Clifford W Fong Eigenenergy
email: cwfong@internode.on.net

Modafinil and modafinil analogues: free radical mechanism of the eugeroic and cognitive enhancment effect

Keywords: Modafinil, modafinil-like analogues, eugeroic effect, cognitive enhancement, free radicals, quantum mechanics

Modafinil and modafinil analogues: free radical mechanism of the eugeroic and cognitive enhancment effect

Introduction

The mechanism of how modafinil exerts its eugeroic or wake-promoting effect is controversial. [START_REF] Zare | Wake-Promoting Agents, Insights into Clinical Use and Molecular Perspectives[END_REF][START_REF] Kim | Practical Use and Risk of Modafinil, a Novel Waking Drug[END_REF][START_REF] Gerrard | Mechanisms of modafinil: A review of current research[END_REF][Zare 2016, Kim 2012, [START_REF] Gerrard | Mechanisms of modafinil: A review of current research[END_REF]] Modafinil is known to weakly but selectively bind with the dopamine transporter (DAT) and possibly exert some of its eugeroic effect by disrupting the transport effect of DAT and hence raising extracellular concentrations of dopamine (DA), which results in wakefulness. Modafinil is thus thought to be a dopamine reuptake inhibitor. DAT knockout mice are known to be unresponsive to modafinil. Modafinil is also thought to be dependent on catecholaminergic (dopaminergic and adrenergic) signaling for its wake-promoting effects. [START_REF] Wisor | Modafinil as a catecholaminergic agent: empirical evidence and unanswered questions[END_REF][START_REF] Wisor | Dopaminergic role in stimulant-induced wakefulness[END_REF][ [START_REF] Wisor | Dopaminergic role in stimulant-induced wakefulness[END_REF][START_REF] Wisor | Modafinil as a catecholaminergic agent: empirical evidence and unanswered questions[END_REF]] However, there is significant evidence that the eugeroic action of modafinil includes other processes beside acting as a dopamine reuptake inhibitor. For example, a structure activity study of modafinil analogues found that DAT inhibition did not correlate with wakefulness-promoting effects in animals, and a number of analogues without any significant inhibition of the DAT still produced wakefulness-promoting effects. [START_REF] Dunn | Wake promoting agents: Search for next generation modafinil, lessons learned: Part III[END_REF][START_REF] Dunn | Wake-promoting agents: Search for next generation modafinil: Part I[END_REF] [Dunn 2012] Another study found that modafinil has a long duration of action, with its wake promoting properties largely arising from dopaminergic activity. [START_REF] Turner | Duration of activity and mode of action of modafinil: Studies on sleep and wakefulness in humans[END_REF] [START_REF] Turner | Duration of activity and mode of action of modafinil: Studies on sleep and wakefulness in humans[END_REF]] Other possible mechanisms for the eugeroic effect besides the dopamine reuptake inhibition include activation of the orexin system. [START_REF] Mereu | The neurobiology of modafinil as an enhancer of cognitive performance and a potential treatment for substance use disorders[END_REF] [START_REF] Mereu | The neurobiology of modafinil as an enhancer of cognitive performance and a potential treatment for substance use disorders[END_REF] However there is also evidence that free radicals may be related to sleep induction as well as cellular damage, and that modafinil has the ability to oppose both of these effects. It is thought that modafinil could directly act on enzymes in the brain's free-radical scavenging system (eg. glutathione peroxidase or superoxide dismutase) and hence directly reduce freeradical levels. This may account for modafinil's known ability to increase the cortical creatine-phosphocreatine pool by directly affecting antioxidant enzymes and decreasing free radical production. ROS affect ATP production, so if modafinil can decrease ROS levels, ATP synthesis can be increased, reducing the need for sleep, as ATP metabolism causes energy reduction and the need for sleep. Cytochrome enzymes in the inner mitachrondrial membrane transport chain may be involved with modafinil's electron accepting ability from superoxide species in this environment. Modifinil is known to suppress the CYP2C9 enzyme and hence reactive oxygen species in the brain, and hence possibly promoting better mitachrondrial function and wakefulness. [START_REF] Gerrard | Mechanisms of modafinil: A review of current research[END_REF][ [START_REF] Gerrard | Mechanisms of modafinil: A review of current research[END_REF] There is evidence that modifinil causes oxidative damage in the amygdala, hippocampus, and striatum of rats at high doses. Modafinil caused behavioural changes, particularly increased locomotor activity, depending on the administered dose. [START_REF] Ornell | Modafinil Effects on Behavior and Oxidative Damage Parameters in Brain of Wistar Rats[END_REF] [START_REF] Ornell | Modafinil Effects on Behavior and Oxidative Damage Parameters in Brain of Wistar Rats[END_REF]] It seems likely that the eugeroic effect of modafinil and modafinil-like analogues has a multi-factor mechanism.

A meta analysis of published studies on sleep deprivation and oxidative stress in the brain was used to test the hypothesis that that sleep is a dynamic-resting state with antioxidative properties. Wakefulness is thought to involve high neuronal metabolism and neuronal electrical potentials, and requires high oxygen levels, and therefore oxidants. Sleep is a state with an increased antioxidant activity which promotes a brain protection against free radicals by lowering oxidant production. ROS and other oxidative stress markers can accumulate in the brain during wakefulness, and so behave as sleep promoters. [START_REF] Villafuerte | Sleep Deprivation and Oxidative Stress in Animal Models: A Systematic Review[END_REF] [START_REF] Villafuerte | Sleep Deprivation and Oxidative Stress in Animal Models: A Systematic Review[END_REF] The neuroprotective antioxidant properties of modafinil may arise from inhibitory interaction with the enzymes (eg mitochondrial CYP-450 family, including CYP2C) responsible for the free radical scavenging in the brain. Increased levels of mitochondrial free radicals may lead to increased levels of adenosine, a sleep promoting factor, in the brain. Mitachondrial activity may be a major factor in sleep inducing activity. [START_REF] Zare | Wake-Promoting Agents, Insights into Clinical Use and Molecular Perspectives[END_REF][ [START_REF] Zare | Wake-Promoting Agents, Insights into Clinical Use and Molecular Perspectives[END_REF] Modafinil has been shown to have cognitive enhancement properties independent of its effect on sleep disorders, a "smart drug". The cognitive enhancement was observed in non-sleepdeprived individuals. [START_REF] Battleday | Modafinil for cognitive neuroenhancement in healthy nonsleep-deprived subjects: a systematic review[END_REF] [START_REF] Battleday | Modafinil for cognitive neuroenhancement in healthy nonsleep-deprived subjects: a systematic review[END_REF]] Modafinil is thought to exert its neurochemical and cognitive altering ability by primarily affecting catecholamines, as well as serotonin, glutamate, gamma amino-butyric acid, orexin, and histamine systems in the brain. [START_REF] Minzenberg | Modafinil: A Review of Neurochemical Actions and Effects on Cognition[END_REF] [ [START_REF] Minzenberg | Modafinil: A Review of Neurochemical Actions and Effects on Cognition[END_REF] We have previously developed a structure activity model of the binding (Ki nM) of 20 modafinil-like analogues to DAT (equation 1), where ΔG desolv,CDS is the free energy of desolvation of the analogues, ΔG lipo,CDS is the free energy of lipophilicity or hydrophobicity in n-octane, DM is the dipole moment in water, and Vol is the molecular volume in water. The general form of this linear free energy relationship (LFER) equation has been previously applied to passive and facilitated diffusion of a wide range of drugs crossing the blood brain barrier, the active competitive transport of tyrosine kinase inhibitors by transporters, cyclindependent kinase inhibitors and HIV-1 protease inhibitors, and the penetration of drugs into tumours. [START_REF] Fong | Permeability of the Blood-Brain Barrier: Molecular Mechanism of Transport of Drugs and Physiologically Important Compounds[END_REF][START_REF] Fong | Drug discovery model using molecular orbital computations: tyrosine kinase inhibitors[END_REF][START_REF] Fong | The extravascular penetration of tirapazamine into tumours: a predictive model of the transport and efficacy of hypoxia specific cytotoxic analogues and the potential use of cucurbiturils to facilitate delivery[END_REF][START_REF] Fong | The effect of desolvation on the binding of inhibitors to HIV-1 protease and cyclin-dependent kinases: Causes of resistance[END_REF][START_REF] Fong | Predicting PARP inhibitory activity -A novel quantum mechanical based model[END_REF][START_REF] Fong | A novel predictive model for the anti-bacterial, anti-malarial and hERG cardiac QT prolongation properties of fluoroquinolones[END_REF][START_REF] Fong | Statins in therapy: Cellular transport, side effects, drug-drug interactions and cytotoxicity -the unrecognized role of lactones[END_REF][START_REF] Fong | Modafinil mechanism of action: Inhibition of the dopamine monoamine transporter DAT by modafinil like and piperidine-based analogues and the role of free radicals in the eugeroic effect[END_REF][ Fong 2016-18] Eq 1 DAT Binding = -4604.8 ΔG desolv,CDS -4588.9 ΔG lipo,CDS -3244.1 DM -2668.9 Vol -26413.5

