New tools for chemical bonding analysis Eduard Matito Donostia - - PowerPoint PPT Presentation

New tools for chemical bonding analysis

Eduard Matito Donostia International Physics Center February 2020. Girona

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Notation

The coordinates of an electron: r ≡ r and the spin (σ=α or β).

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Notation

The coordinates of an electron: r ≡ r and the spin (σ=α or β). x1 ≡ x1 = ( r, σ1).

Eduard Matito MACMoM: New Tools 3 / 87

Notation

The coordinates of an electron: r ≡ r and the spin (σ=α or β). x1 ≡ x1 = ( r, σ1). We assume: 1 ≡ ( r1, σ1) and d1 ≡ d r1dσ1 for the derivates.

Eduard Matito MACMoM: New Tools 3 / 87

Notation

The coordinates of an electron: r ≡ r and the spin (σ=α or β). x1 ≡ x1 = ( r, σ1). We assume: 1 ≡ ( r1, σ1) and d1 ≡ d r1dσ1 for the derivates. N denotes the number of electrons in the system.

Eduard Matito MACMoM: New Tools 3 / 87

Notation

The coordinates of an electron: r ≡ r and the spin (σ=α or β). x1 ≡ x1 = ( r, σ1). We assume: 1 ≡ ( r1, σ1) and d1 ≡ d r1dσ1 for the derivates. N denotes the number of electrons in the system. Wavefunction: Ψ(1, 2, . . . , N)

Eduard Matito MACMoM: New Tools 3 / 87

Notation

The coordinates of an electron: r ≡ r and the spin (σ=α or β). x1 ≡ x1 = ( r, σ1). We assume: 1 ≡ ( r1, σ1) and d1 ≡ d r1dσ1 for the derivates. N denotes the number of electrons in the system. Wavefunction: Ψ(1, 2, . . . , N) Atomic orbitals: φµ(1)

Eduard Matito MACMoM: New Tools 3 / 87

Notation

The coordinates of an electron: r ≡ r and the spin (σ=α or β). x1 ≡ x1 = ( r, σ1). We assume: 1 ≡ ( r1, σ1) and d1 ≡ d r1dσ1 for the derivates. N denotes the number of electrons in the system. Wavefunction: Ψ(1, 2, . . . , N) Atomic orbitals: φµ(1) Molecular orbitals: φi(1)

Eduard Matito MACMoM: New Tools 3 / 87

Notation

The coordinates of an electron: r ≡ r and the spin (σ=α or β). x1 ≡ x1 = ( r, σ1). We assume: 1 ≡ ( r1, σ1) and d1 ≡ d r1dσ1 for the derivates. N denotes the number of electrons in the system. Wavefunction: Ψ(1, 2, . . . , N) Atomic orbitals: φµ(1) Molecular orbitals: φi(1) Vectors are indicated in bold or using an superscripted arrow, e.g.,, n = n = (nx, ny, nz). Exception: ∇ =

• ∂

∂x , ∂ ∂y , ∂ ∂z

• Eduard Matito

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How do we study chemistry?

”The theory behind chemistry, which atoms combine with which at which rate and so forth, is in principle theoretical chemistry deeply, is physics.” [Audience laughs] ”It is not a joke, it is a direct chemists would admit. That’s exactly their point of view, that atoms in deepest level is physics, except that the atoms have some many particles that is very hard to calculate what is going to happen so they have to use a lot of empirical rules to help them... Richard Feynmann Lecture on Quantum ElectroDynamics Auckland (1979)

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The Wavefunction

The simplest (antisymmetric) wavefunction is the Slater determinant: ψK (1, 2, . . . , n) = 1 √ N

• φk1(1)

φk1(2) . . . φk1(n) φk2(1) φk2(2) . . . φk2(n) . . . . . . ... . . . φkN(1) φkN(2) . . . φkN(n)

• Eduard Matito

MACMoM: New Tools 5 / 87

The Wavefunction

The simplest (antisymmetric) wavefunction is the Slater determinant: ψK (1, 2, . . . , n) = 1 √ N

• φk1(1)

φk1(2) . . . φk1(n) φk2(1) φk2(2) . . . φk2(n) . . . . . . ... . . . φkN(1) φkN(2) . . . φkN(n)

• The HF and KS-DFT wavefunction are single-determinant wavefunctions.

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The Wavefunction

The simplest (antisymmetric) wavefunction is the Slater determinant: ψK (1, 2, . . . , n) = 1 √ N

• φk1(1)

φk1(2) . . . φk1(n) φk2(1) φk2(2) . . . φk2(n) . . . . . . ... . . . φkN(1) φkN(2) . . . φkN(n)

• The HF and KS-DFT wavefunction are single-determinant wavefunctions.

The exact wavefunction can be obtained from a linear combination of Slater determinants, Ψ(1, 2, . . . , n) =

• K=1

cK ψK (1, 2, . . . , n) with

• K=1

|cK |2 = 1

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The Born Interpretation

Born: the probability of finding one electron at d1 (dr1 with σ1) is P(1)d1 =

• d2
• d3 . . .
• dn |Ψ(1, 2, . . . , N)|2 d1

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The Born Interpretation

Born: the probability of finding one electron at d1 (dr1 with σ1) is P(1)d1 =

• d2
• d3 . . .
• dn |Ψ(1, 2, . . . , N)|2 d1

regardless the position of the other N-1 electrons.

