22//x x y y 22//x x zz 22//y y xx 22//yy22 22//y y zz22//z z xx 22//z z y y 22//zz22

By diagonalisation of this matrix, we set the off-diagonal terms By diagonalisation of this matrix, we set the off-diagonal terms to zero, and obtain the three to zero, and obtain the three principal axes of curvatureprincipal axes of curvature. These . These principal axes will correspond to symmetry axes, if the critical principal axes will correspond to symmetry axes, if the critical point lies on a symmetry element. The sum of the diagonal point lies on a symmetry element. The sum of the diagonal terms is called the Laplacian of terms is called the Laplacian of , , 22((), and is of fundamental ), and is of fundamental importance.importance.

22(() = ) = 22//xx22 + + 22//yy22 + + 22//zz22

Critical point definitionsJyväskylä Summer School on Charge Density August 2007

At a critical point, the eigenvalues of the Hessian are all real At a critical point, the eigenvalues of the Hessian are all real and are generally non-zero. The and are generally non-zero. The rank rank of the critical point is of the critical point is defined as the number of non-zero eigenvalues, while the defined as the number of non-zero eigenvalues, while the signaturesignature is defined as the algebraic sum of the signs of the is defined as the algebraic sum of the signs of the eigenvalues. These two characteristics are used to label a eigenvalues. These two characteristics are used to label a critical point (critical point (rank,signaturerank,signature). For topologically stable critical ). For topologically stable critical points, the points, the rankrank is always 3. The four possibilities are then is always 3. The four possibilities are then

(3,-3) All curvatures –ve, a local maximum

(3,-1) Two curvatures are –ve and one is +ve is a maximum in a plane and a minimum perpendicular to this plane – a bond cp

(3,+1) Two curvatures are +ve and one is -ve is a minimum in a plane and a maximum perpendicular to this plane - a ring cp

3,+3) All curvatures +ve, a local minimum – a cage cp

The gradient vector fieldJyväskylä Summer School on Charge Density August 2007

This is a representation This is a representation of the gradient vector of the gradient vector field of field of in the in the molecular plane of molecular plane of formamideformamide

The red paths trace the The red paths trace the direction of maximum direction of maximum gradient of gradient of in leaving in leaving the nucleus. For a the nucleus. For a molecule in the gas molecule in the gas phase, the trajectories phase, the trajectories will generally terminate will generally terminate at infinity.at infinity.

In special cases however In special cases however they will terminate at they will terminate at another nucleus – these another nucleus – these special trajectories are special trajectories are known as known as bond pathsbond paths


These special paths are These special paths are associated with the associated with the another topological another topological object, the bond critical object, the bond critical point (shown in blue)point (shown in blue)

Two other trajectories Two other trajectories leave the critical point leave the critical point and terminate at infinity. and terminate at infinity. These are part of a These are part of a family of trajectories family of trajectories defining the defining the zero flux zero flux surface.surface.

((rr))nn((rr) = 0) = 0The scalar product of The scalar product of ((rr))with with nn((rr) the vector ) the vector normal to the surface. normal to the surface. This surfaceThis surfaceencloses each atom and encloses each atom and defines a sub-space – defines a sub-space – the quantum topological the quantum topological definition of an definition of an atom in a atom in a moleculemolecule..

The gradient vector field

The topological atomJyväskylä Summer School on Charge Density August 2007

These interatomic These interatomic surfaces result in a surfaces result in a unique way of unique way of partitioning 3D-space in partitioning 3D-space in molecules into molecules into atomic atomic basins.basins.

The topological atoms The topological atoms are quite transferable are quite transferable and additive, but do not and additive, but do not look very much like the look very much like the balls and spheres of balls and spheres of molecular models !!!molecular models !!!

The simple binary The simple binary hydrides of the second hydrides of the second period elements show period elements show that the relative volumes that the relative volumes of space associated with of space associated with each element is each element is determined by their determined by their relative electro-relative electro-negativities. Surfaces negativities. Surfaces are are truncated at 0.001 au.truncated at 0.001 au.