Where R 2 = 0.518, SEE = 8332, SE(ΔG desolvCDS ) = 1687.8, SE(ΔG lipoCDS ) = 1.413.2, SE(Dipole Moment) = 1679.4, SE(Vol) = 1379.2, F=4.022, Significance=0.020

It is noted that the LFER eq 1 includes substituted modafinil analogues (XC 6 H 4 )CH-S(=O)-CH 2-C(=O)-NR 1 R 2 as well as the sulphide analogues (XC 6 H 4 )CH-S-CH 2-C(=O)-NR 1 R 2 . [START_REF] Fong | Modafinil mechanism of action: Inhibition of the dopamine monoamine transporter DAT by modafinil like and piperidine-based analogues and the role of free radicals in the eugeroic effect[END_REF][START_REF] Cao | SARs at the Monoamine Transporters for a Novel Series of Modafinil Analogues[END_REF][Fong 2018, Cao 2011] However while eq 1 is of interest to how the properties modafinil is related to binding to DAT, the focus of this paper is on the molecular mechanism of how modafinil and its analogues may actually exert their eugeroic and cognitive effects.

Study objectives:

Investigate whether modafinil could be involved in free radical processes when exerting its eugeroic and cognitive effects. Investigate how structural variations to modafinil and its analogues can be used to predict eugeroic and cognitive enhancing capacity to design better modafinil-like analogues.

Results

Free radical eugeroic and cognitive enhancement mechanism of modafinil and modafinil-like analogues

The possibility of modafinil exerting its eugeroic and cognitive behavioural effect by a free radical mechanism has been investigated by examining the behaviour of modafinil after transfer or attachment of an electron in water. This bioreduction process may arise in the brain from superoxide species and or mitochondrial sources or cytochrome sources. The anion radical formed from the attachment of an electron to modafinil is shown to exhibit extension and then cleavage of the diphenylmethyl carbon to sulphoxide (C---S) bond, as shown in the transition state in Figure 1(a). This dissociative electron transfer (DET) or attachment reaction [START_REF] Saveant | Dissociative electron transfer[END_REF][START_REF] Maran | Dissociative electron transfer[END_REF][Malan 2002, [START_REF] Saveant | Dissociative electron transfer[END_REF]] also occurs in the substituted modafinil analogues (C 6 H 4 X) 2 -CH-S(=O)-Z and is driven by the delocalized resonance stabilization of the excess electron over the (C 6 H 4 X) 2 -CH-S(=O)-moiety in the HOMO. There is a decrease of negative charge on the benzylic (C 6 H 5 ) 2 -CH-atom and a decrease of positive charge on the S atom when elongation of the C---S bond occurs upon the attachment of an electron to modifinil in water. The free energy of activation ΔG * act for the water solvated modafinil transition state (TS), C---S bond length 3.02 Å, shown in Figure 1(a) is estimated to be about -15.4 kcal/mol assuming a ΔG for the hydrated electron of -34.6 kcal/mol. [START_REF] Zhan | The Nature and Absolute Hydration Free Energy of the Solvated Electron in Water[END_REF] [START_REF] Zhan | The Nature and Absolute Hydration Free Energy of the Solvated Electron in Water[END_REF]] The ΔG * act for the vertical electron addition to the starting conformation of modafinil in water is -3.9 kcal/mol, followed by the adiabatic relaxation of the radical anion with a ΔG * act of -11.5 kcal/mol, resulting in an overall ΔG * act of -15.4 kcal/mol for the TS shown in Figure 1(a). The elongated benzylic methine C---S bond is 3.02 Å in Figure 1(a) compared to the starting compound with the bond length of 1.93 Å before dissociative electron attachment. The ease of formation of a TS for modafinil by electron attachment is dependent on the particular conformation, particularly the orientation of the -S(=O)-CH 2 -C(=O)NH 2 moiety to the almost coplanar (C 6 H 5 ) 2 CH-moiety. For example, in Figure 1(b) the ΔG * act for the vertical electron addition to the starting conformation of modafinil in water is -3.9 kcal/mol, whereas the conformation in Figure 1(c) has a value of -1.7 kcal/mol. The HOMO in Figure 1(b) more effectively spans the radical anion than it does in Figure 1(c) indicating greater delocalized electron stabilization, and effectively decreased stabilization of 1(c), so increasing the ease of bond elongation of the C---S bond in the vertical TS for conformation 1(c). The conformation in Figure 1(c) where the C---S bond elongation upon adiabatic relaxation is 3.02 Å has a ΔG * act which is lower by -12.9 kcal/mol for the equivalent conformation shown in Figure 1(b) with a C---S bond elongation of 3.02 Å. The enhanced stimulatory effect found by Zhu for 2-[(diphenylmethyl)thio]-N-(3chlorophenyl)acetamide over that for modafinil has been found to be due to the elongation of the S---CH 2 bond upon electron addition, whereas modafinil undergoes elongation of the benzylic methine C---S bond. Figure 4 shows the dissociative electron attachment of 2-[(diphenylmethyl)thio]-N-(3-chlorophenyl)acetamide in water. The S---CH 2 bond length is 2.60 Å, the benzylic methine C(H)---S bond is 1.88 Å in the radical anion TS, compared to values of 1.845 Å and 1.88 Å respectively in the neutral starting compound. It was found that different conformations of the S-CH 2 -C(=O)NH(3Cl-C 6 H 4 ) moiety had little effect on the ease of S---CH 2 bond elongation upon electron attachment. The ΔG * act for the TS state shown in Figure 4 Examination of the substituted mo behaviour upon electron attachment as that for modafinil with elongation of benzylic C bond. This analogue has a high DAT binding affinity ( compared to modafinil (K i = 3800 behaviour might be useful in psychostimulant abuse medication. However Dunn [START_REF] Dunn | Wake promoting agents: Search for next generation modafinil, lessons learned: Part III[END_REF][START_REF] Dunn | Wake-promoting agents: Search for next generation modafinil: Part I[END_REF] reported that fluorenol which is the 9 formed from 9-Fluorenyl-S(=O) itself shows a stronger eugeroic effect but a 59% weaker DAT binding ability than modifinil This result is unusual since fluorenol is a quite different mechanism to that of modafinil substituted modafinil analogue shown in Figure 6 shows the same behaviour upon electron attachment as that for modafinil with elongation of benzylic C high DAT binding affinity (K i = 2.5 nM) [START_REF] Cao | Novel and High Affinity 2-[(Diphenylmethyl)sulfinyl]acetamide (Modafinil) Analogues as Atypical Dopamine Transporter Inhibitors[END_REF][Cao 2016] = 3800 nM), and it was thought that the atypical DAT inhibitory behaviour might be useful in psychostimulant abuse medication.