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The Born Interpretation

Born: the probability of finding one electron at d1 (dr1 with σ1) is P(1)d1 =

• d2
• d3 . . .
• dn |Ψ(1, 2, . . . , N)|2 d1

regardless the position of the other N-1 electrons. From the latter we define the electron density, ρ(1) = N · P(1)

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The Born Interpretation

Born: the probability of finding one electron at d1 (dr1 with σ1) is P(1)d1 =

• d2
• d3 . . .
• dn |Ψ(1, 2, . . . , N)|2 d1

regardless the position of the other N-1 electrons. From the latter we define the electron density, ρ(1) = N · P(1) that integrated over Ω it gives the average number of electrons in Ω. NΩ ≡ Ψ| ˆ N |ΨΩ = Ψ|

• i

a†

i ai |ΨΩ = N · P(Ω) =

• Ω

ρ(1)d1

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The Born Interpretation

Born: the probability of finding one electron at d1 (dr1 with σ1) is P(1)d1 =

• d2
• d3 . . .
• dn |Ψ(1, 2, . . . , N)|2 d1

regardless the position of the other N-1 electrons. From the latter we define the electron density, ρ(1) = N · P(1) that integrated over Ω it gives the average number of electrons in Ω. NΩ ≡ Ψ| ˆ N |ΨΩ = Ψ|

• i

a†

i ai |ΨΩ = N · P(Ω) =

• Ω

ρ(1)d1 This is the basis of electron population analysis.

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The Electron Density

The electron density is the central quantity in DFT and QTAIM.

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The Electron Density

The electron density is the central quantity in DFT and QTAIM. It is an observable obtained from X-ray spectroscopy analysis.

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The Electron Density

The electron density is the central quantity in DFT and QTAIM. It is an observable obtained from X-ray spectroscopy analysis. It depends on three coordinates (vs. wfn that depends on 3N).

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The Electron Density

The electron density is the central quantity in DFT and QTAIM. It is an observable obtained from X-ray spectroscopy analysis. It depends on three coordinates (vs. wfn that depends on 3N). It can be constructed from the molecular orbitals: ρ(1) =

N

• i

φ∗

i (1)φi(1) = N

• i

|φi(1)|2

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The 1-RDM

The first-order reduced density matrix (1-RDM) reads: ρ1(1; 1′) = N

• d2d3 . . .
• dnΨ∗(1, 2, . . . , N)Ψ(1′, 2, . . . , N)

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The 1-RDM

The first-order reduced density matrix (1-RDM) reads: ρ1(1; 1′) = N

• d2d3 . . .
• dnΨ∗(1, 2, . . . , N)Ψ(1′, 2, . . . , N)

The density is actually the diagonal part of the 1-RDM, ρ(1) = ρ1(1; 1)

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The 1-RDM

The first-order reduced density matrix (1-RDM) reads: ρ1(1; 1′) = N

• d2d3 . . .
• dnΨ∗(1, 2, . . . , N)Ψ(1′, 2, . . . , N)

The density is actually the diagonal part of the 1-RDM, ρ(1) = ρ1(1; 1) It does not have any probabilistic interpretation but it can be also written in terms of molecular orbitals: ρ1(1; 1′) =

N

• i

φ∗

i (1)φi(1′)

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The Pair Density

Born’s interpretation can be further extended to include electron pairs, P(1, 2)d1d2 =

• d3 . . .
• dn |Ψ(1, 2, . . . , N)|2 d1d2

P(1, 2) is the probability of finding two electrons, one at 1 and the other at 2, regardless of the position of the other N − 2 electrons.

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The Pair Density

Born’s interpretation can be further extended to include electron pairs, P(1, 2)d1d2 =

• d3 . . .
• dn |Ψ(1, 2, . . . , N)|2 d1d2

P(1, 2) is the probability of finding two electrons, one at 1 and the other at 2, regardless of the position of the other N − 2 electrons. The pair density is obtained from it, ρ2(1, 2) = N(N − 1)P(1, 2)

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The Pair Density

Born’s interpretation can be further extended to include electron pairs, P(1, 2)d1d2 =

• d3 . . .
• dn |Ψ(1, 2, . . . , N)|2 d1d2

P(1, 2) is the probability of finding two electrons, one at 1 and the other at 2, regardless of the position of the other N − 2 electrons. The pair density is obtained from it, ρ2(1, 2) = N(N − 1)P(1, 2) By an analogous procedure we can obtain n-densities (n > 2).

