This journal is c the Owner Societies 2011 Phys. Chem. Chem. Phys., 2011, 13, 20659–20665 20659

Cite this: Phys. Chem. Chem. Phys., 2011, 13, 20659–20665

Aromatic pathways in mono- and bisphosphorous singly Mobius

twisted [28] and [30]hexaphyrinsw

Heike Fliegl,*aDage Sundholm

aand Fabio Pichierri

b

Received 14th June 2011, Accepted 6th September 2011

DOI: 10.1039/c1cp21935a

Magnetically induced current densities and strengths of currents passing through selected bonds

have been calculated for monophosphorous [28]hexaphyrin ((PO)[28]hp) and for bisphosphorous

[30]hexaphyrin ((PO)2[30]hp) at the density functional theory level using our gauge-including

magnetically induced current (GIMIC) approach. The current-density calculations yield quantitative

information about electron-delocalization pathways and aromatic properties of singly Mobius

twisted hexaphyrins. The calculations confirm that (PO)[28]hp sustains a strong diatropic ring

current (susceptibility) of 15 nA T�1 and can be considered aromatic, whereas (PO)2[30]hp is

antiaromatic as it sustains a paratropic ring current of �10 nA T�1. Numerical integration of the

current density passing through selected bonds shows that the current is generally split at the

pyrroles into an outer and an inner pathway. For the pyrrole with the NH moiety pointing

outwards, the diatropic ring current of (PO)[28]hp takes the outer route across the NH unit,

whereas for (PO)2[30]hp, the paratropic ring current passes through the inner CbQCb double

bond. The main diatropic ring current of (PO)[28]hp generally prefers the outer routes at the

pyrroles, whereas the paratropic ring current of (PO)2[30]hp prefers the inner ones. In some cases,

the ring current is rather equally split along the two pathways at the pyrroles. The calculated

ring-current pathways do not agree with those deduced from measured 1H NMR chemical shifts.

1 Introduction

Among the different criteria to assess the aromatic or antiaromatic

character of conjugated molecules, the magnetic criterion

represents a powerful and quantitative approach. In the

presence of an external magnetic field applied in the perpendicular

direction to the molecular plane of planar aromatic molecules,

a net diatropic ring current is generated with the associated

magnetic field opposing the applied one. On the other hand,

the magnetic response of an antiaromatic molecule involves a

net paratropic ring current thus having the induced magnetic

field in the same direction as the external one. The current

pathways and the strength of the induced currents circling

around molecular rings can be unambiguously computed with

the aid of our gauge-including magnetically induced current

(GIMIC) method.1,2 The gauge-independent GIMIC approach has

been applied in aromaticity studies of a variety of molecular

systems.1–17 A recent study showed that GIMIC calculations can

even be used as a noninvasive tool for estimating strengths of

hydrogen bonds.18 The GIMIC method and its applications are

briefly reviewed in a perspective article of this PCCP issue.19

Our recent study on the magnetically induced current

density of doubly twisted hexaphyrins showed that their

aromatic properties follow the generalized Huckel rule,13

which states that Mobius twisted molecules with an even

linking number (LK) are aromatic when having [4n + 2] pelectrons and antiaromatic when the number of p electrons is

[4n].20–22 Planar Huckel-type aromatic molecules are a special

case with LK = 0. For molecules consisting of smaller

molecular rings connected to a macroring such as hexaphyrins,

the ring current can either take the outer or the inner route at

the pyrroles. Alternatively, the ring current is split into two

branches at the pyrroles. For twisted and complex molecular

structures such as the hexaphyrins with Mobius topology, it is

practically impossible to determine the aromatic pathway

using spectroscopic data because internal interactions signifi-

cantly affect the measured nuclear magnetic resonance (NMR)

spectra. Indirect assessment of the ring-current pathway by

nucleus-independent magnetic shielding calculations is also

very hard because the molecules are nonplanar.23–25 Detailed

electron-delocalization pathways of complex molecular struc-

tures are difficult to estimate reliably without performing

explicit current-density calculations.10,13

aUniversity of Helsinki, Department of Chemistry, P.O. Box 55(A.I. Virtanens plats 1), FIN-00014 University of Helsinki, Finland.E-mail: Heike.Fliegl@helsinki.fi, Dage.Sundholm@helsinki.fi

