Theoretical foundations of the bond-orbital projection formalism

A new perspective on quantifying electron localization and delocalization in molecular systems

Theoretical foundations of the bond-orbital projection formalism



JAGIELLONIAN UNIVERSITYDepartment of Theoretical Chemistry

Institut de Química Computacional i Catàlisi

WORKSHOP

GIRONAJanuary 21st, 2016

dr Dariusz Szczepanik

Updated: 2016-02-25


1. Electron density, atoms in molecules, chemical bonds.

2. Localization and delocalization components of the electron density

3. Electron delocalization between atoms.

4. Electron delocalization between bonds.

5. The effectiveness of bond conjugation as an aromaticity criterion.

2/33

Presentation plan

I Hohenberg–Kohn theorem: the ground-state electron density (ED) uniquely determines the potential and thus all physicochemical properties of the molecular system.

Electron density, atoms in molecules, chemical bonds

What about traditional concepts such as atom and chemical bond?

Theoretical foundations of the bond-orbital projection formalism 3/33

Partitioning of the electron density into atomic contributions (charges):


● Bader's charges (QTAIM approach),

● Hirshfeld's charges,

● Politzer's charges,

● Voronoi charges,

● Coulson's charges,

● Mulliken's charges,

● Löwdin's charges,

● Weinhold's (natural) charges,

● Merz-Kollman's charges,

● Breneman's charges,

● Szigeti charges,

and many others...

A difficult choice!


Delocalization of atomic charges – chemical bonding analyses:


● Bond critical points (QTAIM by Bader),

● Coulson's, Wiberg's, Mayer's and Gopinathan-Jug's bond orders,

● Localized molecular orbitals (Boys, Edmiston-Ruedenberg, Pipek-Mezey schemes)

● Natural bond orbitals (NBO),

● Natural orbitals for chemical valence (NOCV),

● Localized orbitals of bond orders (LOBO),

● Iterative double-atom partitioning by orthogonal projectors (IDAP),

● Electron localization function (ELF),

● Localized orbital locator (LOL),

● Single exponential decay detector (SEDD),

● Reduced density gradient (RDG),

and many others...

A difficult choice!


Delocalization of chemical bonds – a multicenter electron-sharing analyses:


● Scanning for multicenter bonding within the framework of the NBO analysis,

● Adaptive natural density partitioning (AdNDP) analysis,

● Multicenter delocalization descriptors (MCI,DI,ESI),

● Bridgeman-Empson's three-center bonding analysis,

● Electron density of delocalized bonds approach (EDDB),

and others...

Is it possible to probe electron localization and delocalization within one theoretical paradigm?


Localization and delocalization components of the electron density

ED(r) EDLA(r) EDLB(r) EDDB(r)

ED(r)

EDLA(r)

EDLB(r)

EDDB(r)

– electron density,

– density of electrons localized on atoms,

– electron density of localized bonds,

– electron density of delocalized bonds,

EDDA(r)

EDDA(r) – density of electrons delocalized between atoms,

..

!

electrons electrons electrons


Electron delocalization between atoms (EDDA)

Electron density layer representing the population of electrons delocalized between atoms, EDDA(r), is crucial for the whole ED-decomposition procedure. It involves the Hilbert-space partitioning scheme and is defined through the following steps:

Definition of the matrix

1.Transformation of non-orthogonal atomic orbitals (AOs) into the represe-

ntation of natural atomic orbitals (NAOs).

2.Solving the eigenproblems of a set of Jug's matrices, representing all

possible bonds (interactions) in a molecule, to obtain two-center bond-

order orbitals (2cBO) and their occupations.

3.Projection of the 2cBO metric onto the subspace of occupied MOs.