Figure 1(a). Adiabatic transition state for dissociative electron attachment of modafinil in water

Fluorenyl analogues of modafinil

Another analogue which shows quite different behaviour to modafinil is the compound 9 (4N-piperazine-1N)-C(=O)-CH 3 [START_REF] Dunn | Wake promoting agents: Search for next generation modafinil, lessons learned: Part III[END_REF][START_REF] Dunn | Wake-promoting agents: Search for next generation modafinil: Part I[END_REF][Dunn 2012] effect compared to modafinil but a potent eugeroic effect. Figure 7 shows the effect of an electron attachment to this compound with elongation of the 9 carbon to S(=O) bond, very similar to the anion radical of modafinil. The HOMO is localized over the fluorenyl-sulphoxide moiety, with the LUMO delocalized over the The free energy of ΔG * act for the bond elongation (2.50 Å shown for the fluorenyl analogue in Figure 7 is estimated to be kcal/mol, similar to that shown for modafinil in Figure 1 shown in Figure 6 shows the same behaviour upon electron attachment as that for modafinil with elongation of benzylic C---S [START_REF] Cao | Novel and High Affinity 2-[(Diphenylmethyl)sulfinyl]acetamide (Modafinil) Analogues as Atypical Dopamine Transporter Inhibitors[END_REF]] atypical DAT inhibitory Another analogue which shows quite different behaviour to modafinil is the compound 9- [Dunn 2012] which shows potent eugeroic effect. Figure 7 shows the effect of an electron attachment to this compound with elongation of the fluorenylhe anion radical of modafinil. The HOMO is , with the LUMO delocalized over the Å compared to the is estimated to be igure 1(a) where the ΔG * act for indicative and not directly the conformations of the modafinil and

Biphenyl analogues of modafinil

Dunn 2012 [START_REF] Dunn | Wake promoting agents: Search for next generation modafinil, lessons learned: Part III[END_REF][START_REF] Dunn | Wake-promoting agents: Search for next generation modafinil: Part I[END_REF] and Louvet 2012 [START_REF] Louvet | Wake-promoting agents: Search for next generation modafinil: Part IV[END_REF] have also reported that biphenyl analogues and diphenylether derivative of modafinil also exhibit similar eugeroic effects and DAT binding to modafinil. These compounds all possess benzylic carbon atoms adjacent to the sulphoxide group (Figure 9). Both of these drugs did not show dissociative electron attachment upon 1 electron attachment, but did show dissociation of the benzylic C---S bond upon 2 electron attachment. It is clear from examination of the HOMOs that the (C 6 H 5 ) 2 CH-and 9-fluorenyl moieties (see Figures 1(a

Sulphide analogue of modafinil

It has been shown that diphenylmethyl p-nitrophenyl sulphide (DNPS) in dimethylformamide undergoes an electrochemical one-electron transfer to form the radical anion of DNPS (as identified by ESR spectrometry), followed by fission of the activated C-S bond to form the pnitrothiophenolate anion and diphenylmethyl radical. The dissociation of the radical anion is the rate determining step of the electrochemical processes. [START_REF] Farnia | Electrochemical reduction mechanism of organic sulphides. Diphenylmethyl p-nitrophenyl sulphide[END_REF] [START_REF] Farnia | Electrochemical reduction mechanism of organic sulphides. Diphenylmethyl p-nitrophenyl sulphide[END_REF]] Since the sulphide modafinil analogues are known to inhibit DAT and show some eugeroic behaviour, it was instructive to see if these drugs also underwent dissociative electron attachment like the parent modifinil. It was found that (C 6 H 5 ) 2 CH-S-CH 2 C(=O)NH 2 also undergoes dissociative electron attachment with elongation of the methine -(H)C---S bond, and the free energy of activation for the TS shown in Figure 10 is estimated to be -25.5 kcal/mol in water which can be compared to a value of -15.4 kcal/mol for the modafinil TS in Figure 1(a). The same dissociative behaviour was observed in dimethylformamide as in water.

Discussion

The effectiveness of eugeroic drugs and their cognition altering neurochemical action is dependent on the bioavailability of the drug, particularly its ability to enter brain and other cells. The ability of modafinil analogues to cross the blood brain barrier and cell membranes, and bind to the DAT, has been previously examined using equation 1. The modafinil-like analogues used to construct eq 1 are closely structurally related to modifinil itself.

