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PAPER
www.rsc.org/pccp | Physical Chemistry Chemical Physics
Non-perturbative magnetic phenomena in closed-shell
paramagnetic molecules
Erik I. Tellgren, Trygve Helgaker and Alessandro Soncini*w
Received 11th December 2008, Accepted 12th March 2009
First published as an Advance Article on the web 16th April 2009
DOI: 10.1039/b822262b
By means of non-perturbative ab initio calculations, it is shown that paramagnetic closed-shell
molecules are characterized by a strongly non-linear magnetic response, whose main feature
consists of a paramagnetic-to-diamagnetic transition in a strong magnetic field. The physical
origin of this phenomenon is rationalised on the basis of an analytical model based on molecular
orbital theory. For the largest molecules considered here, the acepleiadylene dianion and the
corannulene dianion, the transition field is of the order of 103 T, about one order of magnitude
larger than the magnetic field strength currently achievable in experimental settings. However, our
simple model suggests that the paramagnetic-to-diamagnetic transition is a universal property of
paramagnetic closed-shell systems in strong magnetic fields, provided no singlet–triplet level
crossing occurs for fields smaller than the critical transition field. Accordingly, fields weaker than
100 T should suffice to trigger the predicted transition for systems whose size is still well within
the (medium–large) molecular domain, such as hypothetical antiaromatic rings with less than one
hundred carbon atoms.
I.
Introduction
The full ab initio characterisation of closed-shell molecules
interacting with external uniform magnetic fields has mostly
been based on the assumption that the molecular energy W(B)
as a function of magnetic field B can be approximated in
terms of a rapidly convergent polynomial expansion given by
(Einstein summation convention)
1
WðBÞ ¼ Wð0Þ $ wab Ba Bb
2
1
$ Xabgd Ba Bb Bg Bd þ & & &
24
ð1:1Þ
where W(0) is the molecular energy in zero-field, wab is the
second-rank magnetisability and wabgd is the fourth-rank
hypermagnetisability. Since the magnetic interaction energy
is proportional to the magnetic flux—that is, to the product
between the external field strength and the area of the
perpendicular molecular surface—the polynomial ansatz is
clearly a very good one for small molecules subject to experimentally achievable magnetic fields, which at best can be of the
order of 102 T.1
However, with the recent progress in computational
techniques, ab initio quantum chemistry is nowadays expanding its traditional domain of interest to larger systems. For
such systems, the effect of a magnetic field of a few tens of tesla
can, in principle, and in a non-negligible way, change the
very nature of the molecular wave function, with important
Centre for Theoretical and Computational Chemistry, Department of
Chemistry, University of Oslo, Box 1033 Blindern, N-0315, Oslo,
Norway. E-mail: [email protected]
w Present address: Institute for Nanoscale Physics and Chemistry,
Afdeling Kwantumchemie, K.U.Leuven, Celestijnenlaan 200F, B-3001,
Heverlee, Belgium. E-mail: [email protected]
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observable consequences. Indicative examples are the
semiconductor-to-metal transition in carbon nanotubes
caused by a strong field along their axes2,3 and the very recent
observation of room-temperature quantum-Hall effect in diamagnetic graphene sheets.4 In the literature, a few studies have
appeared concerning the perturbative calculation of fourthrank hypermagnetizabilities,5,6 the induced non-linear ring
current in aromatic and antiaromatic molecules,7 and
the electric8 and magnetic9,10 hypershieldings at the nuclei.
However, apart from work on atoms and very small molecules
such as H2 (see ref. 11 and references therein), to the best of
our knowledge, the only quantum-chemical ab initio investigation of the non-perturbative behaviour of closed-shell molecules in strong magnetic fields was reported in ref. 12 by the
present authors.
One well-known problem affecting the convergence of
finite-basis ab initio calculations of magnetic properties is the
gauge-origin dependence of the results and the associated slow
basis-set convergence of the calculated properties. Many
techniques have been proposed to solve this problem, the most
efficient one being based on the use of London orbitals.13–15
Unfortunately, none of these techniques is straightforwardly
applicable to finite-field calculations. Recently, we presented a
scheme for the use of London orbitals with magnetic fields of
arbitrary strength12 and implemented this scheme in the
program LONDON,16 providing a gauge-origin invariant
approach to non-perturbative restricted Hartree–Fock
(RHF) self-consistent field (SCF) molecular calculations in
strong magnetic fields.