1. The occupancy-weighted symmetric orthogonalization (OWSO) of atomic orbitals (AOs) to natural atomic orbitals (NAOs):

A). Transformation from cartesian to pure AOs.B). Partitioning and symmetrization of intra-atomic blocks of the overlap and the Coulson's density matrix (D).C). Löwdin orthogonalization of the intra-atomic blocks of the DM.D). Solving the eigenproblems for intra-atomic blocks of the DM.E). Division of eigenfunctions into the (pre-orthogonalized) natural minimal basis (NMB) and the complementary natural Rydberg's basis (NRB). F). Interatomic Gramm-Schmidt orthogonalization of NRB to NMB.G). Repeating the B,C and D steps, but only for the NRB 'orbitals'.H). A separate interatomic occupancy-weighted orthogonalization of both subsets, I). Re-orthogonalization of both subspaces by repeating the B,C and D steps.

Features of NAOs:

– The effective dimensionality of the AO space is reduced to that of the formal NMB subspace.– For isolated atoms NAOs coincide with natural orbitals (NO).– NAOs mostly retain a well-localized one-center character.– NAOs are intrinsically stable toward basis set extensions.



2. Construction of a set of two-center bond orbitals.

– the Jug's matrix is a (α,β)-diatomic block matrix of type: =

– for duodemponent density matrices, the Wiberg's bond-order (covalency) index reads:

– the eigenvectors of represent the two-center bond orbitals of three different types: bonding >0, non-bonding =0, and antibonding <0

, and both subsets of 2cBOs form a paired-orbital basis.

– electron density in the NAO basis reads



2. Construction of a set of two-center bond orbitals.

Ace

tyle

ne

CC

bo

nd

– bonding 2cBOs

– antibonding 2cBOs

For typical molecular systems with well defined Lewis-like electronic structures the highest-occupied 2cBOs form a set of nearly orthogonal bond orbitals and, consequen-tly, twice the sum of Wiberg's indices between all pairs of covalently bonded atoms approximates the population of electrons delicalized between all atoms (EPDA).



Obviously, for accurate calculations as well as in the case of large molecular systems with non-typical bonds and weak interactions, twice the sum of Wiberg's covalencies sometimes exceeds the exact EPDA due to nonorthogonal overcounting. In such situations we have two choices:

all = ( )n2

I. Restore orthogonality of the highest-occupied 2cBOs within the iterative double-atom partitioning procedure using orthogonal projectors or, more familiar, by transformation to the subset of bonding NBOs.

II. Remove the nonorthogonal electron overcounting by the following projection cascade:

NAO → MO(occupied) → 2cBO(bonding) → MO(occupied) → NAO ,

which is fully equivalent to the following orthogonal similarity transformation:

3. Projection of 2cBOs onto the subspace of occupied MOs



… a little too much! So where is the problem?

0

Num

ber

of π

-ele

ctro

ns

2

4

6

8

4.8

00

4.3

11

6.0

00

4.8

89

6.8

57

4.9

03

7.5

00

4.7

68

14%25%

Projections through the subspace of occupied MOs remove π-electron overcounting...

C5H

5- C

6H

6 C

7H

7+ C

8H

82+




Due to nonorthogonalities both bonding and antibonding 2cBOs are linear combinations of MOocc and MOvir. Therefore, the projection cascade MUST involve both 2cBO subspaces.


!


+

NAO → MO(occupied) → 2cBO(all) → MO(occupied) → NAO

c o m p l e m e n t a r y


0

Num

ber

of π

-ele

ctro

ns

2

4

6

814%

25%

C5H

5- C

6H

6 C

7H

7+ C

8H

82+

4.8

004

.311

5.7

60

6.0

00

4.8

89

6.0

00

6.8

57

4.9

03

5.8

77

7.5

00

4.7

68

5.6

25



NAO → MO(occupied) → 2cBO(all) → MO(occupied) → NAO


ED(r) EDLA(r) EDDA(r)

30.0 2.3(1.7)

27.7



ED(r)

EDDA(r)

EDLA(r)

30.0 30.0 30.0

29.3 24.2 23.4

0.7 5.8 6.6


Electron delocalization between bonds

The EDDA component of the electron density can be further partitioned:

EDLB(r) EDDB(r)EDDA(r) = +

EDLB(r)

EDDB(r)

– electron density of localized (two-center) bonds,

– electron density of delocalized (multi-center) bonds,

EPLB

EPDB

0 – localized 2cBO

1 – delocalized 2cBO


Construction of the matrix

1. Forming two sets of 2cBOs for adjacent bonds Xα–X

β and X

β–X

γ, ie.

and , respectively.