b Tohoku University, G-COE Laboratory, Department of AppliedChemistry, Graduate School of Engineering, Tohoku University,Japan. E-mail: fabio@che.tohoku.ac.jp

w Electronic supplementary information (ESI) available: Cartesiancoordinates and nuclear magnetic shieldings of the studied moleculesas well as magnetic field directions for obtaining the current strengthfunction. See DOI: 10.1039/c1cp21935a

PCCP Dynamic Article Links

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20660 Phys. Chem. Chem. Phys., 2011, 13, 20659–20665 This journal is c the Owner Societies 2011

Here, we apply the GIMIC method to investigate the

magnetically-induced current density of the recently synthesized

formally aromatic and antiaromatic singly Mobius twisted

hexaphyrins with [28] and [30] p electrons, respectively. According

to the generalized Huckel rule, Mobius aromaticity is expected

for twisted ring-shaped molecules with [4n] p electrons, when

the main molecular ring has Mobius topology with an odd LK

value.22,26 Analogously, Mobius twisted molecules with an

odd LK and [4n + 2] p electrons are expected to be Mobius

antiaromatic. The hexaphyrin rings synthesized by Higashino

et al.27 are forced to have a singly twisted Mobius topology

by cross linking three of the pyrrole rings with PQO units.

The first PQO unit increases the number of p electrons that

can participate in the aromatic pathway by two as compared

to hexaphyrins without the corresponding imine hydrogens

as PQO forms covalent bonds with two of the imine nitrogens.

The second PQO unit contributes with three covalent bonds

to the imine nitrogens but it increases the number of p electrons

only by two, because one bond replaces an inner NH bond

of (PO)[28]hp. The cross-linked hexaphyrin with one PQO

unit is singly twisted. It is expected to be aromatic as it has [28]

p electrons. The hexaphyrin PQO complex with two PQO

units is expected to be antiaromatic as it is singly Mobius

twisted having [30] p electrons that can participate in the

conjugated electron-delocalization pathway. The singly

twisted hexaphyrin PQO complex with [30] p electrons is the

first formally antiaromatic singly twisted Mobius molecule

that has been synthesized, whereas the first singly twisted

formally aromatic Mobius molecule was bianthraquinodi-

methane-stabilized [16]annulene,28,29 which however was later

proven to be nonaromatic.10,30

Expanded porphyrins represent an ideal class of molecules

for experimental and computational studies of Mobius aro-

maticity and antiaromaticity because they are very flexible.

The topology of the hexaphyrins can be adjusted by cross

linking the pyrrole rings. The number of p electrons of the

electron-delocalization pathway can be controlled by varying

the number of pyrrole units. Hydrogenation, metal coordina-

tion, or complexation of the imine nitrogens also determines

the number of p electrons of the macroring. The topology of

the expanded porphyrins might also be affected by physical

means such as changes in the temperature or by varying the

solvent.27,31–34

2 Computational methods

The molecular structures were optimized at the density

functional theory (DFT) level using Becke’s three-parameter

functional combined with the Lee–Yang–Parr exchange–

correlation functional (B3LYP).35,36 The molecular structures

were fully relaxed using the Karlsruhe triple-z basis set aug-

mented with polarization functions (def2-TZVP).37 Since the

investigated molecules are large and floppy, no force constant

calculations were performed. Ring-current susceptibilities

yielding ring-current strengths for a given value of the external

magnetic field were calculated at the B3LYP level using a split-

valence quality basis set augmented with polarization func-

tions (def2-SVP).37 The electronic structure calculations were

performed with TURBOMOLE.38–42 The magnetically induced

current densities were obtained using the GIMIC method.1,2

GIMIC is an independent program that uses basis-set information

as well as the perturbed and unperturbed density matrices

from nuclear magnetic shielding calculations as input data.1

Since Gauge-Including Atomic Orbitals (GIAO), also called

London orbitals,40,43–45 are employed in GIMIC calculations,

accurate gauge-independent current densities are obtained

with SVP quality basis sets.