WinXD program – XD2006 T. Koritsanszky et al (2006)

Ni(CO)4





Morphy98 program P. L. A. Popelier et al (1998)

glycine





The partitioning in real space of molecules into atoms by using these interatomic The partitioning in real space of molecules into atoms by using these interatomic surfaces is a fundamental, quantum mechanically rigorous, method. It allows surfaces is a fundamental, quantum mechanically rigorous, method. It allows properties to be calculated for proper open systems, where exchange, e.g. with properties to be calculated for proper open systems, where exchange, e.g. with charge may occur between atoms. The properties calculated by integration within charge may occur between atoms. The properties calculated by integration within these boundaries, O(these boundaries, O(), are chararacteristic of that atom in its chemical ), are chararacteristic of that atom in its chemical environment. Such integrated properties includeenvironment. Such integrated properties include

1. Electron population 1. Electron population NN(() – subtraction of the nuclear charge give the Bader ) – subtraction of the nuclear charge give the Bader atomic charge atomic charge qq((). ). This is an unambiguous method of determining atomic charges This is an unambiguous method of determining atomic charges !!

2. Atomic volume Vol(2. Atomic volume Vol() – the volume of space inside the interatomic surface. ) – the volume of space inside the interatomic surface. Since (for molecules in the gas phase) part of the interatomic surface is terminated Since (for molecules in the gas phase) part of the interatomic surface is terminated at infinity, it is usual to terminate integration at the level 0.001 a.u. Usually this is at infinity, it is usual to terminate integration at the level 0.001 a.u. Usually this is closely similar to the van der Waals volume, and it generally encloses more than closely similar to the van der Waals volume, and it generally encloses more than 99% of the electron population.99% of the electron population.

3. Atomic Laplacian 3. Atomic Laplacian LL(() - this property should vanish, and the actual magnitude is ) - this property should vanish, and the actual magnitude is used as a gauge for the accuracy of integration. used as a gauge for the accuracy of integration.

4. Atomic energy 4. Atomic energy EE(() - Baders analysis provides a unique method for obtaining ) - Baders analysis provides a unique method for obtaining (additive) atomic energies.(additive) atomic energies.

5. Other properties, the atomic dipolar 5. Other properties, the atomic dipolar (() or quadrupolar ) or quadrupolar QQ(() polarisations. ) polarisations. R. F. W. Bader & C. Matta (2004) J. Phys. Chem. A 108, 8385.

The molecular graphJyväskylä Summer School on Charge Density August 2007

Of especial importance, Of especial importance, in terms of chemistry, is in terms of chemistry, is the ability to the ability to unambiguously unambiguously define a define a chemical structure from chemical structure from a topological analysis.a topological analysis.

Shown here is the QTAIMShown here is the QTAIMdefinition of the definition of the structure of the [Bstructure of the [B66HH77]]-- monoanion.monoanion.The bcp’s are shown in The bcp’s are shown in red, the (3,+1) ring cp’s red, the (3,+1) ring cp’s in yellow and the (3,+3) in yellow and the (3,+3) cage cp in greencage cp in green

The bond paths link The bond paths link atoms via their atoms via their associated bcp’s, and associated bcp’s, and indicate their is indicate their is no direct no direct chemical bondingchemical bonding along along the B-B vectors which the B-B vectors which are triply bridged by the are triply bridged by the H atom.H atom.

DFT/B3LYP optimised structure of [B6H7]- , 6-311G** basis

The Poincaré-Hopf ruleJyväskylä Summer School on Charge Density August 2007

The mathematical The mathematical connections between the connections between the topological objects topological objects leads to a simple way of leads to a simple way of checking whether the checking whether the obtained topology is self-obtained topology is self-consistent. consistent.

This is the Poincaré-Hopf This is the Poincaré-Hopf rule for determining the rule for determining the completeness if the completeness if the topological spacetopological space

n – b + r – c = 1n – b + r – c = 1

n = number of (3,-3) n = number of (3,-3) cp’scp’sb = number of (3,-1) b = number of (3,-1) cp’scp’sr = number of (3,+1) r = number of (3,+1) cp’scp’sc = number of (3,+3) c = number of (3,+3) cp’scp’s

13 – 18 + 7 - 1 = 1 !!13 – 18 + 7 - 1 = 1 !!