With respect to the cognition altering neurochemical action of modafinil and analogues, there is little systematic investigation available. The in vivo CNS activity of various modifinil-like analogues has also been evaluated. For the series (C 6 H 5 ) 2 CHS(=O)CH 2 C(=O)NH(C 6 H 4 X) where X = H, 3-Cl, 4-Cl, 4-Et, 3,4-Cl, 4-NO 2 , 4-Br, all these analogues were CNS stimulants, except where X = H. The psychological performances of mice for wakefulness, exploratory activity, depression and anxiogenic and anxiolytic like effects were measured. [START_REF] Zhu | Synthesis and Biological Evaluation of Novel Diphenylthio-Acetamide Derivatives[END_REF][Lari 2013] These results were similar to those previously found by De Risi for the series (C 6 H 5 ) 2 CHS(=O)CH 2 C(=O)NH-R where R = Me, iPr, tBu were found to be stimulants, but where R = Et, piperidine or morpholine these compounds were found to be sedatives. For the series (C 6 H 4 X) 2 CHS(=O)CH 2 C(=O)NH 2 where (X,X) = (4-F,4-H), (4-Cl,4-H) and (4-F,4-H), these analogues were also found to be stimulants. These CNS activities were measured using electrically-evoked tritiated serotonin ([ 3 H]5-HT) efflux from rat cortical slices. [START_REF] Farnia | Electrochemical reduction mechanism of organic sulphides. Diphenylmethyl p-nitrophenyl sulphide[END_REF][De Risi 2008] It is noted that the modafinil-like analogues studied by Lari [START_REF] Lari | Synthesis and psychobiological evaluation of modafinil analogs in mice[END_REF] [START_REF] Lari | Synthesis and psychobiological evaluation of modafinil analogs in mice[END_REF]] and De Risi [START_REF] De Risi | Efficient synthesis and biological evaluation of two modafinil analogs[END_REF][De Risi 2008] all showed similar DET behaviour to modafinil upon electron attachment. These in-vivo results overall suggest that substitutions at the amide N group or at the phenyl groups adjacent to the sulphoxide moiety have an effect on CNS activity, with the majority of these substitutions resulting in stimulatory outcomes. These observations are consistent with a common and dominant cognition altering neurochemical mechanism which may be modulated by smaller steric effects at the amide N atom. The animal cognition altering studies of Lari [START_REF] Lari | Synthesis and psychobiological evaluation of modafinil analogs in mice[END_REF] and De Risi [START_REF] De Risi | Efficient synthesis and biological evaluation of two modafinil analogs[END_REF] on modafinil-like analogues suggests that modafinil-like analogues may have human cognitive enhancement properties similar to those found for modafinil [START_REF] Battleday | Modafinil for cognitive neuroenhancement in healthy nonsleep-deprived subjects: a systematic review[END_REF][Battleday] if the appropriate substitutions are made to the basic modafinil framework. It has been observed that there is a relationship between the administered modafinil dose and the emotional modulation of working memory performance of stressed mice, ie there was a decrease in the efficiency performance threshold of administered modafinil under stress. [START_REF] Pierard | Modafinil-induced modulation of working memory and plasma corticosterone in chronically-stressed mice[END_REF][Pierard] Recent reviews have examined mitochondrial stress transduction and the links between psychosomatic medicine, psychoneuroendocrinology, psychoneuroimmunology, and mitochondrial ROS and the electron transport chain. [START_REF] Picard | Psychological Stress and Mitochondria: A Conceptual Framework[END_REF][START_REF] Picard | Psychological Stress and Mitochondria: A Systematic Review[END_REF][ Picard 2018] It has also been reported that modafinil exhibits antioxidant and neuroprotective properties, while also increasing the cortical phosphocreatine pool, and that there is evidence of the involvement of free radicals. [START_REF] Zare | Wake-Promoting Agents, Insights into Clinical Use and Molecular Perspectives[END_REF][START_REF] Gerrard | Mechanisms of modafinil: A review of current research[END_REF][START_REF] Ornell | Modafinil Effects on Behavior and Oxidative Damage Parameters in Brain of Wistar Rats[END_REF][START_REF] Villafuerte | Sleep Deprivation and Oxidative Stress in Animal Models: A Systematic Review[END_REF][START_REF] Piérard | Effects of a vigilance-enhancing drug, modafinil, on rat brain metabolism: a 2D COSY 1H-NMR study[END_REF][Zare 2016, Gerrard 2007, Ornell 2014, Villafuerte 2015, Pierard 1995] Dimethylsulfoxide (DMSO) is a well known free radical scavenger, and has been recommended as a treatment for endotoxemia and systemic inflammatory response syndrome in horses because of its anti-inflammatory and reactive oxygen species (ROS)-scavenging benefits. DMSO was found to be a scavenger of hydroxyl radicals and an effective inhibitor of platelet aggregation in an in-vivo mouse model of pial arteriolar injury. [START_REF] Rosenblum | Dimethyl sulfoxide (DMSO) and glycerol, hydroxyl radical scavengers, impair platelet aggregation within and eliminate the accompanying vasodilation of injured mouse pial arterioles[END_REF][START_REF] Sprayberry | Robinson's Current Therapy in Equine Medicine[END_REF][Rosenblum 1982, Sprayberry 2015] Hence there is literature evidence that sulphoxides can form free radicals.

It has been shown that diphenylmethyl p-nitrophenyl sulphide (DNPS) undergoes electrochemical dissociative electron attachment in dimethylformamide, with cleavage of the methine C---S bond. [32][Farnia 1978] This observation is consistent with this study which shows that the sulphide analogue of modafinil (C 6 H 5 ) 2 CH-S-CH 2 C(=O)NH 2 also undergoes the methine C---S bond cleavage in water and dimethylformamide.

The dissociative electron attachment behaviour shown by modafinil and modafinil-like analogues is strongly indicative of a free radical mechanism for eugeroic and cognition altering neurochemical action of modafinil and analogues in the CNS. Similarly the eugeroic abilities of 9-fluorenyl derivatives can also be explained by a free radical mechanism. Since the modafinil like analogues are quite different in structure from the 9-fluorenyl derivatives, but both have a common basis in that both possess benzylic carbon moieties that can form stable radicals which can delocalize electron density over the aryl groups.

However, as shown by the DET behaviour of 2-[(diphenylmethyl)sulfinyl]-N-(3chlorophenyl)acetamide, 2-[(diphenylmethyl)thio]-N-(3-chlorophenyl)acetamide, the substitution of appropriate groups at the acetamide moiety can lead to quite different eugeroic capacity when the sulphinyl group is replaced by the sulphide group (see Figure 4). The enhanced stimulatory effect for 2-[(diphenylmethyl)thio]-N-(3-chlorophenyl)acetamide over that for modafinil has been found to be due to the elongation of the S---CH 2 bond upon electron addition, whereas modafinil undergoes elongation of the benzylic methine C---S bond (see Figure 4). It was also shown in Figure 5(a) and (b) that 2-[(diphenylmethyl)sulphinyl]-N-(3-chlorophenyl)acetamide that substituted N-acetamide groups in the appropriate conformation can give completely different DET behaviour. These observations can allow appropriate ab initio design of eugeroic behaviours in modafinil-like molecules. For example, substitution of the -C(=O)NH 2 group of modafinil by a -C(=O)-NH-R group where the R group can accommodate the excess electron after DET (by delocalizing the HOMO over the -C(=O)-NH-R group) can completely alter the resultant dissociative behaviour and the resultant eugeroic effect.