GIMIC calculations provide detailed information about

molecular aromaticity and current pathways.3,4,8,10 The ring-

current susceptibility, denoted in the following as ring-current

strengths, can be used as a reliable measure of the molecular

aromaticity. The current strengths are obtained by numerical

integration of the current density passing through cut

planes perpendicular to selected bonds of the molecular

system. The ring-current strength for benzene calculated at

the B3LYP level is 11.9 nA T�1, which can be used as a

reference value for a typical aromatic molecule. Calculations

of the current strength for selected bonds yield the electron-

delocalization pathways responsible for the current transport

around the connected molecular rings. The sign and magni-

tude of the obtained ring currents indicate whether mole-

cular rings are aromatic, antiaromatic, or nonaromatic thus

having diatropic, paratropic, or vanishing net ring current,

respectively.11

The strongest ring currents are obtained when the external

magnetic field is largely perpendicular to the studied molecular

ring. However, the twisted topology of the investigated mole-

cules makes an unambiguous orientation of the magnetic field

with respect to a selected bond difficult. A careful investigation

of the strength of the magnetically induced current with

respect to the direction of the applied magnetic field has been

performed using the same procedure as described in previous

work.13 The GIMIC calculations with different orientations of

the external magnetic field show how the current transport

and the aromatic properties depend on the direction of the

magnetic field relative to the molecule. The directions of the

employed magnetic fields are given in the ESI.w

3 Results and discussion

3.1 Molecular structures

The optimized molecular structures of monophosphorous

[28]hexaphyrin ((PO)[28]hp) and bisphosphorous [30]hexaphyrin

((PO)2[30]hp) are shown in Fig. 1. The Cartesian coordinates

are given in the ESI.w (PO)[28]hp has one PQO moiety

coordinated to the imine nitrogen atoms of pyrrole rings E

and F and to the Cb atom of pyrrole ring A. Hence, the N–H

bond of pyrrole ring A points outwards with respect to the

Mobius ring centre. The remaining pyrrole rings B, C, and D

are not coordinated to PQO. Pyrrole C has a hydrogenated

imine nitrogen. As a consequence of their coordination to the

PQO moiety, the pyrrole rings A, E and F of (PO)[28]hp

adopt a cone-like orientation, whereas the pyrrole rings B and

D lie below and above the mean plane of pyrrole ring C.

Interestingly, this type of conformation is also maintained in

(PO)2[30]hp, where an additional PQO moiety coordinates

imine nitrogen atoms of pyrroles B, C, and D. In order to

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maintain the singly-twisted conformation, the angle between

the P–N bond and the centre of ring D is significantly out of

plane by 1461. Also the angles with the Ca atoms of ring D

(P–N–Ca) differ considerably from each other being 1361 and

1061. We note that one of the six pentafluoro-phenyl rings of

(PO)[28]hp and (PO)2[30]hp is oriented toward the centre of

the Mobius ring and in (PO)2[30]hp it appears to be stacked

with pyrrole ring B at a distance of 4.3 A. Van der Waals

interactions, which are not taken into account at the B3LYP

level, might affect the molecular structures. However, the

floppy hexaphyrins are large for structural optimization at

ab initio correlation levels considering dispersion effects. The

semi-empirical dispersion correction (+D) of Grimme46,47

would though take van der Waals effects into account in the

B3LYP calculations. However, because it is unclear how the

magnetic properties and current densities are affected by

the semi-empirical van der Waals terms, we did not employ

the +D correction in this work.

The P–N bond distances in (PO)2[30]hp range from 1.684 A

to 1.773 A, the latter corresponding to that of the distorted

P–N bond discussed above, while the P–N bonds in

(PO)[28]hp correspond to 1.700 A and 1.729 A. The P–C

bonds in the two porphyrnoids do not differ significantly from

each other, being 1.774 A and 1.742 A in (PO)[28]hp and

(PO)2[30]hp, respectively.