2. Forming a set of three-center bond orbitals (3cBO), .

3. Expanding bonding 3cBOs in the basis of orthogonalized bonding 2cBOs.

4. Canceling of the phase-opposite 3cBOs.

5. Determining scaling factors .

6. Removing the nonorthogonal electron overcounting.




1. Forming two sets of 2cBOs for adjacent bonds Xα–X

β and X

β–X

γ:

2. Forming a set of three-center bond orbitals (3cBO), representing the bonding between X

α and X

γ through the conjugation center X

β :

We do not take into account the through-space (direct) interactions between X

α–X

β and X

β–X

γ,and hence .



3. Expanding bonding 3cBOs in the basis of orthogonalized bonding 2cBOs.

Orthogonalization: OW SO

= –

3cBO 2cBO 2cBO



4. Canceling out the phase-opposite 3cBOs.

If two linear combinations of 2cBOs are in opposite pha-ses and have nearly degenerated occupation numbers, they do not contribute to bond delocalization. Phase-opposite!If two 2cBOs do not form effectively a 3cBO they do not contribute to bond delocalization.! Effective

3cBO Effective2cBO

To determine if and to what extent both conditions are satisfied we can use an auxiliary vector with the elements defined as follows (we assume that 3cBOs are ordered from the highest to lowest occupied):

where ,

= 1 (effective 3cBO)

= –1 (effective 3cBO but in opposite phase with other 3cBO)

= 0 (effective 2cBO)



4. Canceling out the phase-opposite 3cBOs.

0.980 0.979 0.887 0.032 0.002

1.000 –0.999 1.000 0.999 –0.998

0.980 0.887 0.032 –0.002–0.978

0.002 0.000 0.887 0.030 0.000

Benzene3cBOs:

If < 0 then the m th 3cBO cancels out the p th 3cBO for which

2

2

2



5. Determining scaling factors .

To determine the scaling factor for the i th 2cBO of the Xα–X

β bond we have

to consider conjugations with 2cBOs of all possible adjacent bonds.

6. Removing the nonorthogonal electron overcounting.


*

order– reversed ,

assuming that:


65%

35%


The effectiveness of bond conjugation as an aromaticity criterion

GLOBAL/LOCAL AROMATICITYHÜCKEL's AROMATICITY


Homotropylium cation C8H9+

HOMOAROMATICITY

Cyclononatetraenyl cation C9H9+

MÖBIUS AROMATICITY

Fullerene C6010+

SPHERICAL AROMATICITY3D-AROMATICITY

closo-borane [B7H7]2-

Fuzed closo-borane [B17H13]2-

9.633

25.6

50

METALLOAROMATICITY

19.645

Co(Por)49.118





k


Comparative analysis of selected aromaticity indices


Comparative analysis of selected aromaticity indices



*

*By courtesy of dr Justyna Dominikowska

What distinguishes the EDDB as an aromaticity measure?

