Pag. Chemical Concepts from Density Functional Theory Paul Geerlings, Stijn Fias, Zino Boisdenghien, Thijs Stuyver, Frank De Proft General Chemistry, Vrije Universiteit Brussel, Brussels, Belgium ICCMSE 2016 Athens, March 17 – 20, 2016 21/09/2016 1 Chemistry from the Linear Response Function
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Pag.
Chemical Concepts from Density Functional Theory
Paul Geerlings, Stijn Fias, ZinoBoisdenghien, Thijs Stuyver, Frank De
Proft
General Chemistry, Vrije Universiteit Brussel, Brussels, Belgium
ICCMSE 2016
Athens, March 17 – 20, 2016
21/09/2016 1
Chemistry from the Linear Response Function
Pag.
0. Introduction : the ongoing need for interpretational concepts and models in quantum chemistry
21/09/2016 2
• Chemistry as an experimental science → accumulation of data of countless number of molecules (CA > 100.000.000 Substances!)
Need for unifying theories, concepts, models to rationalize, interpret, predict …… in order to avoid an encylopaedic “science”.
• Nowadays: Computational Chemistry is flourishing (e.g. WATOC Conferences)
Accumulation of accurate data on countless number of molecules, reactions,…
Aren’t we generating a new encyclopaedia??
STILL need for unifying concepts and models to systematize the theoretical data.
Pag.21/09/2016 3
Looking back
• many concepts are MO/VB based cf. HOMO/LUMO, frontier MO theory rationalization of the Woodward Hoffmann rules, …
• Nowadays DFT is the workhorse par excellence for computational studies on medium and large systems
Isn’t there a need for density based concepts?
Part of DFT founded by RG Parr (1978)
CONCEPTUAL DFT
CHEMICAL DFT or CHEMICAL REACTIVITY DFTor
Pag.21/09/2016 4
Today’s talk
1. - A very brief introduction to Conceptual DFT
1. - Concentrating on the linear response function
• Evaluation
• Representation
• Chemistry from the linear response function
- Concepts: inductive and mesomeric effects, aromaticity,…
Alternating positive and negative regions due to conservation of number of electrons
also in 2D plot
21/09/2016 14
Pag.
Extending → the noble gases
21/09/2016 15
Shell structure of atoms
Pag.
χ(r,0) for Ar in the x,y plane
21/09/2016 16
Pag.
A direct application → polarizability calculations
αij= - dr∫ dr' r
i χ r,r'( ) r'
j i,j= x,y,z
• Comparison with high level calculations
• Absolute values deviate, trend is respected
21/09/2016 17
Z. Boisdenghien, C. Van Alsenoy, F. De Proft, P. Geerlings,J. Chem. Theor. Comp. 9, 1007 (2013)
Pag.21/09/2016 18
Z., Boisdenghien, S. Fias, F. Da Pieve, F. De Proft, P. Geerlings Mol. Phys. 113, 1890, 2015
Pag.21/09/2016 19
Extension to open shell atoms/ Spin polarized linear Response
Transition to a spin-polarized version of in the context of spin polarized conceptual DFT
(F. De Proft, E. Chamorro, P. Perez, M. Duque, F. De Vleeschouwer, P.Geerlings,
Chem. Modell. 6, 63-111 (2009))
( )r, r 'χ
N, Ns representation [ ]s sE E N, N , v, v=
vs
=1
2(v
α− v
β)
v =1
2(v
α+ v
β)
sN N Nα β= −
( )r, r 'χ ( )( ) ( )
( )( )
s s
2
NN
N,N N,N
rEr, r '
v r v r ' v r '
δρδχ = =
δ δ δ
usual DFT potential
related to B vs
= µB
B(r)
Pag.21/09/2016 20
Introduction of
χNS
r, r '( ) =δ2E
δv r( )δvs
r '( )
N,Ns
=δρ
s
δv r '( )
N,Ns
s α βρ = ρ − ρ
( )( )
( )( )
( )NS
rrr, r '
v r ' v r '
βαδρδρ
χ = −δ δ
: two terms can be expanded (IPM)
Pag.21/09/2016 21
Li
α − β α β
• Symmetry r, r’
• Middle region negative: sensitivity of to larger than that of
Aρ v∆
βρ
(cf. )N Nα β>
~ ( ) of Be α + β
cf. ground state configuration
~ ( ) of He but contracted
α + β
(higher Z)
Z. Boisdeghien, S. Fias, F. De Proft, P. Geerlings, PCCP, 16, 14614 (2014)S. Fias, Z. Boisdenghien, F. De Proft, P. Geerlings, J.Chem. Phys, 141, 184107 (2014)
Pag.