The purpose of the present paper is to demonstrate,
by means of gauge-origin invariant non-perturbative SCF
calculations, a very simple paradigm: due to strongly nonlinear effects, closed-shell paramagnetic systems become
diamagnetic in high magnetic fields. We show here that the
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perturbative approach based on eqn (1.1) is of little use for
describing these strong non-linearities and that, instead, nonperturbative methods become necessary. We note here that
interesting magnetic behaviours are always associated with
quasi-degeneracies in the electronic spectrum, so that a multireference treatment would be more appropriate for making
quantitative predictions. However, only the SCF approach
presented in ref. 12 is available for the time being and,
especially for the closed-shell systems considered here, the
qualitative features captured by SCF are expected to survive
more accurate approaches accounting for non-dynamic as
well as dynamic correlation. Consequently, all calculations
presented in this paper have been carried out at the RHF level
of theory. We also limit our study to purely electronic effects,
since our present methodology does not allow us to study the
effect of strong magnetic fields on molecular structure in
general. In this work, magnetic fields are expressed in atomic
units (a.u.) of magnetic flux density. Conversion to SI units (T)
5
is readily given by 1 a.u. = !he$1a$2
0 = 2.35 ( 10 T.
Hints of the strong magnetic non-linearity implied by
closed-shell paramagnetism were already given in previous
works.7,12 In this study, those preliminary statements are
confirmed for a whole range of paramagnetic closed-shell
systems. Moreover, a rationalization of the underlying physics
is provided on the basis of a simple analytical model.
II. Ab initio non-linear response of closed-shell
paramagnetic systems
Fig. 1 Scheme representing the eight paramagnetic closed-shell
molecules investigated in this work.
It is well known that paramagnetism is, in most cases, a
temperature-dependent phenomenon, originating from the
unpaired electrons of an open-shell species. However, as early
as in the beginning of the last century, a number of closed-shell
substances were found to have a paramagnetic susceptibility
quite independent of the temperature, one of the earliest
examples being the permanganate ion, MnO4$.17 A general
theory (for closed- and open-shell systems) of temperatureindependent paramagnetism in the weak-field limit was first
introduced by van Vleck.18 At first, the theory for paramagnetism of closed-shell molecules was challenged by a
theorem proving that the magnetizability of closed-shell systems
is always negative, arguing on the basis of a gauge transformation designed to annihilate the paramagnetic contribution to the
total magnetizabilty. However, a fundamental flaw in the proof
was later discovered by Hegstrom and Lipscomb,19 who
confined the validity of the gauge transformation (thus of the
theorem) to systems with less than four electrons. Hence,
closed-shell molecules with at least four electrons can, in
principle, be paramagnetic in the spirit of van Vleck.
We consider here three main classes of systems displaying
some degree of closed-shell paramagnetism (see Fig. 1) and
investigate their response to strong magnetic fields using the
ab initio package LONDON.12,16 The first class consists of
well-known small paramagnetic molecules such as the hydrides
BH 1 and CH+ 2,20–23 and the anion MnO4$.24 The second class
consists of planar [4n]-carbocycles. As is well known,25,26 the
linear magnetic response of these systems is characterized
by paramagnetic ring currents due to p-electrons, which
may or may not result in global closed-shell paramagnetism.
According to a widely accepted definition of aromaticity and
antiaromaticity based on the magnetic criterion, such systems
are here referred to as antiaromatic. Examples of experimentally characterised antiaromatic planar [4n]-carbocycles
are clamped cyclo-octatetraene (COT, n = 2), fully annelated
both with perfluorocyclobuteno groups,27 with bicyclo[2.1.1]hexene groups,28–30 and the dehydro[12]-annulene (n = 3)
fused with bicyclo[2.2.2]octene frameworks,30,31 for which
NMR measurements confirmed the existence of a paramagnetic ring current. We here consider the first three
members of the [4n]-carbocycles series, which represent
ab initio models for the conjugated planar moieties of the
synthetic molecules: cyclobutadiene 4 (n = 1), COT 5 (n = 2)
and [12]-annulene 6 (n = 3). Finally, we consider two organic
dianions that have been experimentally and computationally
characterised as closed-shell paramagnetic systems: the
acepleiadylene dianion C16H102$, 7,32,33 and the corannulene
dianion C20H102$, 8.34 Unless otherwise stated, all geometries
were optimized at the RHF/6-31G** level of theory, imposing
planarity to the annulene structures to mimic the electronic
structure of the above-mentioned synthetic molecules.
5490 | Phys. Chem. Chem. Phys., 2009, 11, 5489–5498
A
Small molecules
Several ab initio computational studies have been reported on
the magnetic properties of BH, CH+20–23 and MnO4$.24 At all
levels of theory, previous investigations on these systems
concur in finding their linear magnetizability to be positive,
characterising them as closed-shell paramagnetic molecules.
For BH and CH+, the component of the magnetizability
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perpendicular to the bond is paramagnetic, and large enough
to dominate the average response. To the best of our knowledge, no investigation of the response of these systems to
strong fields has appeared in the literature, with the exception
of a preliminary study on BH that we reported in a recent
work as a test case for our new methodology.12
In particular, in ref. 12 we found that the energy of BH as a
function of field presents a non-trivial behaviour that cannot
be easily reproduced by the expansion in eqn (1.1). The energy
versus field curve for BH has in fact a sombrero shape, with
minimum at B E ) 0.22 a.u. The field strength defining the
energy minimum characterises the transition of the system
from closed-shell paramagnetic to closed-shell diamagnetic.