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The Pair Density for Single Determinants

The pair density can be easily written in terms of the 1-RDM for single-determinant methods ρ2(1, 2) =

• ρ(1)

ρ1(1; 2) ρ1(2; 1) ρ(2)

• which can be expanded in terms of molecular orbitals

ρ2(1, 2) =

• i
• j

φ∗

i (1)φ∗ j (2) [φi(1)φj(2) − φj(1)φi(2)]

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The Number of Pairs

The expected number of electron pairs in the re- gion A is obtained from the pair density: NAA =

• A
• A

ρ2(1, 2)d1d2 + NA

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The Number of Pairs

The expected number of electron pairs in the re- gion A is obtained from the pair density: NAA =

• A
• A

ρ2(1, 2)d1d2 + NA the last term accounts for the self-pairing that is not included in antisymmetric wavefunctions (Pauli principle: Ψ(1, 1, . . .) = 0).

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The Number of Pairs

The expected number of electron pairs in the re- gion A is obtained from the pair density: NAA =

• A
• A

ρ2(1, 2)d1d2 + NA the last term accounts for the self-pairing that is not included in antisymmetric wavefunctions (Pauli principle: Ψ(1, 1, . . .) = 0). The expected number of electron pairs with one electron at A and another at B reads: NAB =

• A
• B

ρ2(1, 2)d1d2 + NA∩B

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The variance of NA

σ2 [NA] = NAA − NA · NA =

• A
• A

ρ2(1, 2)d1d2 − NA(NA − 1)

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The variance of NA

σ2 [NA] = NAA − NA · NA =

• A
• A

ρ2(1, 2)d1d2 − NA(NA − 1) The uncertainty is minimal when the electron pairs in A are maximal, i.e., when the electrons are independent.

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The covariance

The covariance of electrons populations gives the measure of how much the number of electrons in A and B change together. cov (NA, NB) = NAB − NA · NB σ2 [NA] = cov (NA, NA) rNANB = cov (NA, NA) σ[NA]σ[NB]

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The Covariance Bounds

Non-overlapping regions

− min (NA, NB) ≤ cov (NA, NB) ≤ 0

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The Covariance Bounds

Non-overlapping regions

− min (NA, NB) ≤ cov (NA, NB) ≤ 0

Overlapping regions

− min ` NA\C, NB\C ´ ≤ cov (NA, NB) ≤ NC

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The Covariance Bounds

Non-overlapping regions

− min (NA, NB) ≤ cov (NA, NB) ≤ 0

Overlapping regions

− min ` NA\C, NB\C ´ ≤ cov (NA, NB) ≤ NC

The cov (NA, NB) gives a measure of the number of electron pairs shared between A and B.

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The Electron Sharing Indices

All the electrons in a molecule are either localized in an atom or delocalized with other atoms. The delocalization index (DI): δAB = −cov (NA, NB) The localization index (LI): λA = NA − σ2[NA]

Bader, Stephens JACS 94, 7391 (1975), Bader, Stephens CPL 26, 445 (1974) Fulton JPC 97, 7516 (1993); ´ Angy´ an, Mayer, Loos, JPC 98, 5244 (1994) Fradera, Austen, Bader JPCA 103, 304 (1999) Matito, Sol` a, Salvador, Duran Faraday Discuss. 135, 325 (2007)

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The Electron Sharing Indices

We can decompose all the electrons in a molecule.

N =

• A

NA NA = λA + δA δA = 1 2

• X=A

δAX

Fradera, Austen, Bader JPCA 103, 304 (1999)

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Born interpretation = chemical insight

From Born’s interpretation we obtain average values that can be easily used to calculate other statistics such the covariance and higher-order moments (cumulants). Chemical Insight The key to obtain chemical insight is the Born interpretation of the n-densities, which provides a framework for counting electrons, electrons pairs, etc. Population analysis, delocalization indices, electron localization function and indicator (ELF and ELI), etc. are based on probabilities, expected values and other statistics derived from the Born’s interpretation of the wavefunction.

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The atom: the origin

Two opposite views of the world: Greeks vs. Hebrews. The least expected city: Abdera (ca. 460 BC). The atomists: philosophers that study the existence, looking for the primordial substance. Three people: Leucippus, Democritus and Epicurus. Two principles: the void and the full. Atomic pasive corpuscles: incompressible, compact, hard, indestructible, full and homogeoneous. Atoms are animated by an ethernal constant motion. Atoms have shape, order and position. The soul was made of atoms.

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The atom: the origin

Epicurus adds the weight as a property of atoms. Two giants opposed the atomistic view: Aristotle and Plato (geometric shapes). Clinamen introduced by Lucretius to defend Epicurus’ atom.

When atoms move straight down through the void by their own weight, they deflect a bit in space at a quite uncertain time and in uncertain places, just enough that you could say that their motion has changed. But if they were not in the habit of swerving, they would all fall straight down through the depths of the void, like drops of rain, and no collision would occur, nor would any blow be produced among the atoms. In that case, nature would never have produced anything

Clinamen adds free will (”the atom’s soul” —Ausgustine of Hippo)

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Brownian Motion

Observe what happens when sunbeams are admitted into a building and shed light on its shadowy places. You will see a multitude of tiny particles mingling in a multitude of ways... their dancing is an actual indication of underlying movements of matter that are hidden from our sight... It originates with the atoms which move of themselves (i.e., spontaneously). Then those small compound bodies that are least removed from the impetus of the atoms are set in motion by the impact of their invisible blows and in turn cannon against slightly larger bodies. So the movement mounts up from the atoms and gradually emerges to the level of our senses, so that those bodies are in motion that we see in sunbeams, moved by blows that remain invisible.