How to extend its use for molecules to look for chemical information
condensation at stake in a first approach
21/09/2016 22
M. Torrent-Sucarat, P. Salvador, M. Solà, P. Geerlings, J. Comp. Chem., 29, 1064, 2008
2.2.3. From atoms to molecules
Pag.
An example: : atom condensed values
Numerical methodPresent method
Correlation coefficient between the matrix elements of the two methods:
H C O H2
H -1.22
C 0.68 -4.35
O 0.34 2.99 -3.67
H 0.19 0.68 0.34 -1.22
H2CO
Y = 0.99x - 0.01 R2= 0.96 Slope ∼ 1
H C O H2
H -1.21
C 1.19 -4.82
O 0.42 2.45 -3.28
H -0.40 1.19 0.42 -1.21
N. Sablon, P.W.Ayers, F. De Proft, P. Geerlings, J.Chem.Theor. Comp., 6, 3671, 2010
For other simple molecules (H2O, NH3, CO, HCN, NNO) always a high correlation coefficient is obtained with a very small intercept; the slope varies between 1 and 2.
21/09/2016 23
Pag.
Transmission of a perturbation through a carbon chain
NXχOXχ
Saturated systems
• density response of C atoms on heteroatom perturbation decreases monotonously with distance
• exponential fit: r2= 0.982 ( vide infra) Characterizing and quantifying the inductive effect.
(X= C0, C1, C2 …)
Inductive and mesomeric effects
21/09/2016 24
Pag.
Unsaturated systems
• alternating values• C1, C3, C5 of the chain: minimaC0, C2, C4, C6 : maxima
R= OH, NH2: resonance structures
C1, C3, C5 : mesomeric passive atoms→ follow same trend as alkane structures (inductive effect)
C0, C2, C4, C6 : mesomeric active atoms → effect remains consistently large even after 6 bonds (small decrease due to
superposition of inductive and mesomeric effect)
135
6 4 2
N. Sablon, F. De Proft, P. Geerlings, JPCLett., 1, 1228, 2010
21/09/2016 25
Pag.
Substituted benzenes vs cyclohexanes
• cyclohexane:
1 iC Cχ (i= 2,3,...6)=
• decreases exponentially (inductive effect)• influence of OH small
χ
• benzene: • maxima at C2, C4, C6: mesomerically active atoms (mesomeric effect)• minima at C3, C5 : mesomerically inactive atoms
Reminiscent of (1,4) Para Delocalisation Index. Sola et al,
Chem. Eur. J. 9, 400 (2003) as a criterion of aromaticity
N. Sablon, F. De Proft, P. Geerlings, Chem.Phys.Lett., 498, 192, 2010
21/09/2016 26
• Does Linear response function contain similar information?1,4χ
Pag.
16 non equivalent sixrings studied by Sola et al.
21/09/2016 27
Pag.
• typical benzene-type pattern encountered before
21/09/2016 28
Pag.
→ Linear response function as an “electronic” descriptor of aromaticity→ Unsold’s theorem, resolution of the identity
21/09/2016 29
Pag.
Benzene
• Decreasing σ contribution (cfr. Cyclohexane)
• Zig/zag for π and total
• Anti-aromatic molecules:Cyclobutadiene, COT
“inverted zig zag”
S. Fias, P. Geerlings, P. Ayers, F. De Proft, PCCP, 15, 2882 (2013)
21/09/2016 30
Digging further in aromaticity →→→→ σσσσ,ππππ aromaticity (applying σσσσ,ππππ separation)
Pag.
Inorganic rings
21/09/2016 31
Pag.
Valence Bond Structures
X= NH, O, PH
Resonance Energy (kJ mol-1) and weights W of the four Valence Bond Structures.
Molecule Eres W(I) W(II) W(III) W(IV)
B3N3H6 61.6 0.05 0.05 0.90 0.00
B3O3H3 27.6 0.02 0.02 0.96 0.00
B3P3H6 132.6 0.21 0.21 0.58 0.00
C6H6 161.4 0.47 0.47 0.03 0.03
J. Engelberts, R. Havenith, J. Van Lenthe, L. Jenneskens, P. Fowler, Inorg. Chem., 44, 5266, 2005
21/09/2016 32
Pag.
Borazine
• Typical resonance pattern observed in aromatic systems is recovered if one of the N-atoms is chosen
as the reference atom.
• Purely inductive behaviour (exponential decay of electron delocalization with internuclear distance)
is recovered if one of the B-atoms is chosen as the reference atom.