The observation that 9-Fluorenyl-S(=O)-CH 2 -C(=O)-(4N-piperazine-1N)-C(=O)-CH 3 which has been shown to exert a potent eugeroic effect but is a weak inhibitor of DAT shows a strong dissociation of the C9---S bond upon electron attachment is indicative of a free radical eugeroic effect that is largely independent of acting as a dopamine reuptake inhibitor. Previous structure activity studies of modafinil analogues found that DAT inhibition did not correlate with wakefulness-promoting effects in animals, and a number of analogues without any significant inhibition of the DAT still produced wakefulness-promoting effects. [START_REF] Dunn | Wake promoting agents: Search for next generation modafinil, lessons learned: Part III[END_REF][START_REF] Dunn | Wake-promoting agents: Search for next generation modafinil: Part I[END_REF]

[Dunn 2012]

Conclusions

Evidence is presented that strongly implicates the involvement of free radical species in the eugeroic and cognitive altering ability of modafinil and modafinil-like analogues via a dissociative electron transfer or attachment mechanism. Examination of modafinil and modafinil-like analogues which have been shown in the literature to exhibit wake promoting and cognitive psychobiological properties are shown in this study to produce free radicals when subject to electron attachment. Such observations are consistent with a free radical oxidative stress eugeroic and cognitive altering mechanism for modafinil and related analogues. It has also been shown that appropriate substitution of the acetamide moiety of modafinil can result in quite different DET behaviour, which holds promise of the ability to design and predict eugeroic and cognitive behaviour. In so far as DAT binding is a determinant of eugeroic and cognitive enhancing behaviour of modafinil-like analogues, the LFER with DAT binding can also help identify the molecular properties which govern binding behaviour.

Experimental

All calculations were carried out using the Gaussian 09 package. Energy optimisations for the various drugs were at the DFT/B3LYP/6-31G(d,p) (6d, 7f) level of theory for all atoms. Selected optimisations at the DFT/B3LYP/6-311 + G(d,p) (6d, 7f) level of theory gave very similar results to those at the lower level. Optimized structures were checked to ensure energy minima were located, with no negative frequencies. Anion optimisations, energy and thermodynamic calculations were conducted at the DFT/B3LYP/6-31 + G(d) (6d, 7f) level of theory with optimised geometries in water, using the IEFPCM/SMD solvent model. Similar results were obtained at the DFT/B3LYP/6-311 + G(d,p) (6d, 7f) and DFT/wB97XD/6-31+ G(d) (6d,7F) levels of theory. It was found that the free energy of ΔG * for bond elongations was not only dependent upon the particular conformation of modafinil and the studied analogues, but also on the level of theory. That is, ΔG * determined at the DFT/B3LYP/6-31 + G(d) (6d, 7f) level of theory were lower than those conducted at a DFT/B3LYP/6-31G(d,p) (6d, 7f) level of theory. This observation is consistent with a greater delocalization of the excess electron in the HOMOs using diffuse basis sets, and as shown in the results it is the location of the HOMO that determines bond elongation in the transition state upon electron attachment. The modest dependency on basis set and diffuse functions in DET processes have been previously noted. [START_REF] Soriano | A quantum mechanics-molecular mechanics study of dissociative electron transfer: The methylchloride radical anion in aqueous solution[END_REF][Soriano] However, only the degree (%) of elongation of bonds in the TSs of the DET reactions are dependent on basis sets or functionals. The wB97XD functional is known to better account for long range dispersion effects than the B3LYP functional.

With the 6-31G(d) basis set, the SMD model achieves mean unsigned errors of 0.6 -1.0 kcal/mol in the solvation free energies of tested neutrals and mean unsigned errors of 4 kcal/mol on average for ions. [START_REF] Marenich | Universal Solvation Model Based on Solute Electron Density and on a Continuum Model of the Solvent Defined by the Bulk Dielectric Constant and Atomic Surface Tensions[END_REF] [START_REF] Marenich | Universal Solvation Model Based on Solute Electron Density and on a Continuum Model of the Solvent Defined by the Bulk Dielectric Constant and Atomic Surface Tensions[END_REF]] The 6-31G(d,p) basis set has been used to calculate absolute free energies of solvation and compare these data with experimental results for more than 500 neutral and charged compounds. calculated values were in good agreement with experimental results across a wide range of compounds. [START_REF] Rayne | Accuracy of computational solvation free energies for neutral and ionic compounds: Dependence on level of theory and solvent model[END_REF][START_REF] Rizzo | Estimation of Absolute Free Energies of Hydration Using Continuum Methods: Accuracy of Partial Charge Models and Optimization of Nonpolar Contributions[END_REF][Rayne 2010, [START_REF] Rizzo | Estimation of Absolute Free Energies of Hydration Using Continuum Methods: Accuracy of Partial Charge Models and Optimization of Nonpolar Contributions[END_REF]] Adding diffuse functions to the 6-31G* basis set (ie 6-31 + G**) had no significant effect on the solvation energies with a difference of less than 1% observed in solvents, which is within the literature error range for the IEFPCM/SMD solvent model. HOMO and LUMO calculations included both delocalized and localized orbitals (NBO).

The ease of bond cleavage of the benzylic methine C---S bond of modafinil by dissociative electron attachment was confirmed by observing that this process occurs in the gas phase as well as in water. The electron impact mass spectrometry study of modafinil gives the main ionization fragmentation pathway involving a fragment m/e of 167, ie the (C 6 H 5 ) 2 CH ion. [START_REF] Dubey | A novel study of screening and confirmation of modafinil, adrafinil and their metabolite modafinilic acid under EI-GC-MS and ESI-LC-MS-MS ionization[END_REF] [START_REF] Dubey | A novel study of screening and confirmation of modafinil, adrafinil and their metabolite modafinilic acid under EI-GC-MS and ESI-LC-MS-MS ionization[END_REF] It is noted that high computational accuracy for each species in different environments is not the focus of this study, but comparative differences between various species is the aim of the study. The literature values for DAT Ki used in the multiple regression LFER equation have much higher experimental uncertainties than the calculated molecular properties. The statistical analyses include the multiple correlation coefficient R 2 , the F test of significance, standards errors for the estimates (SEE) and each of the variables SE(ΔG desolCDS ), SE(ΔG lipoCDS ), SE(Dipole Moment), SE (Molecular Volume), as calculated from "t" distribution statistics. Residual analysis was used to identify outliers.

Figures 1 (

 1 Figures 1(b) and 1(c). Vertical transition states for different conformations of modafinil radical anion showing the HOMOs.

Figure 2 .Figure 3 .

 23 Figure 2. 2-[(diphenylmethyl)sulfinyl]-N-(3-chlorophenyl)acetamide Figure 2(a) left, 2-[(diphenylmethyl)thio]-N-(3-chlorophenyl)acetamide Figure 2(b) right

  was -17.0 kcal/mol.

Figure 4 .

 4 Figure 4. Dissociative electron attachment of 2-[(diphenylmethyl)thio]-N-(3chlorophenyl)acetamide in water showing the HOMO mainly located over the S-CH 2 -C(=O)NH(3Cl-C 6 H 4 ) moiety.

Figure 5 .

 5 Figure 5. Dissociative electron attachment of 2-[(diphenylmethyl)sulphinyl]-N-(3chlorophenyl)acetamide in water: (a) the left conformation with the non planar -S-N-(3chlorophenyl)acetamide moiety shows the HOMO mainly located over the -S-N-(3chlorophenyl)acetamide moiety and elongation of the S(=O)---CH 2 bond, (b) the right conformation with the planar -S-N-(3-chlorophenyl)acetamide moiety shows the HOMO mainly located over the diphenylmethylsulphinyl moiety and elongation of the benzylic methine C(H)---S(=O) bond.