✔ Intuitiveness and simplicity

✔ No reference molecule is needed

✔ Ring-size independence

✔ σ/π dissection for planar and non-planar aromatic species

✔ Low computational cost

✔ Small method/basis-set dependence

✔ Wide applicability


This presentation is available at

www2.chemia.uj.edu.pl/~szczepad/UdG.Workshop.pdf


A practical guide to the use of the EDDB method



JAGIELLONIAN UNIVERSITYDepartment of Theoretical Chemistry

Institut de Química Computacional i Catàlisi

WORKSHOP

GIRONAJanuary 28st, 2016

dr Dariusz Szczepanik

Updated: 2016-02-25


1. Electron density of delocalized bonds.

2. Installation, configuration and execution of the RunEDDB program

3. Visualization of EDDB and related functions.

4. Selected examples:

2/40

Presentation plan

4.1. Relative aromaticity along different conjugation paths.

4.2. Multifold aromaticity in all-metal clusters.

4.3. Local and global aromaticity in polycyclic aromatic species.

5. Plans for future development.

Electron density of delocalized bonds

ED(r) EDLA(r) EDLB(r) EDDB(r)

ED(r)

EDLA(r)

EDLB(r)

EDDB(r)

– electron density,

– density of electrons localized on atoms,

– electron density of localized bonds,

– electron density of delocalized bonds.

..

A practical guide to the use of the EDDB method 3/40

arom

atic

stab

iliza

tion

!



EPDB

– one-electron density matrix within the NAO basis

– off-diagonal (α,β)-diatomic block of

– two-center bond orbital (2cBO)

– diagonal matrix of the 2cBO occupations

– diagonal matrix of scaling factors for each 2cBO

(delocalized 2cBO)(localized 2cBO)

One

-det

erm

inan

tCl

osed

-she

ll

For

open

-she

ll sy

stem

s D

DB i

s de

fine

d as

twi

ce t

he s

um

of t

he c

orre

spon

ding

spi

n co

mpo

nent

s


✔ Intuitiveness and interpretational simplicity✔ No parametrization and reference molecule is needed✔ Ring-size independence✔ σ/π dissection for planar and non-planar aromatic species✔ Low computational cost ✔ Small method/basis-set dependence✔ Wide applicability


What distinguishes the EDDB as an aromaticity measure?

References

(1) Electron delocalization index based on bond order orbitals D.W. Szczepanik, E. Żak, K. Dyduch, J. Mrozek, Chem. Phys. Lett. 593 (2014) 154-159.

(2) A uniform approach to the description of multicenter bonding D.W. Szczepanik, M. Andrzejak, K. Dyduch, E. Żak, M. Makowski, G. Mazur, J. Mrozek, Phys. Chem. Chem. Phys. 16 (2014) 20514-20523.

(3) A new perspective on quantifying electron localization and delocalization in molecular systems D.W. Szczepanik, Comput. Theor. Chem. 1080 (2016) 33-37.

The current version of the RunEDDB program is available for Linux systems as a script written in R, and as such it requires the R-package (www.r-project.org).An up to date version of the script can be downloaded from:


Installation, configuration and execution of the RunEDDB program

www2.chemia.uj.edu.pl/~szczepad/RunEDDB (program)

(manual)www2.chemia.uj.edu.pl/~szczepad/RunEDDBman.pdf

To generate the cube files, the original Gaussian (www.gaussian.com) cubegen utility is required as well as the dat2cube bash script available at:

www2.chemia.uj.edu.pl/~szczepad/dat2cube

The method is based on the the NAO representation, which is provided by the NBO software (http://nbo6.chem.wisc.edu). To generate the FILE.49 file (containing the AO→NAO and NAO→MO matrices), an input file to the RunEDDB program, the following NBO keywords have to be used:

$NBO SKIPBO AONAO=W49 NAOMO=W49 $END

If you already have the .chk file you can generate the .49 file using this script:

www2.chemia.uj.edu.pl/~szczepad/chk249

http://www.r-project.org/

http://www.gaussian.com/

http://nbo6.chem.wisc.edu/



1. Download the RunEDDB script:

wget P $HOME www2.chemia.uj.edu.pl/~szczepad/RunEDDB

2. If necessary, edit the first line to provide correct path to the Rscript interpreter. E.g., if the R-package is installed in /aplic/R/3.0.0/ then copy and paste this command:

sed i "1s/.*/\#\!\/aplic\/R\/3.0.0\/bin\/Rscript/" $HOME/RunEDDB

3. Make the script executable:

chmod a+x $HOME/RunEDDB

4. Create an alias to the script in the .bashrc file:

echo "alias runeddb='$HOME/RunEDDB'" >> $HOME/.bashrc

5. Repeat steps 1, 3 and 4 for scripts dat2cube and chk249.

Configuration...