On the Nature of Things (Lucretius, 60 BC).

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The atom on the last 200 years

In 1807 Dalton provided the modern atomic theory (father of chemistry). In 1891 Stoney coined the term electron. In 1913 Bohr formulated his atomic model including orbits and quanta. In 1924 Pauli provided the forth quantum number: the spin. The advent of quantum mechanics brought the concept of wavefunction.

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The Lewis Model

The chemical bond is one of the most fundamental concepts in chemistry. The Lewis model describes the electronic structure of a molecule in terms of electron pairs (2c-2e bonds). These electron pairs can be classified as lone pairs (electrons which are localized in one atom), bonding pairs (electrons that are shared between two atoms), core electrons. However, this intuitive picture of electron distribution does not take into account the quantum nature of electrons.

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Non-standard Chemical Bonds

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An atom in a molecule

Need to characterize an atom inside a molecule → atomic partition. From an atomic partition we can define atomic properties. There is not a unique atomic partition, there have been many suggestions. Two main groups: (i) Hilbert-space based. (ii) Real space partitioning.

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An atom in a molecule: Hilbert-space partition

Schr¨

• dinger Eq. solved by a finite basis set → atomic orbitals.

This partition uses the assignment of orbitals to atomic centers. Mulliken was the first to use it to compute populations. We use the definition of MOs in terms of AOs: φA

i (1) = m

• µ∈A

cµiφµ(1) This assignment provides a partition of the MO: φMO

i

(1) =

• A

φA

i (1) =

• A

m

• µ∈A

cµiφµ(1)

Mulliken, JCP 23, 1833 (1955)

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The Hilbert-space partition

Advantatge: low cost (it is analytical!) Drawback: basis set dependence, not clear assignment (diffuse, polarization functions). Atom Basis set Mulliken QTAIM C DZ

• 0.982
• 0.329

DZP 0.047 0.038 TZ2P+ 0.607 0.047 H DZ 0.246 0.082 DZP

• 0.012
• 0.009

TZ2P+

• 0.152
• 0.011

Table: Bader and Mulliken atomic charges for methane calculated using three different basis sets (DZ: double zeta, DZP: double zeta with polarization and TZ2P+: triple zeta with double polarization and diffuse functions).

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Quantum Theory of Atoms-in-Molecules

Due to Richard F.W. Bader (1931-2012), McMaster University (Ontario). It consists on the topological analysis of the electron density. The density defines the distribution of electrons, and electrons determine the chemistry. Behind the electron density hides the concepts of atoms, bonds, chemical structure and structural stability.

Bader RFW, ACR 18, 9-15 (1985)

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QTAIM: Premises

The atom that has the electron distribution makes the same contribution to the total energy of the system. Quantum subsystems are open systems defined in real space, their boundaries being determined by a particular property of the electronic charge density. QTAIM defines the atom through a partitioning of the real space as determined by the topological analysis of a molecular charge distribution.

Bader RFW, Beddall PM, JCP 56, 3320-3329 (1972)

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Topological analysis of the density

The density is a continuous function. Defined at every point in the space. The critical points fulfill ∇ρ(rc) = 0. The three curvatures (x,y,z) define the nature of the CP. H[ρ(r)] =    

∂2ρ(r) ∂x2 ∂2ρ(r) ∂x∂y ∂2ρ(r) ∂x∂z ∂2ρ(r) ∂y∂x ∂2ρ(r) ∂y2 ∂2ρ(r) ∂y∂z ∂2ρ(r) ∂z∂x ∂2ρ(r) ∂y∂z ∂2ρ(r) ∂z2

    →     

∂2ρ(r) ∂x2

1

∂2ρ(r) ∂y2

1

∂2ρ(r) ∂z2

1

    

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Critical points of a 3D function

There are four types of critical points (CP): H[ρ(r)] →     

∂2ρ(r) ∂x2

1

∂2ρ(r) ∂y2

1

∂2ρ(r) ∂z2

1

     =   λ1 λ2 λ3   (3,-3): Attractors (ACP) all the curvatures at rc are negative. (3,-1): Bond CP (BCP) 2 curvatures are negative and 1 positive. (3,+1): Ring CP (RCP) 1 curvatures are negative and 2 positive. (3,+3): Cage CP (CCP) 3 curvatures are positive. Poncair´ e-Hopf relationship: nACP − nBCP + nRCP − nCCP = 1

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The Atom in the Molecule

An atom: a density maximum surrounded by a zero-flux gradient surface. ∇ρ(r) · n(r) = 0 ∀r ∈ S(r) (1) Gradient lines (red) perpendicular density isocontour lines (black). Blue line is the bond path: connects to nuclei and passes through the BCP.

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The Bond Path: the Controversy

A Bond Path: A Universal Indicator of Bonded Interactions The presence of a bond path [...] provide[s] a universal indicator of bonding between the atoms so linked.