Dual picture of aromatic character of borazine (cfr. ongoing debate in the literature)
N. Sablon, F. De Proft, M. Sola, P. Geerlings, PCCP, 14, 3960 (2012)
21/09/2016 33
Pag.3421/09/2016
Reactions
• Evolution of the �, � LRF upon cyclisation reactions:the trimerization of acetylene to benzene
• Total PLR (Para Linear Response) function maximum at TS• Essentially � in nature � � aromaticity• � contribution only becomes significant when the � framework is returned to a more localized state
Pag.21/09/2016 35
2.3. Beyond the independent Particle Model
• Using a CPKS approach
( ) ( ) ( ) ( ) ( ) ( )ia , jb
1 * *
i a b j
i,a, j,b,
r, r ' 2 M r r r ' r 'σ τ
−σ σ τ τ
σ τ
χ = − ϕ ϕ ϕ ϕ∑∑
• Independent particle:
• Random Phase:
• General CPKS:
( )ia , jb b j ij abM σ τ στ= ε − ε δ δ δ
( ) ( )ia , jb b j ij abM 2 ia | jbσ τ στ= ε − ε δ δ δ + σ τ
( ) ( ) ( )( )ia , jb b j ij ab xcM 2 ia | jb 2 ia | f r, r ' | jbσ τ στ= ε − ε δ δ δ + σ τ + σ τ
( )( ) ( )
2
xcxc
Ef r, r '
r r '
δ=
δρ δρ
W. Yang, A.J.Cohen, F. De Proft, P. Geerlings, J.Chem.Phys. 136, 144110 (2012)
Home made program (S. Fias) ���� Unintegrated Plots
Pag.
Planar Metallic Systems: Unintegrated plots
Aromaticity order2- - + 2+
4 3 2 2 3 4Al > Al Ge ³ Al Ge AlGe < Ge≥ ≤
Expected order (Feixas et al)
Mainly σ aromatic (∼ 65%)
Insertion of C: decreasing aromaticity, sequence unaltered; but increasing σ component
21/09/2016 36
• The use of unintegrated plots
Benzene 2-
4Al
σ
• in plane• perturbation
at top nucleus
π
• 0.5 au above plane• perturbation in that
plane at top nucleus
σ π
Delocalized Nature of the linear Response more pronounced in the σ electron density
σ Aromatic character
S. Fias, Z. Boisdenghien, T. Stuyver, M. Audiffred, G. Merino, P. Geerlings, F. De Proft, J. Phys. Chem. A, 2013, 117, 3556
( )13 241χ + χ
2→
Pag.21/09/2016 37
2.4 An extension to Alchemical Derivatives:exploring Chemical Space
• Alchemical derivatives ( O.A. Von Lilienfeld, R.D. Lins, U. Röthlisberger, Phys.Rev.Lett., 95, 153002 (2005)
MN,Z ,
E
Zβ
α
∂
∂ ℝ
: change in molecular energy upon changing atomic nuclear charge
alchemical potential cf
M MZ ,R v
E E
N N
∂ ∂ = = µ
∂ ∂ electronic chemical potential
2
2
N,Z ,R
E
Z Zγ
α β
∂ ∂ ∂
cf
2
2
v
E
N
∂= η
∂ electronic chemical hardness
alchemical hardness
Pag.21/09/2016 38
Chemical Significance: navigation through the huge Chemical Compound Space for tracing interesting compounds by “Transmutation Reactions”
N=cte
∆Z= ± 1
Chemical Compound Space: set of all possible combinations of chemical elements prone to form stable compounds
P. Kirkpatrick, C. Ellis, Nature, 432, 823 (2004)
Pag.21/09/2016 39
Instead of calculating the energy of each of the “transmutants”, concentrate on the central compound and its alchemical derivatives
2
N,Z , N,Z ,
E EdZ dZ dZ ...
Z Z Zβ γ
α α βα α βα α β
∂ ∂= + + ∂ ∂ ∂ ∑ ∑∑
ℝ ℝ
alchemical potential alchemical hardness ( diagonal and off-diagonal)
electronic part electronic part
( )rdr
r Rα
ρ−
−∫( )
N,...