This follows from the slope of the energy curve changed by
sign, which defines the total magnetic moment induced in the
system. In fact, the slope is negative before the transition field,
defining an induced magnetic moment parallel to the external
field—that is, a paramagnetic state. At the energy minimum
the slope is zero, characterising a state in which the induced
orbital magnetic moment is completely quenched. Beyond the
transition field, the induced magnetic moment opposes the
external perturbation (positive slope), the signature of a diamagnetic system. The basic question addressed in the present
work is thus whether the strongly non-linear behaviour leading
to a paramagnetic-to-diamagnetic transition observed for BH is
common to all paramagnetic closed-shell systems and, if so,
what is its underlying physical origin. In Fig. 2, we have plotted
the RHF ground-state energy against the magnetic field for a
series of London basis sets of increasing quality, for BH, CH+
and MnO4$. Whereas the largest basis for BH and CH+ is
aug-cc-pVDZ for both atoms, the largest MnO4$ basis consists
of an uncontracted Wachters + f basis35,36 on the Mn centre
and aug-cc-pVDZ on the oxygen atoms.
As is evident from Fig. 2, the features previously reported
for the energy versus field diagram of BH are observed for all
systems. The energy lowers when the field is switched on, it
reaches a minimum, and then raises again, displaying a typical
behavior of a paramagnetic (diamagnetic) system for very low
(high) fields. The details of the plots clearly depend on the
specific system and the basis set employed. In particular, we
notice that, for the largest basis sets used here, the critical field,
Bc, at the energy minimum for BH, CH+ and MnO4$ is found
at 0.23, 0.45 and 0.50 a.u., respectively.
However, it is important to note that, for such high fields,
the first triplet excited state also lowers its energy by the pure
spin-Zeeman interaction, so that the energy spectrum could, in
principle, display level crossing (LC) for B o Bc. To check this
possibility, let WS,MS(B) denote the energy of the (S,MS) level
at field strength B and denote by DWT = W1,$1(0) $ W0,0(0)
the triplet excitation energy at zero field. Neglecting spin–orbit
coupling and orbital effects on the triplet state, we then
estimate that the level crossing occurs at the field strength
BLC given by
W0,0(BLC) = W1,$1(BLC) = W0,0(0) + DWT $ gmBBLC,
(2.2)
where the isotropic g-factor is given by g = 2 and the Bohr
magneton by mB = 1/2 a.u. Computing the lowest triplet
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Fig. 2 Plots of the ground state HF energy variation (Eh) as a
function of the external magnetic field (atomic units of magnetic flux
density, !he$1a$2
= 2.35 ( 105 T) for (a) BH, (b) CH+, and
0
$
(c) MnO4 . The diagrams are calculated using a series of uncontracted
basis sets of increasing quality (see text, and the figure legends), all
augmented with London gauge factors.
excitation energy, DWT, in the random-phase approximation
(RPA) in the 6-31G** basis, we obtain, for BH, BLC E
0.32 a.u. (DWT = 0.2903 a.u.), for CH+ BLC E 0.48 a.u.
(DWT = 0.3584 a.u.), and for MnO4$ BLC E 0.006 a.u.
(DWT = 0.0056 a.u.). Comparing these values with the critical
field values reported in Fig. 2, we conclude that the energy
minimum can only be observed as a ground-state property for
BH and CH+.
B
Antiaromatic closed-shell carbocycles
In this work, we investigate three systems belonging to the
antiaromatic [4n]-carbocycle series: cyclobutadiene (n = 1),
flattened cyclo-octatetraene, and [12]-annulene. The results
of the finite-field calculations are plotted in Fig. 3.
Phys. Chem. Chem. Phys., 2009, 11, 5489–5498 | 5491
Fig. 3 Plots of the ground state HF energy variation for planarised (a) C4H4 (c) C8H8 and (d) C12H12 and of the p-electron orbital energy for (b)
5
C4H4 as a function of the external magnetic field (atomic units of magnetic flux density, !he$1a$2
0 = 2.35 ( 10 T). The diagrams are calculated
using a number of uncontracted basis sets of increasing quality (see text, and the figures inset), all augmented with London gauge factors.
Cyclobutadiene represents the only system of the series that is
globally diamagnetic. Even though it is characterised by a
paramagnetic response of its p electrons, this response is
counterbalanced and overwhelmed by the localized diamagnetic response of the s framework. However, as discussed
in ref. 12, the energy versus field curve for this system presents
marked features of strong non-linearity, in agreement with the
fact that its p-electron states can be viewed as a closed-shell
paramagnetic subsystem.