The topology of the singly twisted Mobius molecules was

determined by explicit calculations of the twist (TW), writhe

(WR), and linking numbers (LK), listed in Table 1. The twist

TW, writhe WR, and linking numbers LK were obtained using

the Linking-number code of Rappaport and Rzepa.20 The

calculations were performed along monocyclic pathways. For

both molecules the inner pathway has been used. However, in

a recent study on doubly twisted hexaphyrins it was shown

that the results for TW, WR and LK obtained using the inner

and the outer pathway do not significantly differ.13 The linking

number LK is the sum of two chiral indices twist (TW) and

writhe (WR), which defines the molecular topology. Positive

and negative twist and writhe values correspond to different

chiralities. The TW number correlates with the local atomic

overlap of p orbitals. A strongly twisted molecule has a large

TW number, implying a smaller p orbital overlap. The WR

number provides a measure of the strain of the molecular ring.

Both molecules are singly twisted having LK = 1. Thus,

according to the generalized Huckel rule,20,22 (PO)[28]hp is

expected to be aromatic and (PO)2[30]hp antiaromatic.

(PO)2[30]hp has a WR value of �0.72 as compared to 0.46

for (PO)[28]hp showing that the second PQO group introduces

additional strain into the hexaphyrin ring. For (PO)[28]hp, the

TW andWR numbers of 0.54 and 0.46, respectively, are almost

identical, whereas for (PO)2[30]hp the TW number is 1.72 and

WR is �0.72. The strong twist introduced by the second PQO

group seems to be compensated by a large writhe.

3.2 Magnetically induced current densities

Magnetically induced current pathways and strengths of the

currents passing through selected bonds have been calculated

for the investigated hexaphyrins in the same way as described

in previous work.13 For obtaining information about the

current pathways within (PO)[28]hp and (PO)2[30]hp, a detailed

investigation of the current strength at the involved pyrrole rings

has been performed. The current strengths were obtained by

numerical integration of the current density passing through

planes perpendicular to chemical bonds as illustrated in Fig. 2.

A comparison of the current-strength functions with respect to

the direction of the applied external magnetic field is given

in Fig. 3. In the study of the current pathways, the direction of

the applied external magnetic field has been kept fixed at

the respective maximum or minimum of the current-strength

function. The results for the obtained current pathways around

the whole molecular macrorings at the pyrroles are given in

Tables 2 and 3. The pyrrole rings are labelled A to F as depicted

in Fig. 1.

Investigation of the current pathways at the pyrrole subrings

and around the macrorings of the present hexaphyrins is more

demanding as compared to our previous study of doubly

twisted hexaphyrins.13 The inclusion of the phosphorous

oxides, which are directly coordinated to the pyrrole rings

makes the macroring congested by keeping the pyrrole rings

rather close to each other. The placement of the integration

Fig. 1 Molecular structures of the investigated monophosphorous

[28]hexaphyrin and bisphosphorous [30]hexaphyrin including labelling

A to F of the pyrrole subunits.

Table 1 Calculated twist TW, writhe WR and linking numbers LK for(PO)[28]hp and (PO)2[30]hp in units of p (1801)

Molecule TW WR LK

(PO)[28]hp 0.54 0.46 1.00(PO)2[30]hp 1.72 �0.72 1.00

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planes for calculating the current strength passing through

selected bonds is therefore difficult as current densities belonging

to neighbouring parts of the molecule have to be avoided. The

reported current strengths in Tables 2 and 3 cannot be seen

as precise values as some current contributions from neigh-

bouring parts of the molecule could have been taken into

account or some of the relevant currents might pass outside

the integration planes. However, these current contributions

are though small implying that the obtained current routes are

reliable.

3.2.1 Monophosphorous [28]hexaphyrin. The current-

strength function of the monophosphorous [28]hexaphyrin

((PO)[28]hp) with respect to the direction of the applied external

magnetic field has a maximum value of 14.9 nA T�1, which can

be compared to the current strength for benzene of 11.9 nA T�1

obtained at the same computational level. The obtained current

strength indicates that (PO)[28]hp is aromatic according to the

ring-current criterion.