DFT/B3LYP optimised structure of [B6H7]- , 6-311G** basis

The topology in a crystalJyväskylä Summer School on Charge Density August 2007

The Poincaré-Hopf rule only applies for The Poincaré-Hopf rule only applies for molecules in the gas phase. molecules in the gas phase.

For periodic crystals, the topology is For periodic crystals, the topology is governed by the Morse relationshipgoverned by the Morse relationship

n – b + r – c = 0n – b + r – c = 0n n 1, b 1, b 3, r 3, r 3, c 3, c 1 1

n = number of (3,-3) cp’sn = number of (3,-3) cp’sb = number of (3,-1) cp’s b = number of (3,-1) cp’s faces facesr = number of (3,+1) cp’s r = number of (3,+1) cp’s edges edgesc = number of (3,+3) cp’s c = number of (3,+3) cp’s vertices vertices

In crystals :In crystals :• the atomic basins are finitethe atomic basins are finite• all types of critical points are presentall types of critical points are present• crystallographic symmetry mandates crystallographic symmetry mandates cp’s cp’s

M. Pendás, A. Costales & V. Luaña (1997) Phys. Rev. B. 55, 4275

C. Gatti (2005) Z. Kristallogr. 220, 399.

Atomic polyhedra of Li+I- space group Fm-3m

The ellipticity at the bond critical pointJyväskylä Summer School on Charge Density August 2007

= = 11//22 – 1 – 1

C-C bondC-C bondRho(r)Rho(r) 2.084 2.084Lap(r)Lap(r) -20.791-20.791Eigenvalues Eigenvalues 11 -15.599 -15.599 22 -12.990 -12.990 33 7.797 7.797 0.201 (0.201 (-bond)-bond)

C-H bondC-H bondRho(r)Rho(r) 1.901 1.901Lap(r)Lap(r) -23.379-23.379Eigenvalues Eigenvalues 11 -18.033 -18.033 22 -17.728 -17.728 33 12.382 12.382 0.017 (cylindrical)0.017 (cylindrical)

Units in ÅUnits in Å

DFT/B3LYP optimised structure of benzene , 6-311G** basis

x

y

zR.F.W. Bader et al (1983) J. Am. Chem. Soc. 105, 5061

The Laplacian of the charge densityJyväskylä Summer School on Charge Density August 2007

The Laplacian of rho, The Laplacian of rho, 22((),),provides a measure of provides a measure of the the locallocal charge charge concentration or concentration or depletion.depletion.

+ve values mean local +ve values mean local charge depletion (charge depletion (blueblue))

-ve values mean local -ve values mean local charge concentration charge concentration ((redred))

Often maps of -Often maps of -22((), ), sometimes also called sometimes also called LL, , are drawn, where the are drawn, where the +ve contours imply +ve contours imply charge concentrations !charge concentrations !

Local charge Local charge concentration does not concentration does not imply a maximum.imply a maximum.

DFT/B3LYP optimised structure of benzene , 6-311G** basis


The Laplacian of rho, The Laplacian of rho, 22((),),provides a measure of provides a measure of the the locallocal charge charge concentration or concentration or depletion.depletion.

+ve values mean local +ve values mean local charge depletion (charge depletion (blueblue))

-ve values mean local -ve values mean local charge concentration charge concentration ((redred))

Often maps of -Often maps of -22((), ), sometimes also called sometimes also called LL, , are drawn, where the are drawn, where the +ve contours imply +ve contours imply charge concentrations !charge concentrations !

Local charge Local charge concentration does not concentration does not imply a maximum.imply a maximum.

Experimental Laplacian in plane of RExperimental Laplacian in plane of R22C=NR’ groupC=NR’ group


P. Macchi & A. Sironi (2003) Coord. Chem. Rev. 238-239, 383

The Laplacian The Laplacian 22(() recovers the shell ) recovers the shell structure of atoms - with the general structure of atoms - with the general exception of the transition metals, where exception of the transition metals, where the N shell effects are not observed (4the N shell effects are not observed (4ss density is too diffuse ?) .density is too diffuse ?) .

