Chapter 1

NITROUS OXIDE: ELECTRON ATTACHMENTAND POSSIBLE SCENARIO OF THE REACTIONWITH NS METAL ATOMS

Oksana Tishchenko&, Eugene S. Kryachko&,#, and Minh Tho Nguyen&

&Department of Chemistry, University of Leuven,Celestijnenlaan 200F, B-3001 Leuven, Belgium#On leave from Bogoliubov Institute for Theoretical Physics,Kiev, Ukraine 03143#Corresponding author: eugene@bohr.quantchem.kuleuven.ac.be

Abstract Nitrous oxide (N2O) is known as the “laughing gas” having many bene-ficial actions but also an important “greenhouse gas” whose emission inthe atmosphere should be strongly reduced and strictly controlled. Thepresent review discusses the molecular mechanisms of some chemicalprocesses relevant to its reduction, including the dissociative electronattachment as well as its reactions with lithium and copper atoms, asthe representative ns metal atoms. Novel structural and mechanisticaspects are revealed using high level ab initio quantum chemical calcu-lations.

Keywords: Nitrous oxide, electron attachment, metal atoms, reaction mechanisms,ab initio calculations

1. Prologue: Nitrous Oxide, a Molecule for AllSeasons

The year 1772 witnessed the identification, by some great chemists, oftwo natural and long-lived gases whose intensive use has ever since madeprimordial contributions to the advances of chemical sciences: molecularnitrogen (N2) and nitrous oxide (N2O). While N2 was independentlyisolated in pure form from air by Rutherford, Scheele and Cavendish [1],N2O was actually discovered by Priestley from reduction of NO withiron or iron/sulphur mixtures and dubbed it “diminished nitrous air”

1

2

[2]. Sir Humphrey Davy inhaled N2O during his medical training in1799 as a part of the experimental study on the psychological effect ofa variety of gases and found its ability to promote lightheadness andcalled it as a “laughing gas” [3] (see also Ref. [4]). The dentists HoraceWells and Gardner Colton showed its anaestethic properties in 1844-1846. Nowadays, N2O has a panoply of both positive applications andnegative consequences in our daily life. Let us mention a few of them.

In medicine, N2O is widely used as both anesthetic and analgesicagents mainly in obstetric medicine or as an anesthetic carrier gas onanesthetic machines [5]. It is also injected as a tracer gas to measurethe blood flow [6] and lung volume [7] owing to its biological inertnesswhen present in low concentration.

In radiation chemistry, N2O is usually present in aqueous media as ascavenger for hydrated electrons [8]. Moreover, it is commonly servedas a propellant for pressurized food containers [9] in the food industryor as a possible direct product of nitric oxide (NO) synthetic enzyme inbiology [10]. Nitrous oxide behaves as an oxidant in catalysis to yieldhighly reactive oxygen anion (O−) upon dissociative adsorption on acatalyst surface [11]. Decomposition of N2O is a mean of evaluating thecatalytic activity of various oxide surfaces [12]. N2O plays the role ofan indicator for electron availability at semiconductor surfaces [13] andfor the determination of the surface area of highly dispersed supportedmetal particles, using adsorption measurements [14]. It is also used in thediary industry as a mixing and foaming agent since it is non-flammable,bacteriostatic, colorless (under ambient conditions) and exhibit a slightlysweet odor and taste. It also presents in the interstellar medium [15, 16]and in fact only one known interstellar compound where all five valenceelectrons of the nitrogen atom are engaged [17].

Recently, there has however been considerable concern about the lesslaughing facet of nitrous oxide due to its negative impact on the tro-posphere and stratosphere. In the “Kyoto Protocol” which was signedin 1997 in Japan and confirmed in 2002 in Marrakech, Marocco, andalready ratified by more than 170 countries around the world with theaim to organise the common fight against the climate changes, nitrousoxide was indeed listed as one of the six strong greenhouse gases, alongwith carbon dioxide, methane, fluorohydrocarbons, perfluorohydrocar-bons and hexafluorosulfur [18]. Production of these gases should besignificantly reduced and strictly controlled by participating countries.Nitrous oxide is a minor component of the atmosphere. It is a potentand harmful greenhouse gas whose global warming potential is largerthan that of CO2 by a factor of ca. 310 [18, 19, 20, 21]. N2O is char-acterized by a long lifetime: its half-life in the atmosphere is estimated

Nitrous Oxide: Electron Attachment and Reactions with Metal Atoms 3

as 110-168 years [22]. Thus, although the former gas exists only at thetrace level in the troposphere, its effect on the atmospheric chemistry isexpected to be by far larger than its weights [23]. Due to the existence ofa gap between the Shumann-Runge bands of molecular oxygen and theHartley band of ozone , and the Herzberg continuum of O2, energeticsolar radiation easily reaches the lower stratosphere (at ≈ 20 km, the UVwindow covers the range between 197 and 214 nm [24]). It photolysesN2O as

N2O + hν → N2 + O(1D). (1.1)

Characterized by the unit quantum yield Do = 351 kJ/mol [25], thereaction (1) involves about 90 % of atmospheric N2O. The remainderreacts with electronically excited oxygen atoms:

N2O + O(1D) → 2NO (6%) (1.2)

andN2O + O(1D) → N2 + O2 (4% [26]). (1.3)

An instantaneous lifetime of 123 years is determined by the reactions(1) and (2) [27]. (3) is actually the dominant source of reactive oddnitrogen to the stratosphere [28] whose cycle plays a key role in thedepletion of nonpolar ozone [29, 30]. The reaction (2) yields the nitricoxide radical (NO) [31]. The latter is well known for its role as a destroyerof the ozone layer which protects the Earth from harmful ultravioletradiation [32, 33, 34, 35, 36, 37]. As a consequence, N2O concentrationis expected to affect substantially the global warming and stratosphericozone loss. Kim and Gordon [38] suggested that the N2O isotopomerconcentration in the troposphere results from the light N2O, producedby soil, N2O from oceans, and a back flux of isotopically heavy N2Ofrom the stratosphere. The reactions (1) - (3) are treated as the sourcesof the stratospheric enrichment, except the photolysis of N2O at 185nm [39] whose understanding has been significantly improved thanks toYung-Miller model, which takes the zero-point energy (ZPE) [40] andits current generalization [41] including the transition dipole momentsurface, upper-state dynamics and bending excitation.