The ring current is split at the pyrrole rings into two

branches, as also obtained for the doubly twisted hexaphyrins13

and for free-base porphyrins.1 At the pyrroles A and D, the

C–N–C route is the preferred one, whereas the ring current at

the pyrroles C, E, and F flows mainly via the CQC double

bond. At ring B, the current via C–N–C is 9.4 nA T�1 and the

CQC one is 7.1 nA T�1. The current is thus almost equally split

along the outer and the inner routes of pyrrole B. For ring A,

the current flow of 13 nA T�1 via C–N–C dominates. The

current strength along the inner CQC bond is only 0.7 nA T�1.

Pyrroles E and F are coordinated to PQO via the imine

nitrogens, whereas ring A is coordinated to P at Cb. The

calculations indicate that a weak current of about 1–2 nA T�1

passes via the PQO group from ring A to F, whereas the

current strength of 0.1 nA T�1 between E and PQO is very

small. The obtained results show that the current pathway

around the main macromolecular ring in (PO)[28]hp is not

significantly influenced by the PQO group, almost no current

takes the short cut from pyrrole A to E via PQO. The main

ring current flows via the pyrroles A, E, and F on the outside

of the macromolecular ring as schematically shown in Fig. 4.

3.2.2 Bisphosphorous [30]hexaphyrin. The current-strength

function for bisphosphorous [30]hexaphyrin ((PO)2[30]hp) as a

function of the direction of the applied external magnetic field

is shown in Fig. 3. The minimum of the current-strength

function is �10.3 nA T�1 implying that (PO)2[30]hp is anti-

aromatic according to the ring-current criterion. The ring

current for (PO)2[30]hp branches at the pyrrole rings into

outer and inner pathways. A clear preference for the inner

route via the CQC bond is seen at pyrrole A. For the pyrroles

C and E, the inner current path via the imine nitrogens

dominates, while at the pyrroles B, D, and F the ring current

is almost equally split, showing no preference for the inner or

outer pathway as schematically depicted in Fig. 5.

The calculations show that a current of about 2–3 nA T�1

takes the short cut from A to E across the main molecular

ring via the PQO group. The calculations also indicate that

Fig. 2 Schematic illustration of the placement of the integration

planes for (PO)[28]hp.

Fig. 3 Comparison of the current strengths of (PO)[28]hp and

(PO)2[30]hp calculated as a function of the direction of the external

magnetic field. The largest current strengths are 14.9 nA T�1 and

�10.3 nA T�1, respectively. The integration planes were placed

perpendicular to bond C80–C100 for (PO)[28]hp and bond C54–C66

for (PO)2[30]hp. See ESIw for details.

Table 2 The strengths of the magnetically induced currents (in nA T�1)flowing along the CQC and C–N–C routes of the pyrrole rings of(PO)[28]hp. The A to F labeling of the pyrrole rings is given in Fig. 1.Pyrrole rings with an NH hydrogen are marked with *. The maximum ofthe current-strength function with respect to the angle of the appliedmagnetic field is 14.9 nA T�1

Molecule Ring/Path- C–N–C CQC

(PO)[28]hp A* 12.8 0.7B 9.4 7.1C* 5.5 11.1D 11.1 4.8E 3.3 12.6F 2.2 13.0

Table 3 The strengths of the magnetically induced currents (in nA T�1)passing via the CQC and C–N–C pathways at the pyrroles of(PO)2[30]hp. The A to F labeling of the pyrrole rings is given in Fig. 1.Pyrrole rings with an NH hydrogen are marked with *. The minimum ofthe current-strength function with respect to the angle of the appliedmagnetic field is �10.3 nA T�1