Bader RFW, JPCA 102, 7314-7223 (1998)

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Topological analysis of water

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Topological analysis of water

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Topological analysis of water

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Topological analysis of water

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Topological analysis of water

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Topological analysis of water

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• AIM2000. Molecular graphs and isocontour plots

Molecular representation Isocountor plot with gradient lines

Biegler-K¨

• nig and Sch¨
• nbohm, JCC 23 1489 (2002)

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Input example to obtain wfn file with Gaussian

#HF/6-31G* out=wfn comment 0 1 H 0.0 0.0 0.0 H 0.0 0.0 1.0 H2.wfn

H2.wfn contains the molecular orbitals to construct the wavefunction of H2 at the HF/6-31G(d) level.

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The Non-nuclear Attractors (NNA)

Maxima other than the nuclei (NNA) are not frequent and often due to wavefunction artifacts. For instance, acetylene at the HF/6-31G*. There is, however, some exceptions: Li2 and electrides.

Martin-Pendas PRL 83, 1930-1933 (1999)

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The Laplacian of the electron density

The Laplacian of the density accounts for electron localization. ∇2ρ(r) = ∂2ρ(r) ∂x2 + ∂2ρ(r) ∂y 2 + ∂2ρ(r) ∂z2 ∇2ρ(r) < 0 electron accumulation → localization. ∇2ρ(r) > 0 electron depletion → delocalization.

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Real-Space partitions: Voronoi Cells

Atoms defined as Voronoi polyhedra. Polyhedra constructed by assigning the point to the nearest atom. Provides non-overlapping partition of the real space. Only uses the molecular geometry: atom types are not explicitly considered.

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Real-Space partitions: Hirshfeld Partition

Atoms defined from weight functions. wA(r) = ρ0

A(r)

• B ρ0

B(r)

• A

wA(r) = 1 ∀r ρ0

A is the density of the isolated atom (promolecular density)

The promolecule formed by superposing the densities of the isolated atoms.

B ρ0 B(r).

Overlapping atomic partition. Drawback: The electronic state of the isolated atom can change the result.

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Real-Space partitions: Becke Partition

Uses Becke’s 1988 multicenter integration technique. Adapted by Mayer and Salvador in 2004: fuzzy atom. wA(r) = fA(r)

• B fB(r)
• A

wA(r) = 1 ∀r fA(r) are obtained from empirical atomic radii. Overlapping atomic partition. Becke-rho: uses the BCP as dynamic atomic radii.

Mayer, Salvador, CPL 383 638 (2004) Matito, Sol` a, Salvador, Duran Faraday Discuss. 135, 325 (2007) Salvador, Ramos, JCP 139, 071103 (2013)

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Summary: Atoms in a molecule (AIM)

1.- Mulliken: Hilbert space partitioning.a * AIM → set of orbitals. Overlapping Atoms (OA) * Cheap, exact (analytic). Issue: Basis set dependence (BSD). 2.- QTAIM: 3D-space, based on the density.b * AIM → an attractor surrounded by zero flux surface or by infinity. * Non-OA. No BSD. Issue: Expensive (Not w AIMall) 3.- Fuzzy: 3D-space, based on Becke Multicenter scheme.c * AIM → a sum weights through the space. * Uses weight functions. No BSD. Cheap. Issue: Bond-orders. 4.- Becke-rho: 3D-space, mixed Bader-Becke scheme.d * AIM → boundaries determined by BCPs. * Gives QTAIM-like results at fuzzy expense.

aMulliken, JCP 23, 1833 (1955); bBader, ACR 18, 9 (1985) cMayer, Salvador, CPL 383 638 (2004) dMatito, Sol`

a, Salvador, Duran Faraday Discuss. 135, 325 (2007)

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Population analysis

Assuming M number of basis functions an MO can be expanded: φi(1) ≡ φMO

i

(1) =

M

• µ

cµiφAO

µ (1)

|φi(1)|2 =

M

• µν

cµicνiφ∗

µ(1)φν(1)

and N =

• ρ(1)d1 =

N

• i

ni

• |φi(1)|2 d1 =

N

• i

M

• µν

cµicνini

• φ∗

µ(1)φν(1)d1

=

M

• µν

N

• i

cµicνini φ∗

µ(1)φν(1)d1 = M

• µν

PµνSµν = Tr (P · S) where P and S are the density matrix (in AO) and the overlap matrix, respectively.

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Population analysis: Mulliken

By taking the functions of each atom separately N =

• ρ(1)d1 =
• A
• µ∈A

M

• ν

PµνSµν = Tr (P · S) we can define the Mulliken electron population and gross charge of A: NA =

• µ∈A

M

• ν

PµνSµν QA = ZA − NA where ZA is atomic number of A. Gaussian keyword for electron population analysis: pop=full. Gaussian keyword for L¨

• win pop. and Mayer BO: iop(6/80)=1.

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Population analysis: real space

In the real space we need to perform a numerical integration over the atomic domain of A: NA =

• A

ρ(1)d1 =

• i

ni

• A

|φi(1)|2 d1 QA = ZA − NA where we need the diagonal part of the atomic overlap matrix (AOM): Sij(A) =

• A

φ∗

i (1)φj(1)d1

These analysis are done with the appropriate software One should check the accuracy of the integration performed. The computational cost can be beyond our possibilities.