The concentration of N2O in the atmosphere amounts 315 ppb and itcontinuously increases at an annual rate of 0.8 ppbv [42, 43]. Measure-ments of N2O in polar ice show that its global concentration is highernow than any time during the past 45000 years [44]. After the late iceage, the concentration of N2O was grown and then remained at the levelof ≈ 275 ppb for about 10000 years untill the 19th century. Since then,its concentration has been significantly increased. Atmospheric nitrousoxide is derived from natural and anthropogenic sources (anthropogenic

4

nitrate [45]), but the latter are the main and only ones that can be min-imized by improvements in technology or changes in land use. Severalanthropogenic sources of N2O emissions have been identified, includingbiomass burning, fossil fuel combustion, industrial production of adipicand nitric acids, nylon manufactures and vehicular traffics [46]. Otherwell-known N2O sources are waste water treatment plants, fertilized soils[47], nitrogen-enriched rivers, estuaries and ground water [48].

In this global context, considerable effort has thus been devoted tofind out the ways of reducing the emission of nitrous oxide gas. Numer-ous studies were focused on modelling and understanding of favorableand workable mechanisms regulating the N2O production and balancingit with its destruction. The latter task has opened a wide area of exper-imental and theoretical studies. An interesting proposal for a solutionis to selectively recover N2O on an adsorbent to produce a concentratedstream of N2O during desorption and to use it as an oxidant for or-ganic chemistry [49]. Another possible process is oxidizing phosphines(PR3) by nitrous oxide at or below 100 C [50]. Oxidations of most or-ganic compounds by nitrous oxide, giving N2 as the only by-product, arethermodynamically favorable but are found to be extremely slow exceptat high temperatures. Within this approach, let us however mentiona dramatic procedure put forward by Solutia (former Monsanto) whichconsists in recovering and using N2O as a selective oxidant to convertbenzene to phenol, and then hydrogenate phenol to cyclohexanone. Thelatter is then oxidized by nitric oxide to adipic acid, returning N2O asa by-product and thus closing the N2O-cycle [49]. Reactions of N2Owith radicals [51] and light atoms, such as nitrogen, oxygen, etc. [52]producing N2, were also studied but the corresponding reactions do notappear to be efficient for practical purpose.

Due to the fact that the three-atomic N2O species is isoelectronicwith fulminic acid (HCNO), diazomethane (H2CNN) and hydrazoic acid(HNNN), it belongs to the important class of 1,3-dipoles [53, 54]. Thus,the possibility to use acetylenes and olefins that might exist in differentnatural environments, as dipolarophiles to trap N2O within the frame-work of 1,3-dipolar cycloaddition reactions, has recently been investi-gated [55]. Addition and subsequent elimination of N2 from the primaryfive-membered rings have been found to be facile processes to achieve.

2. Metal Atoms as Reducing AgentsOf the various available proposals, the addition of metals to combus-

tion systems has often been considered as an efficient mean of destroyingN2O, thanks to the large exothermicity and rapidity of the correspond-

Nitrous Oxide: Electron Attachment and Reactions with Metal Atoms 5

ing reactions [56]. As for an alternative, the catalytic decomposition onvarious surfaces involving a dissociative electron attachment to N2O hasbeen put forward [57]. Apparently, an optimum procedure is the use ofheterogeneous catalysts including transition metals, copper in particular[58, 59, 60, 37, 61, 62, 63, 64, 65]. As well known, the oxidation reactionof metal atom with N2O proceeds via the following scheme (M = metal):

N2O(X1Σ+) + M → MO + N2. (1.4)

It is worthwhile mentioning that such reactions are quite interestingper se owing to the fact that metal oxide product (MO) is often formedon energetically accessible lower-lying excited-state potential energy sur-faces (PESs) that gives rise to a strong chemiluminiscence. Moreover,some of these reactions exhibit a rather striked non-Arrhenius behaviorat higher temperatures which could be considered as a clue to its mech-anistic complexity, indicating that either non-adiabatic transitions takeplace between different PESs, and/or other reaction pathway(s) on theground state PES becomes opened [67].

In this review, we will consider the molecular mechanisms of the re-action (1), confining our discussion to metals with ns valence electrons:lithium and copper atoms will be taken as representative examples ofalkali and transition metals, respectively. For mechanistic features ofthe reaction (4) with other metals see theoretical studies [68, 69, 70, 71].

Let us first briefly outline the key features of these reactions observedexperimentally. First of all, it has been revealed that the metal atomscleave the N-O bond in the reaction (4). The reaction (4) is characterizedby a rather large exothermicity of 40.4 kcal/mol (M = Li) [72, 73] and27.1 kcal/mol (M = Cu) [104] thanks to the oxygen atom which, onthe one hand, strongly binds the metal and, on the other hand, is notsufficiently strong bound to N2 in nitrous oxide. Moreover, experimentsclearly demonstrate the existence of two different kinetic regimes [67, 75,76]. One is a typical Arrhenius with the rate constant kA(T ) determinedas

kA(T ) = k(o)A exp(−EA/RT ) (1.5)

where, for M = Cu, the pre-exponential factor k(o)A = 1.70 × 10−10

cm3·mol−1·s−1 (470 K< T < 1190 K) [75] or 2.4±0.6 × 10−10 cm3·mol−1·s−1 (458 K < T < 980 K) [76] and the activation enthalpy amounts toEA = 5129 K = 10.2 kcal/mol [75] or 11.6±0.3 kcal/mol [76]. For thereaction (1) with M = Li, the corresponding parameters are as follows:k

(o)A = 4.8 × 10−10 cm3·mol−1·s−1 (T < 500 K) and 3.4±0.1 kcal/mol

[67].

6

Another regime shows a non-Arrhenius behavior and takes place athigher temperatures: above 500 K for Li, and above 1190 K for Cu. Itsrate constant obeys the following empirical fit:

knon−A(T ) = k(o)non−AT

αexp(−Enon−A/RT ) (1.6)

where for Cu, k(o)non−A = 3.04 × 10−20 cm3·mol−1·s−1, α = 2.97, and

Enon−A = 6.1 kcal/mol [76], and for Li, k(o)non−A = 6.1±0.3 × 10−16

cm3·mol−1·s−1, α = 2.11±0.03, and Enon−A = 2.2±0.15 kcal/mol [77]. Inthis context, two legitimate questions could be posed: what are then themechanisms underlying reaction (1) and how do structurally Arrheniusand non-Arrhenius reaction channels look like?