Molecule Ring/Path- C–N–C CQC

(PO)2[30]hp A* 2.4 �11.0B �3.5 �4.2C �8.5 0.2D �4.9 �4.5E �6.2 �1.0F �2.8 �2.8

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about 2 nA T�1 pass on the other side of the macroring

between pyrroles D and B via the second PQO group. The

PQO groups influence the current pathways of the hexaphyrins

by introducing alternative current transport routes. For the

antiaromatic (PO)2[30]hp, the PQO groups increase the

paratropic ring current via the inner routes of the pyrrole

rings. A relatively weak current of a few nA T�1 also takes the

short-cut routes via the PQO groups. The main ring current

flows via all pyrrole rings. The PQO groups seem to enhance

the antiaromaticity of the hexaphyrin ring, as the ring current

is more paratropic than for the corresponding hexaphyrin with

the two PQO groups replaced by six imine hydrogens.

4 Discussion

The present GIMIC calculations on the singly Mobius twisted

hexaphyrins show that a monophosphorous [28]hexaphyrin

((PO)[28]hp) sustains a net diatropic ring current and that the net

ring current of a bisphosphorous [30]hexaphyrin ((PO)2[30]hp)

is paratropic. Thus, our results confirm the recent experimental

assignment that (PO)[28]hp is aromatic and (PO)2[30]hp is

antiaromatic.27

The calculations show that the PQO moiety in (PO)[28]hp

influences the current pathways only slightly. A weak current

of about 2 nA T�1 is transported over the PQO moiety from

ring A to ring F. It is also in principle possible for the ring

current to take a short cut across the main molecular ring

from pyrrole A to E. However, this is not the case for

(PO)[28]hp at least not with the assumed direction of the

external magnetic field. The main diatropic ring current of

(PO)[28]hp flows on the outside of the pyrrole rings A, F, and

E, which are connected to PQO. At B, C, and D the main

ring current flows on the inside along C–N–C of the pyrrole

rings and on the outside along CQC when the respective

ring contains an NH hydrogen, as expected for porphyrins.

However, significant currents also take the alternative routes

as seen in Table 2.

The addition of a PQO group has the same effect as the

hydrogenation of the imine nitrogens, each coordination to a

nitrogen increases the number of p electrons in the conjugation

pathway by one. The first PQO group coordinates to two

imine nitrogens and one Cb. The second coordinates to three

imine nitrogens of which one coordination replaces an imine

hydrogen, the net increase in p electrons of the second PQO

group is therefore two. The (PO)2[30]hp has all but one imine

nitrogen coordinated to PQO groups, the only hydrogenated

imine nitrogen is pointing outwards. The (PO)2[28]hp has two

hydrogen coordinated imine nitrogens, the one at pyrrole A

points outwards and the second one at C is directed inwards.

Determination of the current pathways in (PO)2[30]hp turns

out to be very complicated. The two PQO moieties introduce

several possibilities for the current to cross the main molecular

ring. The GIMIC calculations show that a current whose

strength is about 3 nA T�1 flows via PQO between pyrroles

A and E and from pyrrole B to D. It is difficult to determine

accurate current strengths, because the twisted topology of the

molecule makes it very challenging to avoid neighbouring

currents. However, the calculations suggest that the paratropic

current density on the inner C–N–C side of the pyrroles

increases due to the complexation with the PQO moiety.

The investigation of possible local aromatic pyrrole rings of

(PO)[28]hp and (PO)2[30]hp is beyond the scope of the present

study. However, in a recent study of doubly twisted hexaphyrins

it seemed that the pyrrole rings do not sustain strong currents

on their own.13 A study of magnetically induced currents

concerning possible local aromatic pyrrole rings should start

with more simple systems. For example, the current pathways

for planar porphyrins are still under discussion.1,48,49

For both investigated molecules the current is split at the

pyrrole rings. The dominating route for the current flow

around the macromolecular ring obtained in the GIMIC calcu-

lations is depicted for (PO)[28]hp in Fig. 4 and for (PO)2[30]hp

in Fig. 5. The study shows that it is mandatory to perform

current-density calculations to understand how magnetically

induced currents flow in twisted hexaphyrins. In the present

study, we have adopted only one direction of the external

magnetic field. Repeating the study using other directions of

the magnetic field could provide a more nuanced picture of the

flow of the magnetically induced currents in Mobius twisted

Fig. 4 Schematic illustration of the current pathway around the

macromolecular ring of (PO)[28]hp. Only the dominating current

pathways at the pyrroles are indicated. The ring current is split into

two branches at all pyrrole rings. The current strengths of the path-

ways are given in Table 2.