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The Exchange Correlation Function

A popular function in DFT is the exchange-correlation density (XCD), ρxc(1, 2) = ρ(1)ρ(2) − ρ2(1, 2) ρxc(1, 2)d1d2 = N For single-determinant wavefunctions, ρxc(1, 2) = ρ(1)ρ(2) − ρ2(1, 2) = ρ1(1; 2)ρ1(2; 1) = |ρ1(1; 2)|2 ≥ 0 We can easily prove that NΩ1Ω2 ≤ NΩ1NΩ2 + NΩ1∩Ω2 so that the maximum number of pairs between two regions is obtained by direct multiplication of its electron averages (plus the population of the intersection of the two regions). The maximum value is only achieved when the electrons are independent.

Bader, Stephens JACS 94, 7391 (1975), Bader, Stephens CPL 26, 445 (1974)

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The Electron Sharing Indices

The XCD gives rise to the Electron Sharing Indices (ESI). The delocalization index (DI): δ(A, B) =

• A
• B

d1d2ρxc(1, 2) The localization index (LI): λ(A) =

• A
• A

d1d2ρxc(1, 2)

Bader, Stephens JACS 94, 7391 (1975), Bader, Stephens CPL 26, 445 (1974) Fulton JPC 97, 7516 (1993); ´ Angy´ an, Mayer, Loos, JPC 98, 5244 (1994) Fradera, Austen, Bader JPCA 103, 304 (1999) Matito, Sol` a, Salvador, Duran Faraday Discuss. 135, 325 (2007)

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Multicenter Bonding

Accounts for electron-sharing between n centers (atoms). It is related to the n-order central moment of the electron population. D(A1, · · · , An) ∼

• ˆ

N − N n

A1,··· ,An Giambiagi, de Giambiagi, Mundim, Struc. Chem. 1, 423 (1990) Ponec, Mayer, JPCA 101, 1738 (1997) Bochicchio, Ponec, Torre, Lain, TCA 105, 292 (2001)

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nc-ESI: What is it good for?

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Agostic Bonds

Feixas, Matito, Maseras, Poater, Sol` a, in preparation

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Conjugation and Hyperconjugation effects

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Aromaticity

Let A={A1,A2,...,An} represent a ring. Iring(A) =

• i1,i2,...,in

Si1i2(A1)Si2i3(A2)...Sini1(An) MCI(A) =

• P(A)

Iring(A) The nth root of these quantities correlates with the TREPE.

Giambiagi, et al., PCCP 2, 3381 (2000); Bultinck, et al. JPOC 18, 706 (2005)

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Multicenter Indices: Formulae

δ(A, B) = −2

• A
• B

d1d2 γ(1, 2) = Cov (N(A), N(B)) δ(A, B, C) = 2

• A
• B
• C

d1d2d3 γ(1, 2, 3) δ(A1, A2, . . . , An) = (−2)n−1 (n − 1)!

• A1
• A2

· · ·

• An

d1d2 . . . dn γ(1, 2, . . . , n) Functions that upon integration give the above ESIs: γ(1, 2) = ρxc(1, 2) = ρ(1, 2) − ρ(1)ρ(2) γ(1, 2, 3) = ρ(1, 2, 3) − ρ(1)ρ(2)ρ(3) − ˆ P1,2,3 (ρxc(1, 2)ρ(3)) γ(1, 2, . . . , n) = (ˆ ρ1 − ¯ ρ1)(ˆ ρ2 − ¯ ρ2) · · · (ˆ ρn − ¯ ρn) they depend on the n-order density (n-density).

Mart´ ın Pend´ as, Francisco, Blanco JCP 127 144103 (2007)

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Computational Details

ESI-3D: http://ematito.webs.com

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OXIDATION STATE

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Oxidation state (oxidation number)

DEFINITIONS: IUPAC (2018, formal): OS of an atom is the charge of this atom after ionic approximation of its heteronuclear bonds IUPAC (old, formal): Charge of TM after removing the L and the electrons sharing with it. Physical or Spectroscopic: Charge of TM that comes from dn and can be measured spectroscopically (e.g. M¨

• ssbauer).

J¨

• rgensen, In Oxidation Numbers and Oxidation States; Springer; Heildeberg, 1969

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Oxidation state

IUPAC (formal): Charge of TM after removing the L and the electrons sharing with it. Physical or Spectroscopic: Charge of TM that comes from dn and can be measured spectroscopically (e.g. M¨

• ssbauer).

Useful references: http://www.edu.upmc.fr/chimie/mc741/PDFs/bond.pdf http://www.chem.umn.edu/groups/harned/classes/8322/lectures/5ElectronCounting.pdf

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Oxidation state: A computational approach

There are many methods to calculate the OS: Bond Valence Sum (BVS) method. Empirical method based on M-L distances (Ri) and reference values (R0, b=0.37˚ A). V =

• i

Ri − R0 b

• From atomic population analysis.