1. Open an interactive session, e.g. :

qlogin clear q compilar.q

2. Load the R module, e.g.:

module load R/R3.0.0_ICS11.1.072

3. To support the cube files load also the appropriate Gaussian module,e.g.:

module load g09d01_pgi10.1_NBO6

4. To get FILE.49 from the Gaussian checkpoint file run the chk249 script:

chk249 FILE.chk

Execution...

What is the general syntax of the command line options for RunEDDB?

4. To perform the EDDB analysis (e.g. benzene / RHF) simply run:

runeddb FILE.49 1:21

If the SGE queue system is used...

runeddb FILE.49 MOspec BONDspec OtherOpts



FILE.49 - file generated by the NBO module and containing the AO→NAO and NAO→MO transformation matrices.

required optional

MOspec – a list of occupied molecular orbitals, separated by commas; the range operator (:) is allowed (e.g. benzene / RHF, σ-MOs):

runeddb FILE.49 7:15,17,18,19

the order is very important!Syntax...

runeddb FILE.49 16,20:22 16,20

For UHF calculations MOspec lists have to be specified separately for alpha and beta molecular spinorbitals (e.g. benzene / UHF, π-MSOs):




BONDspec – a list of atoms and chemical bonds taken into account in the bond-orbital projection procedure. If does not specify or contains the lowercase a character, the program considers all possible bond-conjugation paths in a

molecule, according to the bond-order threshold set to 0.001 (the parameter in the 3rd line of the RunEDDB script):

required optional

runeddb FILE.49 1:21

the order is very important!Syntax...

The uppercase A sets the bond-order threshold to 0:

runeddb FILE.49 1:21 A

runeddb FILE.49 1:21 aequivalent to




Global aromaticity of naphthalene:(all possible bond conjugations)

required optional

the order is very important!Syntax… examples

BONDspec:

a

A

1:18

1:10,11,12,13:18

For medium/large species it is easier to specify atoms and fragments rather than bonds.

7.5e




Local aromaticity of a selected ring in naphthalene:(only kekulean CC conjugations)

required optional


BONDspec:

12,19,23,910,34,410

12341091

91234,410,910

1:4,910

The last case specify in total six conjugations centers, but does not restrict possible bond-conjugations only to one ring (and even to carbon backbone!).

wrong!

3.5e




Local aromaticity of a selected ring in naphthalene:(kekulean + cross-ring CC conjugations)

required optional


BONDspec:

12341091,13,14,11 0,24,210,29,310,39,49

1:2:3:4:10:9

1,2,3,4,10,9 wrong!

4:1:9:2:3:10

The 3rd bond specification list is identical to the 2nd one. This notation allows one to restrict bond conjugations to particular molecular fragment only.

3.6e




required optional

the order is very important!Syntax…

OtherOpts:

EDED

EDLA

EDLB

EDDB

POFF – skip the orthogonal similarity transformation step.

– save the ED(r) function into the ED.dat file.

– save the EDLA(r) function into the EDLA.dat file.

– save the EDLB(r) function into the EDLB.dat file.

– save the EDDB(r) function into the EDDB.dat file.

NMB – restrict the EDDB analysis to the Natural Minimal Basis subspace only; it significantly reduces computational time.

These keywords can appear in the command line in any order.

In the case of spin-unrestric-ted open-shell calculations the corresponding spin-den-sities are saved into the SDXX.dat files in the same directory.