Figure 6 Fluorenyl

 6 Figure 6

Figure 7 9

 7 Figure 7 9-fluorenyl-S(=O)state for dissociative electron attachment in water fluororenyl-S(=O)-moiety

  (a) where the Å was -15.4 kcal/mol. The results are indicative and somewhat dependent on the conformations of the modafinil and -CH 2 -C(=O)-(4N-piperazine-1N)-C(=O)-CH 3 for dissociative electron attachment in water showing the HOMO located over the 9 reported that fluorenol which is the 9-hydroxyfluorene derivative S(=O)-CH 2 -C(=O)-(4N-piperazine-1N)-C(=O)-CH itself shows a stronger eugeroic effect but a 59% weaker DAT binding ability than modifinil t is unusual since fluorenol is vastly structurally different from modifinil indicating to that of modafinil. Since the fluorene moiety can easily

3Figure 8

 8 Figure 8 showing the 9-fluorenyl-S(=O)-CH 2 -C(=O)-(4N-piperazine-1N)-C(=O)-CH 3 , 9fluorenol and protonated 9-fluorenol

  ) and 7) can better stabilize electron density than can the benzylic moieties in Figure9, and the TSs for CH 2 ---S bond dissociation in these two drugs requires a second electron to be attached to activate the CH 2 ---S bond. The free energy of activation ΔG * act for the water solvated ether analogue transition state shown in Figure9is estimated to be about -44 kcal/mol, which compares to the TS shown in Figure1(a) for modafinil of -15.4 kcal/mol.

Figure 9

 9 Figure 9 Showing biphenyl and ether analogues and transition state for two electron dissociative electron attachment to the ether compound.

Figure 10 .

 10 Figure 10. Transition state for dissociative electron attachment of (C 6 H 5 ) 2 CH-S-CH 2 C(=O)NH 2 in water


. It can be viewed as an 8-dimensional subspace of the 32-dimensional representation space postulated in Conformal Geometric Algebra. The larger representation space results from a homogeneous freedom of the origin within E 3 , which does not concern us in this paper.

It is easy to verify that the even subalgebras of Cl λ 1,0 , Cl λ 2,0 and Cl λ 3,0 are indeed isomorphic to R, C and H, respectively.

Eight-dimensional Octonion-like but Associative Normed Division Algebra

We present an eight-dimensional even sub-algebra of the 2 4 = 16-dimensional associative Clifford algebra Cl4,0 and show that its eight-dimensional elements denoted as X and Y respect the norm relation ||XY|| = ||X|| ||Y||, thus forming an octonion-like but associative normed division algebra.

Consider the following eight-dimensional vector space with graded Clifford-algebraic basis and orientation λ = ±1: Cl λ 3,0 = span{ 1, λe x , λe y , λe z , λe x e y , λe z e x , λe y e z , λe x e y e z }.

(1)

Here { e x , e y , e z } is a set of anti-commuting orthonormal vectors in IR 3 such that e j e i = -e i e j for any i, j = x, y, or z. In general the vectors e i satisfy the following geometric product in this associative but non-commutative algebra [START_REF] Doran | Geometric Algebra for Physicists[END_REF]:

e i e j = e i • e j + e i ∧ e j , (2)
with

e i • e j := 1 2 {e i e j + e j e i } ( 3
)
being the symmetric inner product and

e i ∧ e j := 1 2 {e i e j -e j e i } (4)
being the anti-symmetric outer product, giving (e i ∧ e j ) 2 = -1. There are thus basis elements of four different grades in this algebra: An identity element e 2 i = 1 of grade-0, three orthonormal vectors e i of grade-1, three orthonormal bivectors e j e k of grade-2, and a trivector I 3 = e i e j e k of grade-3 representing a volume element in IR 3 . Since in IR 3 there are 2 3 = 8 ways to combine the vectors e i using the geometric product (2) such that no two products are linearly dependent, the resulting algebra, Cl λ 3,0 , is a linear vector space of eight dimensions, spanned by these graded bases.

In this paper we are interested in a certain conformal completion 1 of this algebra, originally presented in Ref. [START_REF] Christian | Quantum correlations are weaved by the spinors of the Euclidean primitives[END_REF]. This is accomplished by using an additional vector, e ∞ , to close the lines and volumes of the Euclidean space, giving K λ = span{ 1, λe x e y , λe z e x , λe y e z , λe x e ∞ , λe y e ∞ , λe z e ∞ , λI 3 e ∞ }.

(

With unit vector e ∞ , this is an eight-dimensional even sub-algebra of the 2 4 = 16-dimensional Clifford algebra Cl 4,0 .

As an eight-dimensional linear vector space, K λ has some remarkable properties [START_REF] Christian | Quantum correlations are weaved by the spinors of the Euclidean primitives[END_REF]. To begin with, it is closed under multiplication. Suppose X and Y are two vectors in K λ . Then X and Y can be expanded in the graded basis of K λ :

X = X 0 + X 1 λe x e y + X 2 λe z e x + X 3 λe y e z + X 4 λe x e ∞ + X 5 λe y e ∞ + X 6 λe z e ∞ + X 7 λI 3 e ∞ ( 6
)
and

Y = Y 0 + Y 1 λe x e y + Y 2 λe z e x + Y 3 λe y e z + Y 4 λe x e ∞ + Y 5 λe y e ∞ + Y 6 λe z e ∞ + Y 7 λI 3 e ∞ . ( 7
) * 1 λ exey λ ezex λ eyez λ exe∞ λ eye∞ λ eze∞ λ I3e∞ 1 1 λ exey λ ezex λ eyez λ exe∞ λ eye∞ λ eze∞ λ I3e∞ λ exey λ exey -1 eyez -ezex -eye∞ exe∞ I3e∞ -eze∞ λ ezex λ ezex -eyez -1 exey eze∞ I3e∞ -exe∞ -eye∞ λ eyez λ eyez ezex -exey -1 I3e∞ -eze∞ eye∞ -exe∞ λ exe∞ λ exe∞ eye∞ -eze∞ I3e∞ -1 -exey ezex -eyez λ eye∞ λ eye∞ -exe∞ I3e∞ eze∞ exey -1 -eyez -ezex λ eze∞ λ eze∞ I3e∞ exe∞ -eye∞ -ezex eyez -1 -exey λ I3e∞ λ I3e∞ -eze∞ -eye∞ -exe∞ -eyez -ezex -exey 1 TABLE I: Multiplication Table for a "Conformal Geometric Algebra 1 " of E 3 .
Here I3 = exeyez, e 2 ∞ = +1, and λ = ±1.