Spin densities → atomic configuration → OS Localization methods (e.g. LOBA or EOS).

http://www.ccp14.ac.uk/solution/bond valence/index.html Thom, Sundstrom, Head-Gordon Phys. Chem. Chem. Phys. 11, 11297 (2009) Ramos-Cordoba, Postils, Salvador J. Comput. Theor. Chem. 11, 1501 (2015)

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Ligand Field Theory. Oh complexes

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A test set of TM complexes

To evaluate the performance of OS methods the following test set of Oh complexes is used: VII [VCl6]4− [V (H2O)6]2+ [V (CN)6]4− [V (CO)6]2+ MnII [MnCl6]4− [Mn(H2O)6]2+ [Mn(CN)6]4− [Mn(CO)6]2+ MnIII [MnCl6]3− [Mn(H2O)6]3+ [Mn(CN)6]3− [Mn(CO)6]3+ FeII [FeCl6]4− [Fe(H2O)6]2+ [Fe(CN)6]4− [Fe(CO)6]2+ FeIII [FeCl6]3− [Fe(H2O)6]3+ [Fe(CN)6]3− [Fe(CO)6]3+ NiII [NiCl6]4− [Ni(H2O)6]2+ [Ni(CN)6]4− [Ni(CO)6]2+ ZnII [ZnCl6]4− [Zn(H2O)6]2+ [Zn(CN)6]4− [Zn(CO)6]2+

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OS from atomic populations

Mulliken charges TFVC charges

Cl− H2OHS H2OLS CN− CO Cl− H2OHS H2OLS CN− CO VII 0.98 1.12 0.05 0.64 1.60 1.76

• 1.60 1.64

MnII 1.10 1.24 1.18 0.10 0.64 1.36 1.64 1.70 1.53 1.53 MnIII 0.93 1.58 1.52 0.35 0.80 1.46 2.05 2.09 1.58 1.67 FeII 0.86 1.22 1.15 0.01 0.51 1.27 1.63 1.77 1.46 1.44 FeIII 0.99 1.64 1.48 0.24 0.66 1.44 2.06 1.98 1.49 1.58 NiII 0.99 1.08

• 0.19 0.31 1.27
• 1.59

1.24 1.30 ZnII 1.02 1.06

• 0.03 0.52 1.25
• 1.45

1.15 1.19

Mulliken population fails in all cases. TFVC gets more than 50% wrong.

Ramos-Cordoba, Postils, Salvador J. Comput. Theor. Chem. 11, 1501 (2015)

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OS from spin densities

Mulliken populations TFVC populations

Cl− H2OHS H2OLS CN− CO Cl− H2OHS H2OLS CN− CO VII 3.06 2.99 2.84 2.79 2.90 2.71

• 2.52 2.44

MnII 4.96 4.88 1.01 1.09 1.06 4.86 4.66 0.95 1.01 0.98 MnIII 4.25 3.85 1.97 2.11 2.16 4.07 3.65 1.84 1.96 2.00 FeII 3.86 3.86 0.00 0.00 0.00 3.72 3.71 0.00 0.00 0.00 FeIII 4.27 4.43 0.89 1.08 1.11 4.17 4.28 0.86 1.01 1.03 NiII 1.87 1.84 1.69 1.66 1.84

• 1.77

1.66 1.63 ZnII 0.00 0.00 0.00 0.00 0.00

• 0.00

0.00 0.00

The spin population suggests the atomic configuration of the TM. From it we can deduce the OS. It only makes sense for open-shell calculations.

Ramos-Cordoba, Postils, Salvador J. Comput. Theor. Chem. 11, 1501 (2015)

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Effective Oxidation State

φA

i (1)

= wA(1)φMO

i

(1) QA

ij

=

• φA

i (1)φA j (1)d1

TrQA = NA QALA = LAΛA EOSA = IA − ZA

wA (atomic weights) define the atom, L contains the effective AOs (EF- FAOs) and Λ the occupancies. NA is the net atomic population. One or zero electrons are assigned to each orbital according to its occupa-

• tion. As a result, an integer number of electrons (IA) is assigned to each

atom. EOS reproduce the correct OS for all the complexes in the test set.

Ramos-Cordoba, Postils, Salvador J. Comput. Theor. Chem. 11, 1501 (2015)

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OS: Calculations

Charges and spin population can be obtained from Gaussian (Mulliken) or real-space partitioning program (QTAIM, Becke-rho, etc.) EOS can be obtained from APOST-3D program. We need to repeat the steps in the slide Computational Details: Fuzzy and Becke-rho. Only one difference: edit yourfile.inp and add the keyword UEFFAO. Recommendations: Define a fragment for each ligand. Add they keyword DOFRAG and define the fragments under the label ## FRAGMENTS #### using the same method than in ESI-3D manual. The best working DFT functionals: UB3LYP and M062X.

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AROMATICITY

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Aromaticity Timeline

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Aromaticity Timeline

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Aromaticity Timeline

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What is Aromaticity?