Results (benzene [singlet] / RHF) …

Atom ED EDLA EDLB EDDB 1 4.208 0.074 3.138 0.996 2 4.208 0.074 3.138 0.996 3 4.208 0.074 3.138 0.996 4 4.208 0.074 3.138 0.996 5 4.208 0.074 3.138 0.996 6 4.208 0.074 3.138 0.996 7 0.792 0.036 0.735 0.021 8 0.792 0.036 0.735 0.021 9 0.792 0.036 0.735 0.021 10 0.792 0.036 0.735 0.021 11 0.792 0.036 0.735 0.021 12 0.792 0.036 0.735 0.021 Total: 30.000 0.656 23.242 6.101 Computational cost: 00:00:01 (660x3cBO)

runeddb FILE.49 7:21 A

B3LYP/6-311+G(d,p)



Results (benzene [triplet] / UHF) …

Atom ED SD EDLA SDLA EDLB SDLB EDDB SDDB 1 4.264 0.178 0.232 0.014 3.390 0.074 0.642 0.091 2 4.169 0.605 0.306 0.184 3.331 0.203 0.533 0.219 3 4.169 0.605 0.306 0.184 3.331 0.203 0.533 0.218 4 4.264 0.178 0.232 0.014 3.390 0.074 0.642 0.091 5 4.169 0.605 0.306 0.184 3.331 0.203 0.533 0.218 6 4.169 0.605 0.306 0.184 3.331 0.203 0.533 0.218 7 0.789 0.004 0.036 0.001 0.731 0.005 0.021 0.000 8 0.804 0.018 0.032 0.005 0.758 0.022 0.015 0.001 9 0.804 0.018 0.032 0.005 0.758 0.022 0.015 0.001 10 0.789 0.004 0.036 0.001 0.731 0.005 0.021 0.000 11 0.804 0.018 0.032 0.005 0.758 0.022 0.015 0.001 12 0.804 0.018 0.032 0.005 0.758 0.022 0.015 0.001 Total: 30.000 2.000 1.887 0.725 24.596 0.586 3.518 0.689 Computational cost: 00:00:02 (1320x3cBO)

runeddb FILE.49 7:22 7:20 A

B3LYP/6-311+G(d,p)


dat2cube FILE.fchk EDXX.dat ngrid nproc

Visualization of EDDB and related functions

required optional

the order is very important!Syntax…

In the current implementation, the dat2cube script requires a formatted check-point file (FILE.fchk) from Gaussian and the EDXX.dat file obtained from the RunEDDB program. By default, dat2cube uses 1 CPU to generate a cube file with the number of points per side equal to 803=512k. It can be changed by specifying parameters nroc and ngrid, respectivety. E.g.

dat2cube FILE.fchk EDDB.dat 100 4

uses 4 CPUs to convert EDDB.dat into the cube file, EDDB.cube, containing 1003 points distributed evenly over a rectangular grid.

Which programs support the Gaussian cube format?

GaussView, (g)Molden, Molekel, VMD, Gabedit, Avogadro, ...


Relative aromaticity along different conjugation paths.

Bis-palladium(II) [36]octaphyrin(1.1.1.1.1.1.1.1)?

wget www2.chemia.uj.edu.pl/~szczepad/octaphyrin.tar.gz

tar xzf octaphyrin.tar.gz




B3LYP/SDD

Along which conjugation path the electron delocalization is the most effective?



All possible bond conjugations

runeddb FILE.49 1:178 a eddb > EDDB.all.log

dat2cube molecule.fchk EDDB.all.dat 80 1

mv EDDB.dat EDDB.all.dat



The conjugated 32 π-electronic (iner) circuit

runeddb FILE.49 1:178 126549101215181926242829363437384342414647495355566361656668691 eddb > EDDB.in.log

dat2cube molecule.fchk EDDB.in.dat 80 1

mv EDDB.dat EDDB.in.dat

1

2 65

49 10

1215

18

19

26

2428

2936

3437

384342

414647

4953

55

56

63

6165 66

68 69



The conjugated 36 π-electronic (mediate) circuit

1

2 65

410

1215

18

19

21

242829

36

3437

384342

414647

4953

55

56

58

61 65 6668 69

runeddb FILE.49 1:178 1265491012151819212224282936343738434241464749535556585961656668691 eddb > EDDB.med.log