The Laplacian allows us to trace the effects The Laplacian allows us to trace the effects of chemical bonding in the total charge of chemical bonding in the total charge densitydensity


The topology of the Laplacian The topology of the Laplacian 22((), ), provides a rationalisation for provides a rationalisation for chemical reactivity and formation of chemical reactivity and formation of complexes.complexes.

In this example, the areas of charge In this example, the areas of charge depletion on the metal atom are depletion on the metal atom are matched by areas of charge matched by areas of charge concentrations on the carbonyl concentrations on the carbonyl ligands – the “lock and key” ligands – the “lock and key” mechanism. mechanism.

F. Cortes-Guzman & R.F.W. Bader (2005) Coord. Chem. Rev. 249, 633

Cr(CO)6

Atomic graph of Cr atom, i.e. the “net” of critical points in L(r) in the valence shell charge concentration - VSCC

Isosurface at L = 0 (encloses charge concentrations)

Topological characterisation of chemical bondsJyväskylä Summer School on Charge Density August 2007

R.F.W. Bader & H. Essén (1984) J. Phys. Chem. 80, 1943D. Cremer & E. Kraka (1984) Angew, Chemie 23, 67

Empirical bond order Empirical bond order nn = exp[ = exp[AA((bcpbcp) – ) – BB], where ], where A,BA,B are empirical constants are empirical constantsbond-length – e.d. relationships : see also A. Alkorta et al (1998) Struct. Chem. 9, 243 andTsirelson et al (2007) Acta Cryst B63, 142


E. Espinosa, I Alkorta, J. Elguerdo & E. Molins (2002) J. Chem.Phys 117, 5529

Study on X-H...F-Y interactions- uses the |Vbcp|/Gbcp indicator

Defines two boundaries I where |Vbcp|/Gbcp = 1 and II where 2() = 0


G. Gervasio, R. Bianchi & D. Marabello (2004) Chem. Phys. Lett. 387, 481

Characterisation of transition metal-metal bonds - 2() < 0, Hbcp ~ 0


R. F. W Bader “A Bond Path – A Universal Indicator of Bonded Interactions” (1998) J. Phys. Chem. A. 102, 7314-7323

It is important to emphasise that bond paths CANNOT be It is important to emphasise that bond paths CANNOT be simply equated with chemical bonds.simply equated with chemical bonds.9

““The adoption of the theory of atoms in molecules requires the The adoption of the theory of atoms in molecules requires the replacement of the model of structure that imparts an replacement of the model of structure that imparts an existence to a bond separate from the atoms it links - the ball existence to a bond separate from the atoms it links - the ball and stick model or its orbital equivalents of atomic and overlap and stick model or its orbital equivalents of atomic and overlap contributions - with the concept of contributions - with the concept of bonding bonding betweenbetween atoms; atoms; two atoms are bonded if they share an interatomic surface and two atoms are bonded if they share an interatomic surface and are consequently linked by a bond path.”are consequently linked by a bond path.”

Topological characterisation of cp’s in regions of low e.d. (e.g. Topological characterisation of cp’s in regions of low e.d. (e.g. in ionic crystals and for transition metal bonds) becomes in ionic crystals and for transition metal bonds) becomes problematical and even controversial.problematical and even controversial.


A. Haaland et al “Topological Analysis of Electron Densities: Is the Presence of an Atomic Interaction Line in an Equilibrium Geometry a Sufficient Condition for the Existence of a Chemical Bond?” (2004) Chem. Eur J. 10, 4416-4421

““Even though the He atom in the Even though the He atom in the inclusion complex inclusion complex He@He@adamantaneadamantane is connected to is connected to the four tertiary C atoms through the four tertiary C atoms through atomic interaction lines with (3,-atomic interaction lines with (3,-1) critical points, the He···1) critical points, the He···ttC C interactions are in fact strongly interactions are in fact strongly antibondingantibonding. This means that the . This means that the conjecture that an AIL between conjecture that an AIL between two atoms in an equilibrium two atoms in an equilibrium structure implies the presence of structure implies the presence of a chemical bond between them is a chemical bond between them is not validnot valid.”.”


