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 Phys. Chem. Chem. Phys., 2009, 11, 5489–5498 | 5489
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 This journal is
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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 This journal is
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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 This journal 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 This journal is
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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 This journal is
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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.
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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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