dat2cube molecule.fchk EDDB.med.dat 80 1

mv EDDB.dat EDDB.med.dat

2259



The conjugated 36 π-electronic (outer) circuit

1

23

410 13

15

18

19

21

24282931

3437

3840

41464750

53

55

56

58

61 6566 71

692259

runeddb FILE.49 1:178 123491013141518192122242829313234373840414647505153555658596165667170691 eddb > EDDB.out.log

dat2cube molecule.fchk EDDB.out.dat 80 1

mv EDDB.dat EDDB.out.dat

9 14

32

51

70




Atom ED EDLA EDLB EDDB

Total: 356.000 140.207 167.939

Computational cost: 00:00:37 (23022x3cBO)

...

...

...

...

...

47.854

EDDB.all.log

EDDB.all.cube

without threshold: 00:07:45 (194472x3cBO)

48.389



The conjugated 32 π-electronic (iner) circuit


Total: 356.000 NA 68.771


...

...

...

...

...

20.419 (43%)

EDDB.in.log

EDDB.in.cube



The conjugated 36 π-electronic (mediate) circuit


Total: 356.000 NA 76.380


...

...

...

...

...

18.510 (39%)

EDDB.med.log

EDDB.med.cube



The conjugated 36 π-electronic (outer) circuit


Total: 356.000 NA 85.511


...

...

...

...

...

14.674 (31%)

EDDB.out.log

EDDB.out.cube



36 π-electronic (mediate) circuit

32 π-electronic (iner) circuit

36 π-electronic (outer) circuit


47.9e (100%) 20.4e (43%)

18.5e (39%) 14.7e (31%)

Along which conjugation path the electron delocalization is the most effective?


wget www2.chemia.uj.edu.pl/~szczepad/allmetaroma.tar.gz


tar xzf allmetaroma.tar.gz



B3LYP/X/Stuttgart+2f

Single,doubleortriplearomaticity?

Multifold aromaticity in all-metal clusters.

B3LYP/6-311+G(d)

Al

AlAl

Al

2-

Cu Cu

Cu

+

Y Y

Y

-

Hf Hf

Hf


runeddb Al42.49 21:27 A > Al42.all.log


2-

21 22 23 24 25

runeddb Al42.49 21:26 A > Al42.sigma.log

runeddb Al42.49 27 A > Al42.pi.log

Al

AlAl

Al

26 27


runeddb Cu3+.49 13:28 A > Cu3+.all.log


Cu Cu

Cu

+

13 14 15 16 17 18

19 20 21 22 23 24

25 26 27 28

runeddb Cu3+.49 13:15,17,20,21,23,24,27,28 A > Cu3+.sigma.log

runeddb Cu3+.49 16,18,19,22,25,26 A > Cu3+.pi.log


runeddb Y3.49 13:17 A > Y3.all.log


13 14 15 16 17

runeddb Y3.49 13:16 A > Y3.sigma.log

runeddb Y3.49 17 A > Y3.pi.log

Y

Y

Y -

runeddb Hf3.49 13:18 A > Hf3.all.log

13 14 15 16 17

runeddb Hf3.49 13:16 A > Hf3.sigma.log

runeddb Hf3.49 17 A > Hf3.pi.log

Hf Hf

Hf

18

runeddb Hf3.49 18 A > Hf3.delta.log



Al Hf42- Cu 3

+ Y -3 3

EDDB ( σ )

EDDB ( π )

EDDB

EDDB ( δ )

3.42 (3.43) 1.92

1.50 0.00

MCI* ( σ )

MCI* ( π )

MCI* ( δ )

1.69 1.87

0.00

MCI* 3.56

Aromaticity:

1.98 (1.98) 1.95

0.02 0.00

1.88 0.01

0.00

1.89

5.78 (5.89) 4.00

1.78 0.00

4.58 2.96

0.00

7.54

6.16 (6.26) 2.60

1.78 1.78

4.45 2.96

2.95

10.37

σ+π σ σ+π σ+π+δdouble single double triple

* m

ultip

lied

by 1

0


Global and local π-aromaticity in polycyclic aromatic species.

wget www2.chemia.uj.edu.pl/~szczepad/glolocaroma.tar.gz


tar xzf glolocaroma.tar.gz



Does (and to what extent) the orthogonal similarity transformation affects local aromaticity?