Fig. 5 Schematic illustration of the current pathway around the

macromolecular ring of (PO)2[30]hp. Only the dominating current

pathways at the pyrroles are indicated. The ring current is split into

two branches at all pyrrole rings. The current strengths of the path-

ways are given in Table 3.

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hexaphyrins. Such extensive calculations would though yield

limited new insights.

5 Summary and conclusions

Magnetically induced current densities, current pathways,

and current strengths of recently synthesized singly Mobius

twisted monophosphorous [28]hexaphyrin ((PO)[28]hp) and

bisphosphorous [30]hexaphyrin ((PO)2[30]hp) compounds27

have been investigated and analyzed using the GIMIC method.1

The aromatic character of the investigated mono- and bispho-

sphorous hexaphyrins has been assigned according to the ring-

current criterion. Calculations of the ring-current strengths

passing around the macrorings show that (PO)[28]hp is aro-

matic sustaining a net diatropic ring current of 14.9 nA T�1

and that (PO)2[30]hp is antiaromatic with a net paratropic ring

current of �10.3 nA T�1, which agree with experimental

findings.27 However, a detailed study of the current pathways

yielded electron-delocalization pathways that significantly

differ from those proposed by the experimentalists.27 Explicit

calculations of the current pathways in (PO)[28]hp and

(PO)2[30]hp show that the ring currents are split at each

pyrrole ring into two branches. Some current flows via C–N–C

and the rest passes through the CQC bond of the pyrroles, as

also previously found in the studies on doubly twisted Mobius

hexaphyrins13 and on free-base porphyrins.1 The paratropic

currents of the antiaromatic (PO)2[30]hp generally prefer the

inner pathway, whereas the diatropic currents of the aromatic

(PO)[28]hp are stronger along the outer route. Current-density

calculations have shown that diatropic currents generally flow

along the outer surface of molecules, whereas the paratropic

ones usually appear in the interior of the molecular rings.11

These simple rules do have a few exceptions for the studied

hexaphyrins. The only reliable means to determine the current

pathway seems to be explicit calculations of the current

strengths.

The current pathway deduced from experiment follows the

traditional aromatic pathway of porphyrins i.e., the outer route

at the pyrroles with a hydrogenated imine nitrogen and the

inner one where the imine nitrogen is uncoordinated. However,

calculations on free-base porphyrins showed that the ring

current of porphyrins is evenly split along the two pathways

when the imine nitrogen is uncoordinated, whereas for the

hydrogenated ones roughly one third of the ring current passes

via the inner route.1 The current-pathway pattern of the twisted

hexaphyrins is a generalization of the one obtained for free-base

porphyrins. The strengths of the currents along the possible

branches can be estimated by numerical integration of the

current density passing through the bonds.

The influence of the PQOmoiety on the dominating current

pathways has also been investigated. We found that the PQO

moiety allows alternative current pathways, but for both

molecules a rather weak current takes the route over the

PQO group(s). For (PO)[28]hp the main current flows on

the outside of the pyrrole rings, while for (PO)2[30]hp the

paratropic current prefers mainly the inner pathways. The

paratropic currents of (PO)2[30]hp also take short-cut routes

across the main molecular ring. Due to the PQO complexa-

tion, the paratropic current density increases on the inside of

the main molecular ring as compared to the nonphosphonated

but completely hydrogenated hexaphyrin.

Acknowledgements

This research has been supported by the Academy of Finland

through its Centers of Excellence Programme 2006–2011. F.P.

thanks the Global COE program of Tohoku University for

financial support. We thank Prof. Rzepa for a copy of the

Linking-number program. We also thank CSC—the Finnish

IT Center for Science—for computer time.

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