Because of the separability of the s–p one-electron states
based on symmetry, it is possible to plot an estimate of the
p-electron energy as function of field, approximating it by the
sum of the p-electron orbital energies. The results are reported
in Fig. 3b, where we can see that the pure closed-shell
paramagnetism of the p electrons is reflected in the characteristic sombrero-shaped curve. For the larger members of the
[4n]-carbocycles series, the paramagnetic response of the p
electrons dominates the linear magnetic response. In fact,
exactly as for the small closed-shell paramagnetic molecules
considered in the previous section, the energy curves for the larger
[4n]-annulenes C8H8 and C12H12 display the typical sombrero
shape, characterizing the paramagnetic-to-diamagnetic
5492 | Phys. Chem. Chem. Phys., 2009, 11, 5489–5498
transition of their RHF closed-shell ground state as a function
of the external field strength; however, for these larger
molecules, the critical field is about one order of magnitude
smaller than for the previous systems. We find Bc E 0.032 a.u.
for COT, and Bc E 0.016 a.u. for the [12]-annulene.
Clearly, the magnitude of the critical field Bc decreases with
system size. In particular, since orbital magnetism is proportional to the external magnetic flux, we can expect Bc to vary
roughly as the inverse of the area of the molecule. Therefore,
since the area of a regular N-sided polygon with sides of length
l is given by
AN ¼
! p " N2 ‘2
N‘2
*
cot
;
N
4
4p
ð2:3Þ
it follows that, if we take a ring six times larger than C12H12,
the energy minimum should be observed for a critical field of
about 100 T, which is within reach of current experimental
techniques. Therefore, according to this back-of-the-envelope
estimate, the ratio between the critical fields of C12H12 and
C8H8 is given by Bc,12/Bc,8 E A8/A12 E 0.43, comparable with
our best basis-set ab initio results Bc,12/Bc,8 = 0.5. For these
systems, the corresponding singlet–triplet level-crossing field is
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estimated to be BLC E 0.02 a.u. for COT (DWT = 0.0179 a.u.)
and BLC E 0.07 a.u. for C12H12 (DWT = 0.1395 a.u.), so that
the diamagnetic transition will only occur for the ground state
of the latter.
C Antiaromatic closed-shell polycycles: acepleiadylene
and corannulene dianions
We finally consider the non-perturbative magnetic response of
the acepleiadylene and corannulene dianions. Both systems
have been identified as closed-shell paramagnetic molecules by
means of ab initio calculations.33,34 The bowl-shaped structure
for the corannulene dianion was taken from ref. 34. The
results of the RHF calculations of the energy as a function
of applied perpendicular field are plotted in Fig. 4, where we
recognize the features characterising the paramagnetic–
diamagnetic transition observed for the smaller systems. The
magnitude of the transition field does not vary much with the
quality of the basis set. It is of the same order of magnitude as
for the antiaromatic rings considered in the previous section,
which is not surprising considering the similar spatial extent of
these systems. We find Bc = 0.023 a.u. (about 5400 T) for the
acepleiadylene dianion, similar to the COT value. Note that,
as reported in ref. 33, the paramagnetic ring current is mostly
localised on the seven membered ring, which is of about
the same size as COT. For the corannulene dianion, we find
Bc = 0.015 a.u. (about 3500 T).
The results for these polycyclic molecules appear to be
particularly interesting for engineering larger closed-shell
paramagnetic systems. It is well known that the introduction of odd-member ring defects—in particular, pentagonal
rings—in a carbon honeycomb lattice leads to an increase of
its antiaromaticity,37 which, by the magnetic criterion, corresponds to an increase of its paratropicity.37–39 Hence, graphene
flakes doped with acepleiadylene or corannulene dianion
moieties might be expected to give rise to extended closedshell paramagnetic structures, in which the paramagnetic–
diamagnetic transition occurs for relatively small critical fields.
The singlet–triplet level-crossing field for these polycyclic
systems is estimated to be BLC E 0.043 a.u. for C16H102$
(DWT = 0.05187 a.u.) and BLC E 0.035 a.u. for C20H102$
(DWT = 0.05158 a.u.). For both systems, therefore, the
diamagnetic transition is expected to occur in the ground
state.
III. An analytical model for the diamagnetic
transition
Why should we expect paramagnetic closed-shell species to
become diamagnetic in strong magnetic fields? To answer this
question, we consider a very simple model. Two distinguishing
features common to all closed-shell paramagnetic systems
consist of (i) a totally symmetric ground state (ii) the existence
of a low-lying excited state that has the symmetry of a
magnetic dipole operator (i.e., of a rotation about the external
field). All molecules considered here trivially fulfill the first
requirement and can also be shown to fulfill the second
requirement. We approximate here the lowest-lying states of
the molecular energy spectrum in terms of excitations among
frontier molecular orbitals (MOs), so that the lowest-lying
excited state corresponds to an excitation between the highest
occupied MO (HOMO) and the lowest unoccupied MO
(LUMO).
For instance, six-electron diatomic hydrides such as BH and
CH+ can be described in terms of a non-degenerate HOMO,
consisting of the sp hybrid (the lone pair) centered about
the heavier atom and pointing along the bond axis (say, the
z axis), and of a doubly degenerate LUMO, given by the
perpendicular p orbitals centered on the heavier atom
(px and py on carbon and boron). If we consider a magnetic
field perpendicular to the bond (e.g., along the y direction), the
molecular Hamiltonian can be written as
H = Hel + HB
Fig. 4 Plots of the ground state HF energy variation (Eh) as a
function of external magnetic field (atomic units of magnetic flux
density, !
he$1a$2
= 2.35 ( 105 T) for (a) acepleiadylene dianion
0
C16H102$ and (b) corrannulene dianion C20H102$. The diagrams are
calculated using a number of uncontracted basis sets of increasing
quality (see text, and the figure legends), all augmented with London
gauge factors.