In order to elucidate the mechanisms of the reaction (4), three theo-retical models have been proposed. Within the electron transfer (ET)model [78, 79, 80, 81, 82, 83, 84], the electron transfer from M to N2O,which occurs at a definite M-N2O distance via the harpoon mecha-nism [85], predetermunes (4). The activation energy EA then dependson the ionization energy IEM , equal to 5.392 eV for Li and 7.72 eVfor Cu [86], although such dependence has recently been questioned[82, 83, 84, 87]. Furthermore, as recently shown [88], the electron affinity(EA) of the ground state N2O molecule is firmly negative and equal toca. -6 kcal/mol (vide infra). This implies that the electron hopping fromM to the ground state N2O must be ruled out. The model of the directabstraction [78, 79] is based on the suggestion that M directly abstractsthe oxygen from N2O.

The third model is the resonance interaction (RI) model proposed byFontijn and co-workers [89, 90, 91, 92, 93, 94] who examined empiricalcorrelations between the reaction rate constant and activation energyEA, on the one hand, and, on the other, the sums of ionization en-ergy and s-p promotion energy of M(PEM ) (for instance, the PECu =87.2 kcal/mol). The IR model assumes that the reaction of any of theaforementioned metal atoms M with N2O is governed by an activatedcomplex (AC) composed of three resonating structures S1, S2, and S3

described by the corresponding wave functions ψ1, ψ2, and ψ3. S1 isprimarily formed by the covalent interaction between the oxygen atomof N2O and the s bonding orbital of M. The structure S2 is analogousto S1 where the covalent interaction is realized throughout the p bond-ing orbital of Z whereas the third one, S1, is suggested to be ionic andcomposed of the anion N2O− interacting with cation M+. Therefore,the total wavefunction ΨAC of the activated complex is expressed as a

Nitrous Oxide: Electron Attachment and Reactions with Metal Atoms 7

linear combination of those three wave functions

ΨAC =3∑

i=1

ciψi (1.7)

with real variational parameters ci, i = 1,2,3 subject to the normalizationcondition

∑3i=1 c

2i = 1, chosen under the assumption of the mutual or-

thogonality of the wave functions corresponding to the different resonat-ing structures. ΨAC is then defined as the eigenfunction correspondingto the lowest eigenenergy of the 3 × 3 Hamiltonian matrix

⎛⎝H11 H12 H13

H21 H22 H23

H31 H32 H33

⎞⎠

where the diagonal elements are the following: H11 = V is the barrierof the formation of S1 treated as a model parameter, H22 = PE + V/2,and H33 is approximated as

H(1)33 = IEZ − EAN2O − e2

Ro− αe2

2R4o

+B

Rδo

(1.8)

or

H(1)33 = IEZ − EAN2O − e2(δ − 1)

δRo(1.9)

including particularly the attractive Coulomb and polarization poten-tials and the repulsive Born-type potential. In Eqs (8) and (9), e is theproton charge, Ro is the sum of the univalent radii of Z+ and O−, αis the sum of their polarizabilities, and finally, δ is the averaged Bornexponent.

3. The Ground State N2O MoleculeThe molecule of nitrous oxide is formed by three second-period atoms

and contains 16 valence electrons. In its electronic ground state, N2Ois a linear asymmetrical molecule N-N-O, characterized by the pointsymmetry group C∞v and thus assigned to its X1Σ+ irreducible repre-sentation [95-101]. It can be also characterized by two Lewis resonancestructures [99] (see also Ref. [101]):

N N O

� ���� ���� +N N O

���� ����+�

�

The bond lengths r(N-N) and r(N-O) of the ground state N2O moleculeare collected in Table 1. N2O(X1Σ+) has a relatively small dipole mo-ment of 0.161 D [119, 120]. Its molecular orbital (MO) electronic config-uration is (1σ)2(2σ)2(3σ)2(4σ)2(5σ)2 (6σ)2(1π)4(7σ)2(2π)4 [121]. The

highest occupied 1π and 2π MOs consist of π-type orbitals lying or-thogonal to the molecular axis [122]. The former has a strong bondingcharacter whereas the latter is of a dominant non-bonding characterthough, precisely, there exists a positive and constructive interferencebetween the nitrogen atoms and the destructive one with the oxygenthat gives an overall non-bonding characters [123, 124]. The highestσ MO, 7σ, has a predominant N-O bonding character [121, 125]. Thelowest unoccupied MO (LUMO) of N2O (X1Σ+) is antibonding 3π.

The infrared spectrum of the ground-state N2O molecule, surveyed inTable 2, exhibits three strong fundamental absorption bands: ν1 (asym-metric stretch, Σ+), ν2 (bending, Π) and ν3 (symmetric stretch, Σ+),which reveal a superimposed rotational fine structure [126]. What isimportant for a further discussion in Section 4 is that the bending modeν2 is doubly degenerate. This degeneracy of the ν2 vibrational mode canbe lifted upon interaction of N2O with surrounding, paricularly with ametal atom M.

The dipole-moment derivatives of N2O have recently been accuratelydetermined in Ref. [127]. Spectroscopic and force constants are corre-spondingly shown in Tables 3 and 4. The recent NMR shielding con-stants of the title molecule are collected in Table 5.

The PESs of the lower-lying electronic states of N2O were extensivelystudied by Peyerimhoff and Buenker [133] , Hopper et al. [125], Hop-per [121], Chutjian and Segal [134], and recently, by Balint-Kurti andco-workers [135] and Hwang and Mebel [109]. Notice that in addition tothe ground-state NNO molecule, there exists two N2O isomers. One is acyclic isomer with the symmetry C2∞ which was obtained by Klapotkeand Schulz [136, 137] in their MP2/6-31+G(d) study of the dissociationof the N4O molecule and later was thoroughly investigated by Galbraithand Schaefer [138]. The most recent CCSD/TZ-ANO [112, 113] calcu-lations place the C2∞ isomer of N2O by 64.8 kcal/mol above the NNOglobal minimum and predict its bond lengths r(N-N) = 1.188 A andr(N-O) = 1.536 A and the bond angle � NNO = 67.3o. The magneticproperties of the cyclic N2O isomer have been studied by Wullen andKutzelnigg [139]. The other isomer N-O-N with the D∞h symmetry hasrecently been found by Crawford and Klapotke [110] and Wang and Har-court [112] who located it at 110.7 kcal/mol higher the global minimumand calculated N-O bond length of 1.202 A [112].