B3LYP/6-311+G(d,p)

What is the influence of non-local conjugation effects on local aromaticity?


runeddb C6H6.49 17,20,21 1:2:3:4:5:6 > C6H6.global.log


runeddb C10H8.49 27,30,32:34 5:6:1:2:7:8:9:10:4:3 > C10H8.global.log

1 2

345

6

12

34

56

789

10 runeddb C10H8.49 27,30,32:34 1:2:3:4:5:6 > C10H8.local.log

Ben

zene

Nap

htha

lene



runeddb C16H10.49 39,43,46,49:53 1:2:3:4:5:6:11:12:13:14:15:16:17:18:22:23 > C16H10.global.log

1211

12

6

3 4 515

14

22 23

181716

runeddb C16H10.49 39,43,46,49:53 1:2:3:4:5:6 > C16H10.local.A.log

A

B

C

D13

runeddb C16H10.49 39,43,46,49:53 12:13:14:15:22:23 > C16H10.local.B.log

runeddb C16H10.49 39,43,46,49:53 11:12:13:14:15:16:17:18:22:23 > C16H10.local.C.log

runeddb C16H10.49 39,43,46,49:53 2:3:11:12:13 > C16H10.local.D.log

Flu

oran

then

e



6.501

3.211

3.211

5.327

11.828

3.139

0.9104.828

6.414

12.387 (+5%)

(-1%)

(-9%)

(-2%)

3.139(-2%)

Numbers inside rings stand for the (total) local EDDB populations. The outermost numbers are the (total) global EDDB populations Percentages in brackets refer to the effect of linking of benzene ring to naphthalene.

-40% compared to benzene



3.994

5.327

4.290

4.004 5.367

7.018 (+10%)

(+11%)

(+37%)

Numbers inside rings indicate the sum of the corresponding atomic contri-butions to the global EDDB population. Percentages in brackets show the effect of non-local conjugations on local aromaticity of particular ring..

4.290(+37%)

(+24%)

3.994(+24%)

(0%) +340%

The non-local conjugation effects are especially important in the cases of fused aromatic rings. But the the global EDDB populations should be used with a special caution due to overlap overcounting.



Numbers above structures are (total) local EDDB populations. Numbers inside rings stand for the sum of the selected (coloured) atomic contributions to the local EDDB population. Numbers in brackets refer to the corresponding populations before orthogonal similarity transformation.

4.672(4.832)

0.156(0.000)

96.8%3.2%

0.061(0.000)

0.197(0.000)

91.8%

6.3%

1.9%

2.881(3.153)

The more effective conjugation between cyclic subunits, the higher „diffusion” of local EDDB populations due to orthogonal similarity transformation.

4.828 3.139


Method

39/40

Plans for future

(1) Energy decomposition based on the EDDB method.

(2) Post-HF correlated wavefunctions generalization.

Program

(1) A new Fortran 90 implementation of the method.

(2) Support for other quantum-chemical programs

(3) Different representations of the minimal-basis AOs

(4) Automatic dissection into σ- and π-components.

(5) Built-in cube generating module.

(6) New functionalities (e.g. detection of hydrogen bonds)


A complete presentation of the EDDB method is available at

www2.chemia.uj.edu.pl/~szczepad/UdG.Workshop.pdf

Theoretical foundations of the bond-orbital projection formalism

Documents

Transcript of Theoretical foundations of the bond-orbital projection formalism