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(3.4)
where Hel is the electrostatic Hamiltonian and HB represents
the magnetic interaction (in atomic units):
HB = BLa $ wdaaB2,
(3.5)
where a is the direction of the magnetic field (here y) of
strength B. The magnetic Hamiltonian contains two contribuP
tions: first, the orbital Zeeman term BLa, where La = ila,i is
Phys. Chem. Chem. Phys., 2009, 11, 5489–5498 | 5493
the component of the angular-momentum operator in the direction of the field (centered on the heavy atom); second, a diamagnetic term $wdaaB2, where wdaa is one half of the diamagnetic
P
susceptibility operator—for example, wdyy = $ i(x2i + z2i )/8.
Thus, denoting the HOMO by fH and the LUMO by fLx +
2px and fLy + 2py, we find that the Hamiltonian matrix
elements between the ground state C0 = |f1f!1f2f!2fHf!H|
and
the doubly degenerate first excited states
C1x
=
2$1/2|f1f!1f2f!2(fHf!Lx
+
fLxf!H)|
and
$1/2
!
!
!
!
C1y = 2
|f1f1f2f2(fHfLy + fLyfH)| are given by:
pffiffi
hC0 jHjC1x i ¼ 2hfH jly jfLx iB;
ð3:6Þ
hC0 jHjC1y i ¼
pffiffi
2hfH jly jfLy iB;
ð3:7Þ
since the matrix elements of Hel and of the diamagnetic part of
HB, eqn (3.5), between the ground and degenerate excited
states are zero by symmetry. On the other hand, the transition
is clearly rotationally allowed.
The HOMO is a combination of the hybrid spz,X on the
heavy atom (where X is B or C) and 1sH on hydrogen:
|fHi = N(a|1sHi + b|2sXi + c|2pz,Xi)
2
2
2
N = (a + b + c + 2abS1s,2s + 2acS1s,2p)
(3.8)
$12
(3.9)
where S1s,2s and S1s,2p are the relevant overlap integrals. If we
choose the quantization axis for L along the y axis, it is
possible to write the irreducible tensor representation of
the operators and orbitals appearing in the matrix elements
eqn (3.6) and (3.7) as ly + l0, 2py + p0, 2pz + 2$1/2(p1 + p$1)
and 2px + $i2$1/2(p1 $ p$1). An estimate of the Hamiltonian
matrix elements between the ground state and the first
excited singlet states eqns (3.6) and (3.7) can then be easily
provided:
hC0|H|C1xi E 12Nchp1 + p$1|l0|p1 $ p$1iB
= imB
hC0|H|C1yi E 12Nchp1 + p$1|l0|p0iB = 0
(3.10)
(3.11)
with m = Nc.
Similar arguments can be made for the other closed-shell
paramagnetic systems considered here, to identify those states
that contribute strongly to the mixing in a strong magnetic
field. For instance, from the theory of ring currents in planar
conjugated p systems, it is known that the paramagnetic ring
currents in [4n]-annulenes stem from the rotationally allowed
HOMO–LUMO transition.25,26 Considering a fully symmetric
DNh [N]-carbocycle in a minimal basis of one pz,i atomic
orbital per carbon center, we can write the N symmetry
adapted p MOs as:
$
%
X
2pl
jcl i ¼ Nl
exp i
ð3:12Þ
r jpz;r i
N
r
Nl ¼
(
$
%)$12
2pl
Nþ2
Srs cos
;
ðr $ sÞ
N
r4s
X
ð3:13Þ
where, for even N, l = 0, )1, )2,. . ., )N/2, and the overlap
integrals are given by Srs = hpz,r|pz,si. It follows that l is a
5494 | Phys. Chem. Chem. Phys., 2009, 11, 5489–5498
good rotational quantum number, so that for the one-particle
states we can write:
lz|cli = l|cli
(3.14)
From these simple symmetry-based considerations, it follows
that all [4n]-carbocycles of full D4nh rotational symmetry have
an orbitally degenerate ground state, since the HOMO and
LUMO are rotational pairs with l = ) n, belonging to a
doubly degenerate irreducible representation. According to
the Jahn–Teller theorem,40 vibronic coupling removes this
degeneracy by reducing the symmetry to at least D2nh, corresponding to the structures considered here: C4H4 of D2h
symmetry, C8H8 of D4h symmetry, and C12H12 of D6h
symmetry. This symmetry reduction leads to new one-particle
states, resulting from a well-defined mixing pattern between
the states of the fully symmetric parent41 so that each occupied
angular momentum shell eqn (3.12) with l r n is solely mixed
with a virtual shell having l 0 = N/2 $ l. Thus, in closed-shell
[4n]-annulenes, the HOMO (cH) and LUMO (cL) are nondegenerate, resulting from a mixing between the doubly
degenerate HOMO and LUMO of the fully symmetric
open-shell parent:
1
cH ¼ pffiffi ðcn þ c$n Þ;
2
i
cL ¼ $ pffiffi ðcn $ c$n Þ:
2
ð3:15Þ
ð3:16Þ
Omitting from the notation all closed-shell orbitals that do not
contribute to the coupling, we may write the ground and first
excited states as C0 = |cHc!H| and C1 = 2$1/2|cHc!L + cLc!H|,
respectively. The coupling by the magnetic Hamiltonian in
eqn (3.5) (with a = z) can then be written as:
hC0|H|C1i = hcH|lz|cLiB = inB + imB
(3.17)
We can thus proceed to write a very general effective coupling
Hamiltonian, valid for all closed-shell paramagnetic molecules
in strong magnetic fields as:
$
%
$D $ w0 B2
imB
H¼
;
ð3:18Þ
$imB
D $ w1 B2
where 2D is the energy gap between the two states in zero-field,