3.1 Electron attachment to N2ODue to a possibility of charge transfer from metal to N2O, a knowledge

of the electron affinity (EA) of N2O and the PES of its anion is a neces-

Nitrous Oxide: Electron Attachment and Reactions with Metal Atoms 9

Table 1.1. The experimental and theoretical bond lengths (in A) of the ground-statenitrous oxide.

Reference r(N-N) r(N-O)

Experimental data

Rotational spectroscopy [102-104] 1.1282 1.1842Cotton and Wilkinson [105] 1.129 1.188Bruning et al. [57] 1.126 1.186Teffo and Chedin [107] 1.127292 1.185089CRC book [108] 1.1284 1.1841Computational data

MP2/6-31G(d) [109] 1.172 1.193MP2/6-311++G(d) [110] 1.163 1.181MP2/6-311+G(2d) [109] 1.157 1.186B3LYP/6-311+G(2d) [109] 1.126 1.184B3LYP/6-311+G(2d,2p) [111] 1.122 1.188B3P86/6-311+G(2d,2p) [111] 1.120 1.181SVWN/6-311+G(2d,2p) [111] 1.129 1.178BLYP/6-311+G(2d,2p) [111] 1.140 1.202BP86/6-311+G(2d,2p) [111] 1.1392 1.196MP2/6-311+G(2d,2p) [111] 1.157 1.186BP/TZP [112, 113] 1.145 1.120VWN/TZVP [112] 1.1375 1.1855BP/TZVP [[113] 1.145 1.197GGA-BP/TZVP [112] 1.1453 1.2004B3LYP/DZP++ [118] 1.141 1.202B3P86/DZP++ [118] 1.138 1.195BHLYP/DZP++ [118] 1.119 1.190BLYP/DZP++ [118] 1.161 1.218BP86/DZP++ [118] 1.158 1.210LSDA/DZP++ [118] 1.150 1.193CISD/TZ2P [114] 1.1073 1.1809CISD(T)/TZ2P [115] 1.15053 1.2068CCSD(T)/TZVP [106] 1.137 1.193QCISD/aug-cc-pVDZ [116] 1.138 1.201QCISD(T)/aug-cc-pVDZ [116] 1.150 1.204CCSD(T)/cc-pVTZ [115] 1.13285 1.18960QCISD/aug-cc-pVTZ [116] 1.121 1.189QCISD(T)/aug-cc-pVTZ [116] 1.134 1.193CCSD(T)/cc-pVQZ [116] 1.12909 1.18703CCSD(T)/ANO[5421] [117] 1.1311 1.1874CCSD(T)/ANO [112] 1.1303 1.1896

sary step. A retrospective look on this problem indicates that the EA ofN2O was a controversial issue, as it can be seen from Table 6, collectingreported spectroscopic and theoretical data. Let us briefly summarize

10

Table 1.2. Fundamental and harmonic (in parentheses) vibrational frequencies of theX1Σ+ N2O (in cm−1). Calculated frequencies are referred to the harmonic unless theotherwise is noticed.

ν1 ν2 ν3

Experimental data

Herzberg [126] 1284.9 588.78 2223.76Bruning et al. [57] 1277 589 2224Teffo and Chedin [107] 1271.4 (1298.3) 588.8 (596.3) 2223.8 (2282.1)Computational data

MP2/6-311++G(d) [110] 1286 538 2243CISD/DZP [138] 1352 2402CCSD/DZP [138] 1290 2296CCSD(T)/DZP [138] 1269 2222CISD/DZP++ [138] 1345 2400CCSD/DZP++ [138] 1281 2292CISD/TZ2P [138] 1354 (1345.9) [114]) 645.5 [114] 2417 (2419.5) [114]CCSD/TZ2P [138] 1288 2307CCSD(T)/TZ2P [115] 1256 579 2221VWN/TZVP [112] 1339.72 618.67 2338.12BP/TZVP [113] 1280.67 589.48 2253.07GGA-BP/TZVP [112] 1280.65 589.70 2253.07QCISD/aug-cc-pVDZ [116] 1282 586 2311QCISD(T)/aug-cc-pVDZ [116] 1265 564 2233CCSD(T)/cc-pVTZ [115] 1271.0 (1297.2) 594.4 (601.1) 2224.9 (2282.8)QCISD/aug-cc-pVTZ [116] 1297 615 2340QCISD(T)/aug-cc-pVTZ [116] 1274 590 2258CCSD(T)/cc-pVQZ [115] 1303.5 602.1 2287.9CCSD(T)/ANO[5421] [117] 1272.0 (1299.1) 590.2 (604.3) 2227.3 (2285.9)

the main available results that disagree with each other on this quantity.The first experimental estimations date back to the early 1970, when avalue of 0.27±0.17 eV was derived from the electron capture detectorexperiments [140] and a lower limit for the EA of -0.15±0.1 eV was ob-tained from collisional dissociation or collisional ionization spectra [141].The positive value of 0.22 eV currently recommended in, among othercompilations, the CRC-Handbook [142] was taken from a 1976 estima-tion of Hopper et al. [125] which was based on results of molecularbeam experiments. A negative vertical electron affinity of -2.23 eV anda dissociation limit of 0.42 eV relative to the N2 + O− asymptote werealso derived in this study. This implies that the anionic system gainsa stabilization of 2.45 eV following geometry relaxation. These authors[125] also carried out ab initio MCSCF-CI calculations using a double-zeta quality basis set augmented by diffuse functions to determine theadiabatic EA. Nevertheless, these calculations, quite extensive by the

Nitrous Oxide: Electron Attachment and Reactions with Metal Atoms 11

Table 1.3. Spectroscopic constants of the X1Σ+ N2O (in A and cm−1).

Expt. CCSD(T)/cc-pVTZ CISD/TZ2P[107] [115] [114]

X11 -15.138 -15.069 -14.14X12 -27.207 -26.858 -32.57X13 -14.328 -14.264 -12.88X22 -4.319 -3.766 -4.05X23 -5.374 -5.204 -6.10X33 1.112 1.211 1.30G33 -0.575 -0.631 -0.54Be 0.42112 0.417488 0.42988106De 0.17610(=Do) 0.1691 0.167431012He -0.0156(13) (=Ho) -0.01417 -0.0278α1 0.003444 0.003437 0.00308α2 0.001925 0.001836 0.00211α3 -0.0005697 -0.0005655 -0.000541

standard of that time, indicated that both vertical and adiabatic EAsare negative by -4.1 and -1.7 eV, respectively, even though the differencebetween these values seems to be comparable with the experimentalestimate. Accordingly, the N2O− anion has a bent form with NNO an-gle of 132.7o and the N-O and N-N distances significantly stretched ascompared to those in the neutral species. In a 1983 theoretical study,Yarkony [68] reported that the 2A

′state of N2O− is actually stable by

0.39 eV with respect to ground state product limit. Such a positive EAcame no doubt from the fact that only Hartree-Fock calculations wereused in these calculations.