R. F. W Bader & D. E. Fang “Properties of Atoms in Molecules: Caged Atoms and the Ehrenfest Force” (2005) J. Chem. Theory Comput 1, 403-414

““The presence of … The presence of … atoms linked by a bond atoms linked by a bond path implies not only the path implies not only the absence of repulsive absence of repulsive Feynman forces on the Feynman forces on the nuclei but also the nuclei but also the presence of attractive presence of attractive Ehrenfest forces acting Ehrenfest forces acting across the interatomic across the interatomic surface shared by the surface shared by the bonded atoms.”bonded atoms.”


C. Evans, L. J. Farrugia, M, Tegel (2006) J. Phys Chem A 110, 7952-7961

Fe(1)-C(1) = 1.9449(2) C(1)-C(2)=1.4294(3) Fe(1)-C(2) = 2.1237(2) C(1)-C(3)=1.4288(3) Fe(1)-C(3) = 2.1326(3) C(1)-C(4)=1.4304(3) Fe(1)-C(4) = 2.1343(3) av Fe-C(O) = 1.7983(3)

No bond path observed between Fe-CNo bond path observed between Fe-C

Independent of level of theory Independent of level of theory Independent of multipole modelIndependent of multipole model





A normal mode analysis of the vibrational spectrum A normal mode analysis of the vibrational spectrum indicates that the indicates that the (Fe-TMM) stretch has a large force (Fe-TMM) stretch has a large force constant of 3.7 mdyn/Å. Conclude that a model with a constant of 3.7 mdyn/Å. Conclude that a model with a single bond from Fe to Csingle bond from Fe to C is inadequate, and that is inadequate, and that

significant significant -interactions with the C-C bonds occurs.-interactions with the C-C bonds occurs.DH Finseth et al (1976) J Phys Chem 80, 1248-1261

The NMR barrier to rotation of the TMM ligand is The NMR barrier to rotation of the TMM ligand is generally found to be quite high, again indicative of generally found to be quite high, again indicative of significant Fe significant Fe -interactions-interactionsMD Jones & RDW Kemmitt (1987) Adv Organomet Chem 27, 279-309

An MO analysis of the DFT wavefunction agrees with An MO analysis of the DFT wavefunction agrees with the qualitative Hoffmann EHMO scheme. The frontier the qualitative Hoffmann EHMO scheme. The frontier orbitals primarily responsible for the Fe-TMM bonding orbitals primarily responsible for the Fe-TMM bonding (15(15ee and 16 and 16aa11) imply significant Fe-C) imply significant Fe-C interactions. interactions.R. Hoffmann (1977) J. Am. Chem. Soc. 99, 7546-7557.


R.F.W. Bader & M. E. Stephens (1975) J. Am. Chem Soc. 97, 7391-7399X. Fradera et al (1999) J. Phys. Chem. A 103, 304-314

Delocalisation indices from DFT wavefunctionDelocalisation indices from DFT wavefunctionDelocalisation indices provide a measure of the total Fermi Delocalisation indices provide a measure of the total Fermi correlation shared between atoms, correlation shared between atoms, i.e.i.e. number of shared pairs. number of shared pairs.

(Fe-C) = 2.082 (Fe-C) = 2.082 ~ 2 pairs of e ~ 2 pairs of e-- shared between ring and Fe shared between ring and Fe

Integrated charge over the common interatomic surfaces of A and B


(r) eÅ(r) eÅ-3-3

bcp 0.662bcp 0.662rcp 0.655rcp 0.655

The average MDA of CThe average MDA of C in the direction of Fe is 0.12 Å, from thermal parameters in the direction of Fe is 0.12 Å, from thermal parameters

at 100Kat 100K

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Topological Analysis of Electron Density Louis J Farrugia Jyväskylä Summer School on Charge Density August 2007