and w0/1 + hC0/1|wdaa|C0/1i o 0. Without loss of generality, we
assume in the following that m Z 0. The ground- and excitedstate energies as functions of the external field strength, B, are
straightforwardly obtained by diagonalising eqn (3.18), giving
1
W0=1 ðBÞ ¼ $ ðw0 þ w1 ÞB2
2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
,
½2D þ ðw0 $ w1 ÞB2 .2 þ 4m2 B2 ;
2
ð3:19Þ
where the minus sign is used in the ground-state energy W0(B)
and the plus sign in the excited-state energy W1(B).
Let us now investigate how the energies of the ground and
excited states depend on the magnetic field, B, which may take
on negative as well as positive values. We first consider briefly
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m2 o $2Dw0), m = 0.374 (nonmagnetic case with m2 = $Dw0),
and m = 0.6 (paramagnetic case with m2 4 $Dw0). In the
paramagnetic case, the ground-state energy takes on a
sombrero-like shape when plotted against B. This characteristic shape arises as a result of two competing mechanisms:
a diamagnetic mechanism, which raises the energy quadratically with B, and a paramagnetic mechanism, which lowers the
energy due to coupling with the first excited state.
Let us now identify the characteristic field, Bc, where the
system goes through an energy minimum, changing from
paramagnetic to diamagnetic. The conditions for the existence
of stationary points other than B = 0 can be determined by
setting W 0 0(B) = 0 with B a 0. If w0 a w1, this leads to the
following expression for the critical fields:
Fig. 5 Ground and excited state energies, eqn (3.19), arising from the
two-level model, eqn (3.18), plotted as function of magnetic field, for
the arbitrary parameters D = 0.01, w0 = $7.0, w1 = $4.0, and for
different values of the coupling parameter m. Note that as the linear
magnetizability of the two-level model becomes paramagnetic
(m2/2D 4 |w0|, see (d), the ground state energy is described by the
sombrero shape observed in ab initio calculations.
the case of no interaction between the two states, m = 0. The
energies eqn (3.19) may then be written as:
2
Wm=0
0/1 (B) = 8D $ w0/1B ,
(3.20)
which
pffiffiffiffiffiffiffiffiffiffiffibecome
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi degenerate and cross each other at B ¼
2D=ðw1 $ w0 Þ if |w0| 4 |w1|. As expected, both states are
diamagnetic in this case, with a minimum at B = 0. This
situation is illustrated in Fig. 5a, for a system with (atomic
units) D = 0.01, w0 = $7.0, and w1 = $4.0.
When m a 0 and the two states interact, the two states avoid
each other and no crossing may occur: the ground state is
lowered W0(B) r miniWim=0(B), while the excited state is
lifted W1(B) Z maxiWim=0(B). As seen from eqn (3.19), the
interacting and noninteracting energies are identical for zero
field and approach each other asymptotically in the limit of an
infinitely strong field. Differentiating W0/1(B) with respect to B
at B = 0, for the first and second derivatives (i.e., the magnetic
dipole moment and magnetizability at zero field) we obtain:
$W00/1(0) = 0
$W000=1 ð0Þ ¼ 2w0=1 )
(3.21)
m2
:
D
ð3:22Þ
Thus, while the excited state is always diamagnetic (positive
curvature) at zero field, the ground-state is paramagnetic
(negative curvature) for large coupling constants m2 4
$2Dw0. For paramagnetic ground states, the energy first
decreases around B = 0 with increasing magnitude of the
magnetic field; eventually, however, the energy must increase
with B until it approaches the lowest of the curves, $w0B2 and
$w1B2. For a paramagnetic state, the energy must therefore go
through a minimum as B increases from zero to infinity. In
Fig. 5, we have plotted W0/1(B) against B for a system with
D = 0.01, w0 = $7.0, w1 = $4.0, increasing the coupling
parameter from m = 0 (the uncoupled diamagnetic case
discussed above) to m = 0.2 (coupled diamagnetic case with
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sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
$2w0 w1 ½m2 þ Dðw0 $ w1 Þ. þ jmðw0 þ w1 Þj w0 w1 ðm2 þ 2Dðw0 $ w1 ÞÞ
;
Bc ¼ )
w0 w1 ðw0 $ w1 Þ2
ð3:23Þ
whereas, for w0 = w1, we obtain:
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi%
ffiffiffiffi$
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi%
ffiffiffi
sffi$
1
m2
m2
Bc ¼ )
$ 2D
þ 2D :
2jmj
jw0 j
jw0 j
ð3:24Þ
We discuss further only eqn (3.23) since the case w0 = w1 is
possibly not the most frequent. Furthermore, it can be easily
checked that the conditions obtained from eqn (3.23) are
stronger than those obtained from eqn (3.24) in the sense that
their fulfillment (e.g., the conditions for a real Bc) always
implies the fulfillment of the equivalent conditions derived
from eqn (3.24).