In the later 1990s, the electron affinity of N2O was further evaluatedin three theoretical works [145, 118, 116]. Yu et al. [145] considered in1992 this quantity for a series of triatomic species and paid a special at-tention to the N2O/N2O− case. Electronic energies of ions and neutralswere calculated by the G2 approach using geometry optimization at the(U)MP2/6-31(d) level. With the N-N and N-O bond lengths of 1.128and 1.674 A, and a NNO angle of 134.6o, a negative value of 0.13 eVwas derived. The N-O distance of 1.674 A, which is rather long with re-spect to the standard N-O distances, was no doubt due to the absence ofdiffuse functions in the basis set inherent in the geometry optimizationstep of the G2 procedure. In 1997, a density functional theory studyby Tschumper and Schaefer [118] indicated that nitrous oxide has pos-itive electron affinity ranging from +0.3 to +0.8 eV. Nevertheless, thenegative EA value was recently confirmed in a 1999 study by McCarthyet al. [116] using essentially quadratic configuration interaction QCISD

12

Tab

le1.

4.Forc

eco

nst

ants

ofth

egro

und

state

N2O

mole

cule

(in

Jaco

biin

tern

alco

ord

inate

s,aJ;A

,ra

d).

Expt.

Com

puta

tional

CC

SD

(T)/

CC

SD

(T)/

CC

SD

(T)/

DV

RT

Z2P

cc-p

VT

ZA

NO

[5421]

[107]

[128]

[129]

[130]

[114]

[115]

[117]

[131]

f rr(N

N)

18.2

512

182362

18588

18.1

904

21.9

14

18.1

2625

18.1

4833

18.6

53

f rR

.0276

1.0

286

1.1

34

1.0

240

1.6

35

0.9

6047

0.9

5093

0.7

18

f RR(N

O)

11.9

596

11.6

662

11.8

02

12.0

308

12.0

96

12.0

2091

12.0

7914

11.6

93

f αα

.6659

0.6

664

0.6

6550

0.6

660

0.7

63

0.6

8257

0.6

8766

0.6

61

f rrr

133.5

984

-132.4

326

-150.4

8-1

34.6

650

-161.1

-137.9

6941

-136.1

88

-151.2

62

f Rrr

6.8

725

-9.8

418

-4.5

4-1

.9200

-1.0

69

-2.5

3165

-3.5

356

-2.5

48

f RR

r.4

979

2.4

512

-5.9

0-0

.4728

-0.9

28

-0.2

9464

-0.3

7687

2.9

06

f RR

R98.8

309

-96.3

270

-94.7

4-1

02.9

000

-104.3

-98.0

0215

-99.7

19

-94.7

46

f αα

r-1

.5796

-2.5

674

-2.9

18

-1.6

812

-1.7

22

-1.6

9056

-5.9

44

f αα

R-1

.5374

-1.1

012

-0.7

84

-1.5

744

-1.6

55

-1.4

4902

0.7

55

f rrrr

691.3

727

674.7

480

52.5

6745.2

912

931.1

811.3

6004

796.7

0949.1

84

f Rrrr

46.4

463

65.6

076

-51.6

010.4

292

5.6

99

10.1

1118

25.2

32

-12.0

17

f RR

rr

-3.4

847

8.7

712

171.9

6-8

.6808

5.0

77

0.0

797

-2.0

384

-12.7

19

f RR

Rr

-7.6

910

-20.7

588

9.8

4-1

2.8

936

12.1

717.7

7822

24.2

76

-27.2

99

f RR

RR

634.9

213

590.3

304

623.2

8614.9

256

614.0

587.1

6909

584.6

3571.3

93

f αα

rr

1.8

083

12.6

700

-1.4

43.0

024

0.2

37

1.1

0549

30.7

55

f αα

Rr

5.1

053

7.2

216

0.2

05.0

150

4.1

17

4.0

413

4.1

31

f αα

RR

1.4

348

-6.1

652

3.9

03.1

852

2.4

91

2.3

9417

-18.7

65

f αα

αα

1.8

947

2.2

824

2.6

520

1.8

120

2.4

14

1.9

8497

1.8

2216

8.1

59

Nitrous Oxide: Electron Attachment and Reactions with Metal Atoms 13

Table 1.5. NMR shielding constants (in ppm) of the ground state N2O molecule (N1-N2-O), calculated with the qz2p basis set at the experimental equilibrium geometry[132].

HF CCSD CISD(T)14N1 -33.7 7.6 15.714N2 62.4 102.7 110.517O 172.3 200.3 201.3

Table 1.6. Experimental and Theoretical Electron Affinities (in eV) of N2O.a,b

EA Method Ref.

0.27±0.17 electron capture detector [140]> −0.15±0.10 neutral beam ionization potentials [141]

0.22±0.1 (-2.23±0.20) from electron affinity of radical /enthalpy of formation of anion relationship [125]

0.22±0.1 collision induced dissociation threshold [143]< 0.76±0.10 (ca. 1.5) laser photoelectron spectroscopy [144]

-1.7 (-4.1) ab initio MCSCF-CI calculations [125]0.39 ab initio Hartree-Fock calculations [68]-0.13 ab initio calculations using G2 approach [145]

0.3-0.8 density functional calculations [118]-0.0205 -0.149 ab initio QCISD and QCISD(T) calculations [116]

-0.03±0.1 ab initio calculations up to CCSD(T) level [88]aVertical EAs are given in parentheses.bPartly reproduced from Ref.[142].

and QCISD(T) methods with aug-cc-pVDZ and aug-cc-pVTZ basis sets.Indeed, these calculations led to values for EA in the range of -0.021 and-0.149 eV.

Yu et al. [145] also attempted to figure out the reason for the disagree-ment between experiment and theory and suggested that it apparentlycame from the presense of an energy barrier of about 0.15 eV to thedecomposition of N2O− into N2 and O− which was not taken into ac-count by Hopper et al. [125] in the interpretation of their experimentalresults for the threshold energy of the collisionally induced reactions ofthe anions.