Unhopefully, aromaticity remains an ill − defined concept. Unlike other similar quantities like bond ionicity or bond order aromaticity refers to not one, but several properties not necessarily mutually related. However, in practice, organic chemists still use this concept to elucidate phenomena such as chemical stability/reactivity, bond length equalization/alternation, among others. Saying that aromaticity is a multidimensional phenomena (an accepted fact), sometimes hinders the understanding of where certain aromaticity indices are

• failing. Two important goals in aromaticity are the study of the domain of

application of each aromaticity index, as well as the definition of universal low-cost aromaticity measure.

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Aromaticity descriptors

Energetic: Isodesmotic reactions Magnetic: Ring currents, NICS and Λ Geometrical: HOMA Electronic: FLU, PDI, Multicenter

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The HOMA

The harmonic oscilator model of aromaticity (HOMA). Only relies on geometrical data. Easy to compute: HOMA = 1 − 257.71 n

n

• i

(Ropt − Ri)2 = 1 − 257.71 n

• Ropt − R

2 +

n

• i
• Ri − R

2

• =

1 − (EN + GEO) Ropt available: C-C, C-N, C-O, C-P, C-S, N-N, N-O

HOMA can be computed with ESI-3D using the keyword $GEOMETRY

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Magnetic Aromaticity criteria: NICS

A magnetic field directed perpen- dicular to the plane of an aromatic system induces a ring current in the delocalized electrons of the ring. It influences the chemical shifts of 13C and 1H in molecules. NICS: Nuclear Independent Chemical Shift. Calculates the (minus) absolute magnetic shielding at the center of the ring. The more neg- ative, the more aromatic. Λ: Diamagnetic Suceptibility Exaltation. The difference between the measured magnetic susceptibility and the calculated from group ad- ditivity tables.

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Magnetic Aromaticity criteria: Overview

NICS Λ are easy to calculate. They are both ring-size dependent. NICS is computationally expensive. Ring currents are better but complicated to calculate and expensive. NICS is by far the most popular aromaticity index.

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How to compute NICS

In Gaussian include keyword NMR and Bq in the points of interest.

#HF/6-31G* NMR NICS automatic generation 0 1 C 0.00000000 1.38617405 0.00000000 ... Bq 0.00000000 0.00000000

• 1.00000000

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Aromaticity descriptors

Currents Magnetic Structural, electronic

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Electronic Aromaticity Indices

We are concerned with the calculation of the local aromaticity of a given molecule which possesses at least one ring structure. Let us a suppose such ring structure consists of n atoms, represented by the following string A={A1,A2,...,An}, whose elements are ordered according to the connectivity of the atoms in the ring. For such system we can calculate the following electronic aromaticity indices: 1 FLU 2 PDI 3 Iring 4 MCI All these electronic aromaticity indices will be compared afterwards with the well-known NICS and HOMA descriptors.

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The FLU index.

FLU(A) = 1 n

n

• i=1

V (Ai) V (Ai−1) α δ(Ai, Ai−1) − δref (Ai, Ai−1) δref (Ai, Ai−1) 2 where A0 ≡ An and V (A) is the atomic valence that for a closed-shell system reads as follows, and α is a simple function to ensure the first term in the Eq. is always greater or equal to 1, V (A) =

• B=A

δ(A, B) α = 1 V (Ai−1) ≤ V (Ai) −1 V (Ai) < V (Ai−1) δ(A, B), δref (A, B) are quantities that account for the electron sharing of A and B; the latter is taken from an aromatic molecule which has the pattern of bonding A − B. 1

1Matito, Duran, Sol`

a JCP 122, 014109 (2005); Bird Tetrahedron 41, 1409 (1985)

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The PDI index.

Based on the works of Fulton and Bader, which showed that benzene has larger para-related atoms electron sharing than meta-related one, the index uses the para-related atoms electron sharing as a measure of aromaticity for six-membered rings: 1 PDI(A) = δ(A1, A4) + δ(A2, A5) + δ(A3, A6) 3

1Poater, Fradera, Duran, Sol`

a CEJ 9, 400 (2003)

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Aromaticity and Multicenter Indices

Let A={A1,A2,...,An} represent a ring. Iring(A) =

• i1,i2,...,in

Si1i2(A1)Si2i3(A2)...Sini1(An) MCI(A) =

• P(A)

Iring(A) The nth root of these quantities correlates with the TREPE.

Giambiagi, de Giambiagi, Figuereido, PCCP 2, 3381 (2000) Bultinck, Ponec, Van Damme, JPOC 18, 706 (2005)

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TREPE for annulenes. The normalized Iring index.

Cioslowski, Matito, Sol` a, JPCA 111, 6521 (2007)

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TREPE for annulenes. The normalized MCI index.

Cioslowski, Matito, Sol` a, JPCA 111, 6521 (2007)

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Testing Aromaticity: TS of Diels-Alder

FLU and HOMA are not valid for reactivity (references!).

Matito, Poater, Duran, Sol` a, THEOCHEM 727 165 (2005)

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Testing Aromaticity: Inorganic molecules

Feixas, Jim´ enez-Halla, Matito, Poater, Sol` a, JCTC 6, 118 (2010)

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