The condition for the existence of non-zero critical fields can
thus be translated into the condition of reality of eqn (3.23),
leading to
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
jmðw0 þ w1 Þj w0 w1 ½m2 þ 2Dðw0 $ w1 Þ.
ð3:25Þ
2
$ 2w0 w1 ½m þ Dðw0 $ w1 Þ.40;
which is fulfilled if
m2
4 jw0 j:
2D
ð3:26Þ
Thus, stationary points other than B = 0 always exist for
closed-shell paramagnetic systems. Next, the condition for the
critical field, eqn (3.23), to be a minimum reads:
W000 ðBc Þ¼
8w0 w1
qffiffi2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
0 $w1 Þ
ðw0 $w1 Þ jmj m þ2Dðw
w w
2
0 1
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
#
m2 þ2Dðw0 $w1 Þ
2
jmðw0 $w1 Þj
$2ðm þDðw0 $w1 ÞÞ 40
w0 w1
"
ð3:27Þ
Again, it can be seen that eqn (3.27) is always fulfilled when
condition eqn (3.26) holds.
Hence, the model predicts that closed-shell paramagnetic
systems always turn diamagnetic in a strong field; also, it
provides an estimate of the critical field eqn (3.23). In Fig. 5,
the behaviour of the two-level model for a few illustrative
values of the coupling parameter m are shown, illustrating the
Phys. Chem. Chem. Phys., 2009, 11, 5489–5498 | 5495
appearance of the sombrero-shaped ground-state energy curve
when m2/2D 4 |w0|, that is, when the system becomes
closed-shell paramagnetic. Interestingly, for m2/2D E |w0| in
Fig. 5c, we obtain a diagram that is very similar to the
cyclobutadiene ab initio energy plot in Fig. 2a. Using
eqn (3.19) to fit the cc-pVDZ total energy curve of butadiene,
Fig. 6 On the left column, the energy curves obtained by fitting the best ab initio ground state energy data points (reported as black dots) for (a)
BH, (c) CH+, (e) C16H102$ and (g) C20H102$ with eqn (3.19) (solid line) are compared to the energy curves obtained by fitting the same data with a
sixth-order polynomial (dotted line) and an eighth-order polynomial (dashed line). On the right column, the critical field Bc is plotted as a function
of the effective gap 2D|w0|/m2 for (b) BH, (d) CH+, (f) C16H102$ and (h) C20H102$ using eqn (3.23) with the gap D varying between zero and Dmax =
m2/2|w0|, and the other three parameters fixed to their best ab initio values. The plots can be understood as ‘‘phase diagrams’’, in that they define
regions of existence of paramagnetic and diamagnetic ‘‘phases’’ in parameter space. The superimposed gridlines identify the exact values of the
reduced gap and of Bc characterizing the ab initio energy minimum.
5496 | Phys. Chem. Chem. Phys., 2009, 11, 5489–5498
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we obtain m2/2D E 4.8 a.u. and |w0| E 5.2 a.u., so that
m2/2D E |w0|, confirming that, due to its strong p-electron
paratropicity, the onset of global paramagnetism is indeed
quite close in cyclobutadiene.
The two-level model also proves very useful for obtaining an
approximate analytical fit to the ab initio data, accurate for a
large range of field strengths. In Fig. 6 (left column), for
instance, we report the energy versus field ab initio data points,
together with a least-squares fit to the data of a sixth-order
polynomial (dotted line), of an eighth-order polynomial
(dashed line), and of the ground-state energy of the two-state
model eqn (3.19) (solid line). The two-state model invariably
produces an accurate fit to the data, whereas the sixth- and
eighth-order polynomial fits based on the Taylor perturbative
expansion eqn (1.1) perform poorly, especially for the smaller
systems, where larger ranges of magnetic field strengths are
considered.