Seeking to unravel the mystery of the electron attachment to N2O,the authors of Ref.[88] have recently undertaken a thorough study ofthe ground state PES of N2O−, which allowed a resolution of this long-standing controversy and a new look on the dissociative electron attach-ment to N2O. A novel entity - a T-shaped ground state anion A2 (Figure1) has been revealed and fully characterized. This cyclic anion turns outto be more stable than the open-chain A1 form by 5.3 kcal/mol, lying,

14

O

NN

N N

O

N N

O

2.7032.698*(2.720)

1.1201.110*(1.123)

1.3021.310*(1.330)

140.0131.9*(133.2)

1.1201.144*(1.158)

1.6311.686*(1.673)

134.5129.1*(130.1)

A1

ATS

A2

N N O

O

1.1651.200*(1.209)

N1

N N

1.137* 1.193*

1.546*

1.194*

N2

2.68 eV

Electronattachment

Electronattachment

Energy, kcal/mol

0

10

20

60

NTS

80.1

62.6

0.7

12.0

6.0

Figure 1.1. The potential energy profile of the dissociative electron attachment toNNO, calculated at a variety of computational levels (reading from the top down):MP2/TZV, CCSD(T)/TZVP indicated by the asterisk, and QCISD(T)/aug-cc-pVDZ(in parentheses) [?]. Bond lengths are given in A, bond angles in o.

Nitrous Oxide: Electron Attachment and Reactions with Metal Atoms 15

at most, 0.7 kcal/mol above the ground state neutral N1 molecule, andcan thus be regarded as the adiabatic anion. In view these results, adia-batic electron affinity of nitrous oxide, evaluated as the energy differencebetween A2 and N1 forms, amounts to only -0.03 eV.

Let us consider the properties of A2 in some detail. First of all, itis characterized by the N· · ·O distances of 2.7-2.8 A and can be consid-ered as a weakly-bound complex of N2 and O−. A population analysisindicated that in fact the negative charge is almost entirely located onthe oxygen atom. Asymptotically, this ground state A2 anion corre-lates with a higher-lying cyclic 1A1 isomer of neutral nitrous oxide (seethe preceding Section). Since in Cs symmetry, both A1 and A2 formsbelong to the representation 2A′, they are smoothly connected withoutsurface crossing via a transition state structure ATS (Figure 1). Thelatter represents a stationary point on the ground state PES with a singleimaginary frequency describing the normal mode of predominant N-Ostretching character mixed with NNO bending vibration. It is placed by6.0 and 12.0 kcal/mol above the anion A1 and neutral N1, respectively,and is characterized by the following geometrical parameters: r(N-N)1.12-1.16 (cf. 1.17-1.21 in A1), r(N-O) 1.63-1.69 (cf. 1.30-1.33 in A1),� NNO 131-140o (cf. 129-135o in A1). Although this ATS structurewas earlier reported by Yu et al., it has in contrast been interpretedas a transition state governing the dissociation of A1 to the productfragments N2 and O−.

In summary, following the electron attachment, the nitrous oxidesystem undergoes a rather facile dissociation process throughout thecyclic isomer to the final fragments N1→A1→ATS→ A2→N2+O−,and characterized by an overall barrier of 12.0 kcal/mol. The presenceof ATS allows a number of experimental findings to be rationalized,either in gas phase reactions or transformations at metal surfaces [88].

Hopper et al. used in their study the following thermodynamic cy-cle to estimate the adiabatic EA of N2O: EA(N2O) = -Do(NN-O) +EA(O) + Do(NN-O−). While the dissociation energy Do(NN-O) andthe electron affinity of oxygen EA(O) were accurately known, the valuefor dissociation energy Do(NN-O−) of 0.43±0.1 eV, was only estimatedfrom fitting of the reaction cross sections, that led to an error in Do. Ourcalculations [88] indicate that the Do(NN-O−) value derived by Hopperet al. was overestimated by 0.2 eV which in turn overestimated the sta-bility of the N2O− anion by placing it below the linear neutral. As aconsequence, a small positive EA(N2O) was proposed. It is clear thatthe long-standing discrepancy between theory and experiment on elec-tron affinity of nitrous oxide came from a less accurate evaluation of thedissociation energy of the anion Do(NN-O−).

16

An absolute error of 0.2 eV appears quite reasonable and lies withinthe expected “chemical accuracy”, but it is large enough, in the presentcase, to induce a qualitative change of the value and thereby a long-standing disagreement.

4. Reactions with Lithium and CopperOn the basis of the mechanism of dissociative electron attachement to

N2O, outlined in Section 3.1, it would be reasonable to suggest that itmight play an important role in predetermining the pathways of chemicalreactions of the type (1) of metals with N2O, where the latter acts as anelectron acceptor in the initial stage. Two remarkable examples of thereaction (1) with M = Li and Cu will be thoroughly examined in thepresent Section [106, 146]. Their PES profiles are vividly displayed inFigures 2 and 3 (Ref. [146] and [106]).

4.1 Entrance channel and transition statesThe entrance channel of the reaction (1) with M = Li and Cu is repre-

sented by complex I, which is bound by 2.8 and 0.24 kcal/mol in cases ofLi and Cu, respectively. I is characterized by the interfragment M· · ·Odistances of 2.098 A (ILi) and 3.695 A (ICu) which are smaller than thecorresponding sums of the van der Waals radii. A comparison of themolecular geometries and vibrations of I with the N2O (X1Σ+) showsthat no appreciable charge transfer from the given metal atoms to N2Otakes place. A departure from I along the reaction pathway, parallel tothe bending doubly degenerate mode ν2, leads to the transition struc-tures, TS1 or TS2, that results from lifting the degeneracy of ν2 due tothe interaction of N2O with M (cis or trans depending on the mutualdisposition of the N-N and M-O bonds). We thus may suggest that thereaction coordinate of (1) in the transition region is mainly determinedby the mode ν2. This is in accord with the earlier study on the Mg+ N2O reaction by Yarkony [68]. It is therefore worth to analyze thebehavior of the lower-lying doublet PESs in this region as a function ofthe angle � NNO. Their sections, corresponding to M = Li are shown inFigure 4.