rZ
d rr R
β
α
∂ρ ∂
−∫
Pag.21/09/2016 40
Connection with linear response function
( ) ( )( ) ( )2 2
'
N N
v r v r 'E E dr dr '
Z Z Z Zv r v rα α ββ
∂ ∂∂ δ= ∂ ∂ ∂ ∂δ δ ∫ ∫
( )1 1
r, r ' dr dr'r R r Rα β
= χ− −
∫
Pag.21/09/2016 41
Evaluation of the alchemical potential and hardness
Coupled Perturbed Kohn Sham Theory (cf linear response function)
M.Lesiuk, R. Balawender, J. Zachara, J.Chem.Phys., 136, 034104(2012)
Prediction of energies of molecules with charge of the central nuclei changed by ± 1
( ) ( ) ( )iN
i
ii 1
1 EE Z 1 E Z 1
i! Z=
∂± = + ±
∂ ∑
‖
0E
Cf CH4
(HF)
0E 40.214 au= −E
14.752 auZ
∂= −
∂
2
2
E 3.023 au
Z
∂= −
∂
3
3
E 0.135 au
Z
∂= −
∂
( ) ( )0 0 4E Z 1 E NH - 56.500++ = =
( ) ( )0 0 4E Z 1 E BH - 26.950−− = =
cf “exact” 56.565−
26.987−
Pag.21/09/2016 42
For the complete series of six N2 transmutation reactions (B3LYP/cc-pVTZ) an error between “vertical” and alchemical transmutation energies of 0.03 a.u. was obtained
N2 → CO case: 0.004 au ~ 2.5 kcal mol-1
Not “chemical accuracy” yet but the ordering of the energy of the compounds comes out correctly
Straightforward procedure for looking at neighbouring structures when exploring chemical space
Recent quantum chemical interest in Molecular Design
(cf W. Yang et al, JACS, 128, 3228 (2006))
Pag.21/09/2016 43
A more involved application: transmutation of benzene
( ) nn C H N azines− → →
( ) ( )n n
C C B N azoborines− → − →
iso-electronic transmutations
Azoborines • n-sequence correctly reproduced
• sequence of isomers n=1 (3)n=2 (11) n=3 (3)
correctly reproduced
Transmutation energies and alchemical derivates effective tools for stability prediction and exploring CCS
Present work: CC → BN substitution in fullerenes, graphene, …
R. Balawender, M.A. Welearegay, M. Lesiuk, F. De Proft, P. Geerlings, J. Chem.Theor.Comp., 9,5327,(2013)
• Extension: to non adjacent atoms � long bond order (T. Morkawa, S. Narito, D.J. Klein, Chem. Phys. Lett., 402, 554 (2005)
Now: K2 (G) = ∆(0) K2(GΘrs)
=
Prs
P( )2
=∆r, s(0)
∆(0)
2
~ Trs(0)
K(GΘrs)
K(G)
2
= ∆rs,rs
(0)
Transmission � Pauling Bond Order
Pag.21/09/2016 50
Consider Benzene: Kekulé Structure for (GΘr,s) say (GΘ1,4)
� If no curly arrows drawing can be made of the electron displacements between the two atoms
�Pauling long bond order is zero � transmission probability of MED is zero
�Trs Ξ T14 ≠ 0 � transmission
G Θ 1,4
G
Displacement of Electron Pairs in the Kekulé structures of G from r�s (contact atoms)
Pag.21/09/2016 51
The use of curly arrows opens up a connection to the widely used concepts of resonance effects in reactivity theory
Selection RuleAtom-AtomPolarizability
Pag.21/09/2016 52
Application in Photo-, Thermo and Redoxswitches
- Oligo (phenylenevinylene)quinone redox switch
(Q – OPV)
- Conductance ratio: 40, S. Tsoi et al., A.C.S. Nano, 2, 1289 (2008)
T. Stuyver, S. Fias, F. De Proft, P. Geerlings, Chem. Phys. Lett., 630, 51 (2015)T. Stuyver, S. Fias, F. De Proft, P. Geerlings, J. Phys. Chem.C., 119, 26390 (2015)
Pag.21/09/2016 53
Pag.
4. Conclusions
• Conceptual DFT offers a broad spectrum of reactivity descriptors in line with the need for unifying concepts.
• The “missing” second order derivative, the “linear response function”, comes within reach with various techniques of increasing complexity. It is a tool to see how ∆v perturbations are propagated through an atom or molecule. Its physical relevance becomes apparent, thanks to various representations of the kernel for atoms revealing atomic shell structure, and extensions in the context of spin polarized Conceptual DFT
• The computational results on molecules reveal that important chemical information can be retrieved from the linear response function: from inductive and mesomeric effects to the aromatic character of organic and inorganic rings including its evolution during reactions. In the context of alchemical derivatives it opens the gate for exploring chemical compound space in an efficient way.
•An important step towards the link between the linear response function and electrical conductivity has been taken, retrieving the curly arrow approach of organic chemistry.
21/09/2016 54
Pag.
Acknowledgements
Acknowledgements
Prof. Frank De Proft, Dr. Stijn Fias, Dr. Zino Boisdenghien, Dr. Freija De Vleeschouwer
Dr. Nick Sablon
Prof. Paul W. Ayers (Mc Master University, Hamilton, Canada)
Prof. Weitao Yang (Duke University, USA) and Dr. A. J. Cohen (Cambridge, UK)
Dr. R. Balawender and Drs. M. Welearegay (Warsaw, Poland)
Dr. M. Torrent – Sucarrat (Brussels, Girona)
Prof. C. Van Alsenoy (Antwerp, Belgium)
Prof. M. Sola (Girona), Prof. Gabriel Merino (aromaticity)
Fund for Scientific Research-Flanders (Belgium) (FWO)