One quantitative measure of the quality of the fitted energy
functions provided by eqns (1.1) and (3.19) can be obtained by
comparing the relevant diagonal component of the linear
magnetizability tensor computed by taking the first derivative
at zero field of the fitted expressions, with the exact value
calculated by means of response theory. To produce the most
accurate estimate for linear response by finite-field methods in
a highly non-linear system, it is clearly expedient to consider a
set of data points corresponding to very small fields in the
optimization procedure. We consider here only four systems—
namely, BH, CH+, the acepleiadylene dianion, and the
corannulene dianion—taking only the first four values for
each set of data points corresponding to the plots in Fig. 2
and 4. The results are reported in Table 1. Bearing in mind
that the range and number of points were not optimised for
accuracy, it is clear from Table 1 that the expression in
eqn (3.19) invariably delivers more accurate zero-field second
derivatives than does eqn (1.1). In addition, the two-state
model provides a better description of the energy variation
in the high-field regime.
A suggestive explanation for the difficulties of finding
accurate polynomial representations of the energy curve is
provided p
byffiffiffiffiour
ffiffiffiffiffiffiffi model energy curve. The Taylor series of the
function 1 þ x, expanded around x = 0, has the radius of
convergence |x| o 1. From this it follows that a Taylor
expansion of our model energy has a radius of convergence
given by:
B2conv ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
ð
ðDdw þ m2 Þ2 þ D2 dw2 $ Ddw $ m2 Þ:
dw2
ð3:28Þ
Table 1 Linear magnetizability (positive diagonal component) for BH,
CH+, C16H102$ and C20H102$ calculated as (i) a linear response
function (w>,response) (ii) a second derivative of the ab initio fitted twolevel model energy (w>,model) and (iii) a second derivative of the ab initio
fitted eighth-order polynomial (w>,poly8), using an (aug)-cc-pVDZ basis
of London orbitals
Molecule
w>,response
w>,model
w>,poly8
BH
CH+
C16H102$
C20H102$
7.154
10.330
38.398
70.419
7.151
10.315
38.394
70.419
7.100
10.218
38.342
70.253
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(with dw = w0 $ w1). For the systems studied here, the radii of
convergence are 0.07 (BH and CH+), 0.15 (MnO4$), 0.09
(C4H4), 0.04 (C8H8), 0.03 (C12H12), 0.02 (C16H102$), and 0.01
(C20H102$). As the radius of convergence is approached from
below, the perturbative ansatz in eqn (1.1) will provide slower
and slower convergence. Beyond the radius of convergence,
Bconv, the perturbation series diverges and fails to provide even
slow convergence. Judging from the molecules in this work,
the onset of divergence appears to occur at slightly smaller, but
comparable, fields than the change from paramagnetism to
diamagnetism, i.e. Bconv t Bc.
Finally, it is interesting to note that the critical field
eqn (3.23) plotted as a function of the reduced gap 2D|w0|/m2
can be understood as an effective ‘‘phase diagram’’, which
separates the diamagnetic and paramagnetic ‘‘phases’’ as
functions of the applied field and the effective gap. In Fig. 6
(second column), we have plotted these diagrams, where the
gap varies between zero and Dmax = m2/2|w0|, while the
remaining three parameters are set to their optimal cc-pVDZ
values (aug-cc-pVDZ for BH and CH+). The superimposed
grid lines correspond to the exact values of the reduced gap
and of the critical field characterizing the energy minimum
associated with the real system.
Furthermore, we note that the reduced gap is the ratio
between the diamagnetic and paramagnetic components of the
two-state-model linear magnetizability, so that both magnetic
phases can exist only if the effective gap is smaller than one.
Accordingly, these plots show that the closer the effective gap
value is to one—that is, the less paramagnetic the system
is—the smaller is the critical field at which the molecule turns
diamagnetic. Thus, if it were possible to modulate the effective
energy gap (i.e., to decrease the value of the positive linear
magnetizability) by means of some additional perturbation
(electric fields, substituent effects, etc.), it would be possible to
tune the value of the transition field.
IV.
Conclusions
We have investigated the non-linear magnetic behaviour of a
set of organic and inorganic paramagnetic closed-shell molecules in strong magnetic fields. We found that all the systems
considered are characterized by a similar behaviour in strong
fields, in that the paramagnetic system turns diamagnetic when
the field is larger than a critical value Bc. Moreover, a simple
two-level model was developed to rationalize this phenomenon. The model provides both a useful energy expression
to fit ab initio data and a demonstration that such behaviour
should be expected for any paramagnetic closed-shell molecule. Because of their small size, the systems considered here
require magnetic fields at least one order of magnitude larger
than what can be achieved experimentally to undergo the
diamagnetic transition. However, the same transition should
occur for experimentally attainable fields if larger systems
(containing about 100 atoms) are considered, due to the
enhancement of the effective magnetic flux. Interestingly, a
straightforward analysis of the simple model hereby proposed
suggests that it might be possible to tune the transition field by
means of external perturbations capable of modulating the
paramagnetic linear response of the closed-shell molecule.
Phys. Chem. Chem. Phys., 2009, 11, 5489–5498 | 5497
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