In the reactant region (θ ≈ 180o), the ground state PES correspondsto the covalent state of the reactants with the unpaired electron re-siding on the 2s MO of Li, whereas the lowest excited state can beregarded as the charge-transfer ionic state with the unpaired electronon the nitrous oxide’s LUMO. The product region (θ < 155o) is char-acterized by a lower-energy charge-transfer ionic state. Hence, we thinkthat these two 2A′ PESs interact with each other and likely undergo a

Nitrous Oxide: Electron Attachment and Reactions with Metal Atoms 17

Li( S ) + N O(X )2 1 +

1/2 2 �

LiO(X ) + N (X )2 1 +� �2 g

7.5

Reaction coordinate

Energ

y,kcal/

mol

I

1TS

TS2

II

Li ( S) + N O ( )+ 1

2

-A2 ( A )

2

1

Li( S ) + N O( A ), cyclic2 1

1/2 2 1

Electrontransfer

4.56.0

38.5

Nitrogen

Oxygen

Lithium

III

Figure 1.2. Potential energy profiles of the reaction (1). Relative CASSCF-MP2(11/12)/6-311G(d) energies are given in kcal/mol.

18

weak avoided crossing in the region of covalent repulsion (θ ≈ 160o), asdemonstrated in Figure 3. Since the energy separation at this θ value isca. 30 kcal/mol, these reaction pathways are confined to the lowest adi-abatic PES. In other words, this would exclude the pathway where N2Odirectly “harpoons” the lithium atom and implies that the electron andoxygen atom transfers occur nearly simultaneously. This lends a supportfor a conclusion drawn in ref. [67] where a short-range electron transfermechanism on a single PES has been surmised as a second pathway.

TS1 is placed energetically lower than TS2: 4.5 vs. 6.0 kcal/mol forLi and 8.8 vs. 22.1 kcal/mol for Cu. We suggest that if the experimen-tally observed departure from Arrhenius plot is linked to an opening of ahigher-energy channel, associated with a trans-TS, then such energy dif-ferences may pinpoint the reason why the reaction with lithium exhibitsa non-Arrhenius behavior starting from 600 K, whereas the reaction withcopper shows a similar regime above 1190 K. In the latter case, the al-ternative rection pathway with the activation barrier of 22.1 kcal/mol(for the reaction enthalpy being 16.3 kcal/mol) is much less accessibleand embarks on to interfere perceptibly with the minimum-energy path-way at higher temperatures. Being transition structures, TS1 and TS2

have a single imaginary frequency at 1481i (TSLi1 ), 2090i (TSLi

2 ), 593i(TSCu

1 ), and 567i (TSCu2 ) cm−1. It is associated with the normal mode

characterized by a predominant NNO bending character, mixed with theLi-O and N-O stretches for Li and with N-O stretch for Cu and func-tioning as the N-O bond-breaker and M-O bond-maker. It is worthwhilementioning an existence of a third reaction pathway starting at the 1A1

cyclic form of nitrous oxide and correlates with the T-shaped complexIII. Due to its large electron affinity of 2.68 eV [88], the electron transferalong this pathway is likely to occur non-adiabatically, resembling to acertain extent the “harpoon” mechanism. Since the cyclic form of N2Olies, as estimated in earlier works, by 2.55 - 2.72 eV [121, 88] above itslinear isomer, such pathway seems to be rather hardly accessible underconditions examined by Plane [67].

4.2 Product channelsThe product (ionic) portion of PESs describing the reaction (1) com-

prises two energy-minimum structures: one of which is the linear struc-ture II residing in its global minimum, and the other, in case of M = Li,is a T-shaped structure III placed at 6.0 kcal/mol higher in energy. IIis formed by a linear arrangement of the molecular nitrogen and metaloxide, where the latter is bound to N2 mainly by metal, being sepa-rated from the former by only 2.202 A (IILi) and 1.844 A (IICu). The

Nitrous Oxide: Electron Attachment and Reactions with Metal Atoms 19

50.5 (45.8)

8.8

22.1

Cu 3d + N O ( )( A )+ 10 2( ) 2

�A2 1

Cu(4s S ) + N O(X )2 1 +

1/2 2 �

CuO(X ) + N (X )2 1 +� �2 g

41.8

20.2 (20.8)

Reaction coordinate

En

erg

y,

kcal/

mo

l

~ 170

I

II

2TS

TS1

Saddle24TS

III

II

3TS

II

III19.2 (20.3)

12.8 (15.9)

CuO(X ) + N (X )2 1 +� �2 g

(8.7)

(15.2)

(37.3)

6.0 (7.5)

13.2 (12.7)

Nitrogen

Oxygen

Copper

Figure 1.3. Potential energy profiles of the reaction (1) with M = Cu. RelativeCCSD(T)/TZVP and B3LYP/TZVP (in parentheses) energies (in kcal/mol), takenat the B3LYP/TZVP geometries, are indicated.

-0.72

-0.71

-0.7

-0.69

-0.68

-0.67

-0.66

-0.65

-0.64

-0.63

145 150 155 160 165 170 175 180

Ene

rgy

+19

1,a.

u.

�, degree

12A�

22A�

0

-0.72

-0.71

-0.7

-0.69

-0.68

-0.67

-0.66

-0.65

-0.64

140 145 150 155 160 165 170 175 180

Ener

gy+

191,

a.u.

�, degree

12A�

22A�

0

Figure 1.4. The CASSCF-MP2 sections of the ground and first excited PESs of thereaction (1) with M = Li as functions of the NNO angle. The remaining geometricalparameters are fixed at their optimized values in TS1 (left) and TS2 (right).

20

metal-nitrogen bond is appreciably stronger for M = Cu, as reflected bythe frequency of the (M-N) stretching vibration of 164 cm−1 (IILi) and352 cm−1 (IICu). There are obviously two identical linear structuresII, linked via the T-shaped transition state TS3 placed at 4.4 kcal/mol(TS3

Li) and 12.8 kcal/mol (TS3Cu) above the relevant complex II.

The T-shaped complex IIILi resembles the structure of the ground-state NNO− A2 interacting with the metal cation. This is apparentlya consequence of the fact that molecular orbitals of lithium oxide aremainly constructed from those of oxygen anion. Therein, the lithiumoxide is bound to N2 mainly by oxygen atom, being separated fromboth nitrogens by a longer distance of 3.027 A. In case of M = Cu, thecorresponding T-shaped structure represents either a transition stateTS4 or a saddle point Saddle2, depending on the N· · ·O distance. Theformer links two identical T-shaped complexes IIICu, in which CuOis bonded to molecular nitrogen mainly by copper atom. The latter,Saddle2, connects transition structures TS3

Cu and TS4Cu. These subtle

patterns of the ionic part of PES of the reaction (1) with copper aredisplayed in insert of Figure 3. In fact, the reaction pathway that passesthrough the structures IICu → TS3

Cu → TS4Cu → IIICu, establishes

another channel of the reaction (1) which might also contribute to itsnon-Arrhenius behavior. Notice that the structures on the ionic part ofthe PESs in both Li and Cu cases can be partitioned into two classes:the one with metal approaching the weak bond between O and N2 inA2 whereas the other originates from metal atom approaching O of A2

perpendicular to N-N bond. After all the studied reaction pathwaysbarrierlessly exit to the product limits CuO (X 2Π) + N2 (X 1Σ+

g ) orLiO(X 2Π)+N2(X 1Σ+

g ).The existence of a finite lifetime linear complexes II seems to be nicely

confirmed by the earlier experimental photoelectron spectrum by Wrightet al. [147] who studied the product channel of the lithium reaction (1).These authors observed two broad bands with the corresponding verticalionization energies of 8.8±0.2 and 9.4±0.1 eV, respectively. Based onthe results presented above, it is reasonable to suggest that the observedspectrum is associated with ionization of a mixure containing II andseparated LiO, rather than solely that of the latter. Indeed, II is boundby 7.5 (CASSCF-MP2/6-311G(d)) and 6.3 kcal/mol (CASSCF-MP2/6-311+G(d)) with respect to the ground state separated products. Hence,the weak onset of a lower-energy band at 7.6 eV [147] should likely be in-terpreted as the first adiabatic energy of II, which, due to a stabilizationof a positive charge within this complex, is expected to be lower thanthat of LiO (8.44-9.0 eV, [148, 149, 150, 151, 152, 153, 154, 155]). This isaffirmatively demonstrated by (U)MP2/6-311G(d) and (U)CCSD(T)/6-

Nitrous Oxide: Electron Attachment and Reactions with Metal Atoms 21

N N Li O2.202 1.7121.106

OLiN N

1.1062.087 2.143

II

II+

Figure 1.5. CASSCF(11/12)/6-311G(d) equilibrium structures of the global-minimum complex II and its cation II+. Bond lengths in A.

311G(d) calculations that yield the first adiabatic ionization energy forII of 7.6 eV perfectly matching the experimental estimate [147]. Theequilibrium ground state geometry of the cation II+ (Figure 5) markedlydiffers from that of the neutral, exhibiting the Li· · ·O bond length of2.143 A (vs. 1.712 A in II) and the Li· · ·N length of 2.087 A (vs. 2.202A in II); the lithium atom is now closer to nitrogen. This causes signifi-cant changes in normal vibrations. In II+, the low frequency vibrationsresemble those of a strong hydrogen bonded complex indicating thatthe lithium atom moves in a rather symmetric single-minimum poten-tial well, whereas in the parent neutral structure II, it is mainly bondedto oxygen. In particular, while the vibration at 453 cm−1 in cationdescribes a migration of Li between O and N, the mode at 175 cm−1

corresponds to a N· · ·O stretch. These features are likely relevant to arather broad nature of the corresponding bands in experimental photo-electron spectrum.

5. EpilogueA thorough theoretical investigation of the lower-lying PES for disso-

ciative electron attachment of nitrous oxide and its reaction with lithiumand copper atoms provides us with a clear view on the molecular mech-anisms of such processes, whose key features can be summarized as fol-lows:

Addition of an electron to the linear NNO ground state leads firstto an open-chain bound anion lying 6.0 kcal/mol above neutral, whichis further transformed upon cyclization with a barrier of 6.0 kcal/molto a T-shaped anionic complex A2. The latter is more stable than theopen isomer and thus can be considered as the adiabatic anion, lying atmost 250 cm−1 above the neutral ground state. This process of dissocia-

22

tive electron attachment plays an important role in predetermining themechanistic features for the reaction (1) which, as we have shown for M= Li and M = Cu, exhibit a rather similar topology of the ground statePES. Starting from the reactant complex I which foreshadows the reac-tant channel of this reaction, excursions along the NNO in-plane bendingmode lead to two transition state linkers TS1 and TS2. The existence ofboth transition structures, and hence, a two-channel reaction pathway,is stipulated by two possible directions of the NNO in-plane bendingvibration within complex I, that brings reactants to the molecular ar-rangement, suitable for a charge transfer. Although the nature of bothtransition structures is rather similar, they differ from each other ener-getically. At lower temperatures, the former activation barrier plays adominant role in determining the reaction kinetics. It starts to interplayappreciably with the higher activation barrier above 600 K, where thelatter channel mainly controls the curvature of the 1/T plot. The energyseparation between two lower-lying PES’s (2A′), calculated for M = Lito be equal to about 30 kcal/mol in both transition regions allows thedominance of a process by non-adiabatic electron transfer to be ruledout.

The lower-lying product portion of PES is found to be far from ex-haustion by the presence of solely N2 and lithium oxide, but instead, itis populated by a variety of structures which can be considered as origi-nating from a weakly-bound T-shaped anion interacting with metal. Inboth lithium and copper cases, the linear structure II attains the globalminimum of the total PES, being 7.5 and 20.2 kcal/mol, respectively,more stable than the corresponding product limits. For M = Li, theexistence of such structure also provides us with an explanation for theobservation of broad bands and relatively low ionization energies in theearlier experimental photoelectron spectrum [147].

Needless to say again that both Li and Cu atoms are having ns un-paired electrons implying doublet electronic states. The scenario forreactions involving earth alkaline metals (Mg, Ca, K, etc.) and transi-tion metals, interacting in higher spin states, is expected to be different.Nevertheless, there is a solid reason to believe that most correspondingprocesses are governed by an avoided crossing of lower-lying electronicstates, giving rise to the key transition state structure. Much still re-mains to be done to reveal the behavior of these chemically fundamentaland socially pertinent reactions. Let us hope that in a foreseeable future,nitrous oxide could fully recover its primary status as “the laughing gas”(“no laughing matter” [8]), a gas for all seasons!

REFERENCES 23

AcknowledgmentsThe authors are indebted to the government of the Flemish Commu-

nity of Belgium, the Fund for Scientific Research (FWO-Vlaanderen)and the KULeuven Research Council for continuing financial support.We also thank Chris Vinckier for valuable discussion and support.

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