And using the definition ||X|| := √ X • X † (where † represents the reverse operation [START_REF] Doran | Geometric Algebra for Physicists[END_REF]) they can be normalized as

||X|| 2 = 7 ∑ µ = 0 X 2 µ = 1 and ||Y|| 2 = 7 ∑ ν = 0 Y 2 ν = 1 . ( 8
)
Now it is evident from the multiplication table above (Table I) that if X, Y ∈ K λ , then so is their product Z = XY:

Z = Z 0 + Z 1 λe x e y + Z 2 λe z e x + Z 3 λe y e z + Z 4 λe x e ∞ + Z 5 λe y e ∞ + Z 6 λe z e ∞ + Z 7 λI 3 e ∞ = XY. (9)
Thus K λ remains closed under arbitrary number of multiplications of its elements. This is a powerful property. More importantly, we shall soon see that for vectors X and Y in K λ (not necessarily unit) the following norm relation holds:

||XY|| = ||X|| ||Y|| . ( 10
)
In particular, this means that for any two unit vectors X and Y in K λ with the geometric product Z = XY we have

|| Z || 2 = 7 ∑ ρ = 0 Z 2 ρ = 1 . (11)
Now, in order to prove the norm relation (10), it is convenient to express the elements of K λ as dual quaternions.

The idea of dual numbers, z, analogous to complex numbers, was introduced by Clifford in his seminal work as follows:

z = r + d ε, where ε ̸ = 0 but ε 2 = 0 . ( 12
)
Here ε is the dual operator, r is the real part, and d is the dual part [3][4]. Similar to how the "imaginary" operator i is introduced in the complex number theory to distinguish the "real" and "imaginary" parts of a complex number, Clifford introduced the dual operator ε to distinguish the "real" and "dual" parts of a dual number. The dual number theory can be extended to numbers of higher grades, including to numbers of composite grades, such as quaternions:

S 7 S 3 K λ Q z = q r + q d ε q d ε q r IR 4
Q z = q r + q d ε , ( 13
)
where q r and q d are quaternions and Q z is a dual-quaternion (or in Clifford's terminology, Q z is a bi-quaternion).

Recall that the set of unit quaternions is a 3-sphere, which can be normalized to a radius ϱ r and written as the set S 3 = { q r := q 0 + q 1 λ e x e y + q 2 λ e z e x + q 3 λ e y e z ||q r || =

√ q r q † r = ϱ r } . ( 14
)
Consider now a second, dual copy of the set of quaternions within K λ , corresponding to the fixed orientation λ = +1:

S 3 =
{ q d := -q 7 + q 6 e x e y + q 5 e z e x + q 4 e y e z ||q d || =

√ q d q † d = ϱ d } . ( 15
)
If we now identify λ I 3 e ∞ appearing in (5) as the duality operator -ε, then (in the reverse additive order) we obtain

ε ≡ -λ I 3 e ∞ with ε † = ε and ε 2 = +1 (since e ∞ is a unit vector in K λ ) (16)
and q d ε ≡ -q d λ I 3 e ∞ = q 4 λ e x e ∞ + q 5 λ e y e ∞ + q 6 λ e z e ∞ + q

7 λ I 3 e ∞ , (17)
which is a multi-vector "dual" to the quaternion q d . Note that we write ε as if it were a scalar because it commutes with each element of K λ in (5). Comparing ( 14) and ( 17) with ( 5) we can now rewrite K λ as a set of paired quaternions:

K λ = { Q z := q r + q d ε ||Q z || = √ Q z Q † z = √ ϱ 2 r + ϱ 2 d , 0 < ϱ r < ∞, 0 < ϱ d < ∞ } . ( 18
)
Now the normalization of Q z necessitates that for that condition to hold every q r must be orthogonal to its dual q d :

||Q z || = √ Q z Q † z = √ ϱ 2 r + ϱ 2 d ⇐⇒ q r q † d + q d q † r = 0 , ( 19
)
or equivalently, (q r q † d ) s = 0; i.e., q r q † d must be a pure quaternion (for a pedagogical discussion of (19) see section 7.1 of Ref. [START_REF] Kenwright | A beginners guide to dual-quaternions: what they are, how they work, and how to use them for 3D character hierarchies[END_REF]). We can see this by working out the geometric product of Q z with Q † z while using ε 2 = +1, which gives

Q z Q † z = ( q r q † r + q d q † d ) + ( q r q † d + q d q † r ) ε . ( 20
)
Now, using definitions ( 14) and ( 15), it is easy to see that q r q † r = ϱ 2 r and q d q † d = ϱ 2 d , reducing the above product to

Q z Q † z = ϱ 2 r + ϱ 2 d + ( q r q † d + q d q † r ) ε . (21)
It is thus clear that for Q z Q † z to be a scalar q r q † d + q d q † r must vanish, or equivalently q r must be orthogonal to q d . But there is more to the normalization condition q r q † d + q d q † r = 0 than meets the eye. It also leads to the crucial norm relation (10), which is at the heart of the only possible four normed division algebras R, C, H and O associated with the four parallelizable spheres S 0 , S 1 , S 3 and S 7 , with octonions forming a non-associative algebra in addition to forming a non-commutative algebra [START_REF] Hurwitz | Über die Composition der quadratischen Formen von beliebig vielen Variabeln[END_REF]. To verify this, consider a product of two distinct members of the set K λ , Q z1 Q z2 = (q r1 q r2 + q d1 q d2 ) + (q r1 q d2 + q d1 q r2 ) ε , (

together with their individual definitions

Q z1 = q r1 + q d1 ε and Q z2 = q r2 + q d2 ε . ( 23
)
If we now use the fact that ε, along with ε † = ε and ε 2 = 1, commutes with every element of K λ defined in [START_REF] Hurwitz | Über die Composition der quadratischen Formen von beliebig vielen Variabeln[END_REF] and consequently with all q r , q † r , q d and q † d , and work out

Q † z1 , Q † z2 and the products Q z1 Q † z1 , Q z2 Q † z2 and (Q z1 Q z2 ) † as Q † z1 = q † r1 + q † d1 ε , ( 24
)
Q † z2 = q † r2 + q † d2 ε , ( 25
)
Q z1 Q † z1 = ( q r1 q † r1 + q d1 q † d1 ) + ( q r1 q † d1 + q d1 q † r1 ) ε , ( 26
)
Q z2 Q † z2 = ( q r2 q † r2 + q d2 q † d2 ) + ( q r2 q † d2 + q d2 q † r2 ) ε , (27) and (
Q z1 Q z2 ) † = ( q † r2 q † r1 + q † d2 q † d1 ) + ( q † d2 q † r1 + q † r2 q † d1 ) ε , (28)
then, thanks to the normalization condition q r q † d + q d q † r = 0 of (19), the norm relation (10) is not difficult to verify. To that end, we first work out the geometric product (Q z1 Q z2 )(Q z1 Q z2 ) † using expressions ( 22) and (28), which gives

(Q z1 Q z2 )(Q z1 Q z2 ) † = { (q r1 q r2 + q d1 q d2 ) ( q † r2 q † r1 + q † d2 q † d1 ) + (q r1 q d2 + q d1 q r2 ) ( q † d2 q † r1 + q † r2 q † d1 )} + { (q r1 q d2 + q d1 q r2 ) ( q † r2 q † r1 + q † d2 q † d1 ) + (q r1 q r2 + q d1 q d2 ) ( q † d2 q † r1 + q † r2 q † d1 )} ε . ( 29
)
Now the "real" part of the above product simplifies to (32) as follows:

{ (Q z1 Q z2 )(Q z1 Q z2 ) † } real = q r1
q r2 q † r2 q † r1 + q d1 q d2 q † r2 q † r1 + q r1 q r2 q † d2 q † d1 + q d1 q d2 q † d2 q † d1 + q r1 q d2 q † d2 q † r1 + q d1 q r2 q † d2 q † r1 + q r1 q d2 q † r2 q † d1 + q d1 q r2 q † r2 q † d1 (30) = q r1 q r2 q † r2 q † r1 + q d1 q d2 q † d2 q † d1 + q r1 q d2 q † d2 q † r1 + q d1 q r2 q † r2 q † d1 (31

) = ϱ 2 r1 ϱ 2 r2 + ϱ 2 d1 ϱ 2 d2 + ϱ 2 r1 ϱ 2 d2 + ϱ 2 d1 ϱ 2 r2 . ( 32
)
Here (31) follows from (30) upon inserting the normalization condition (19) in the form q r q † d = -q d q † r into the second and third terms of (30), which then cancel out with the sixth and seventh terms of (30), respectively; and (32) follows from (31) upon inserting the normalization conditions ||q|| 2 = qq † = ϱ 2 for the real and dual quaternions specified in ( 14) and (15), for each of the four terms of (31). Similarly, the "dual" part of the product (29) simplifies to

{ (Q z1 Q z2 )(Q z1 Q z2 ) † } dual =
{ q r1 q d2 q † r2 q † r1 + q d1 q r2 q † r2 q † r1 + q r1 q d2 q † d2 q † d1 + q d1 q r2 q † d2 q † d1 + q r1 q r2 q † d2 q † r1 + q d1 q d2 q † d2 q † r1 + q r1 q r2 q † r2 q † d1 + q d1 q d2 q † r2 q † d1 } ε (33) = 0 . (34)

IR 4 0FIG. 1 :

 41 FIG.1: An illustration of the 8D plane of K λ , which may be interpreted as an Argand diagram for a pair of quaternions.

  Einstein Centre for Local-Realistic Physics, 15 Thackley End, Oxford OX2 6LB, United Kingdom (Dated: 17 November 2018)

	Joy Christian *

We can see this again by inserting into (33) the normalization condition (19) in the form q r q † d = -q d q † r and the normalization conditions ||q|| 2 = qq † = ϱ 2 for the quaternions in ( 14) and (15), which cancels out the first four terms of (33) with the last four. Consequently, combining the results of (32) and (34), for the left-hand side of (10) we have

On the other hand, once again using the same normalization condition (19), for the right-hand side of (10) we have

Thus, comparing the results in ( 35) and (36), we finally arrive at the relation

which is evidently the same as the norm relation (10) in every respect apart from the appropriate change in notation.

Without loss of generality we can now restrict space K λ to a unit 7-sphere by setting the radii ϱ r and ϱ d to 1 √ 2 :

q r = q 0 + q 1 λ e x e y + q 2 λ e z e x + q 3 λ e y e z , and q d = -q 7 + q 6 e x e y + q 5 e z e x + q 4 e y e z , (39) so that Q z = q 0 + q 1 λe x e y + q 2 λe z e x + q 3 λe y e z + q 4 λe x e ∞ + q 5 λe y e ∞ + q 6 λe z e ∞ + q 7 λI 3 e ∞ .

(40)

Needless to say, since all Clifford algebras are associative division algebras by definition, unlike the non-associative octonionic algebra the 7-sphere we have constructed here corresponds to an associative (but non-commutative) algebra.

We may now view the four associative normed division algebras in the only possible dimensions 1, 2, 4 and 8, respectively, as even sub-algebras of the Clifford algebras


Bruno Lanson
email: bruno.lanson@obs.ujf-grenoble.fr

Victor A Drits

Feng Qi

Alain Manceau

Qi Feng

Structure of synthetic Na-birnessite: Evidence

come

INTRODUCTION

Birnessite is the most common mineral of the phyllomanganate family. It can form under a variety of physico-chemical conditions and is consequently present in different geological environments such as soils [START_REF] Taylor | The mineralogy and chemistry of manganese in some Australian soils[END_REF][START_REF] Chukhrov | Iron and manganese oxide minerals in soils[END_REF][START_REF] Cornell | Transformation of hausmannite into birnessite in alkaline media[END_REF], marine manganese nodules and micro-nodules [START_REF] Burns | Mineralogy of ferromanganese nodules[END_REF][START_REF] Glover | Characterization of a marine birnessite[END_REF][START_REF] Chukhrov | Birnessite characterization[END_REF][START_REF] Chukhrov | Crystal structure of birnessite from the Pacific ocean[END_REF][START_REF] Chukhrov | Supergenic manganese hydrous oxides[END_REF][START_REF] Drits | Manganese minerals of Fe-Mn nodules from the sediments of the central part of Pacific Ocean and their post-sedimentation transformation[END_REF], and Mn-rich ore deposits (Usui and Mita 1995). Further, birnessite possesses unique surface charge [START_REF] Healy | The effect of crystal structure on the surface properties of a series of manganese dioxides[END_REF][START_REF] Murray | The surface chemistry of hydrous manganese dioxide[END_REF], cation exchange [START_REF] Balistrieri | The surface chemistry of δMnO 2 in major ion seawater[END_REF][START_REF] Goff | Synthesis, ion exchange and electrochemical properties of lamellar phyllomanganates of the birnessite group[END_REF], and redox [START_REF] Stone | Sources and reactivity of reductant encountered in aquatic environments[END_REF] properties, which makes it highly reactive with respect to sorption phenomena [START_REF] Paterson | The influence of structure on Ba and K uptake by a synthetic phyllomangante[END_REF]Tu et al. 1994). It is also easily synthesized under laboratory conditions [START_REF] Bricker | Some stability relations in the system Mn-O 2 -H 2 O at 25° and one atmosphere total pressure[END_REF][START_REF] Stälhi | Ueber Manganate (IV) mit Schichtenstruktur[END_REF]Giovanoli et al. 1970aGiovanoli et al. , 1970b;;[START_REF] Golden | Ion exchange, thermal transformations, and oxidizing properties of birnessite[END_REF][START_REF] Strobel | Structural chemistry of phyllomanganates: Experimental evidence and structural models[END_REF][START_REF] Cornell | Transformation of hausmannite into birnessite in alkaline media[END_REF]) and therefore has been often used as a model manganese oxide in environmental chemical studies [START_REF] Stone | Reductive dissolution of manganese (III) and manganese (IV) oxides by organics[END_REF][START_REF] Stone | Reductive dissolution of manganese (III/IV) oxides by substituted phenols[END_REF][START_REF] Manceau | X-ray absorption spectroscopic study of the sorption of Cr(III) at the oxide-water interface[END_REF]Xyla et al. 1992;[START_REF] Bidoglio | X-ray absorption spectroscopy investigation of surface redox transformations of thallium and chromium on colloidal mineral oxides[END_REF].

Two major synthetic analogues of sodium birnessite (NaBi) have been described in the literature. A three-layer hexagonal modification was synthesized recently under hydrothermal conditions [START_REF] Chen | Hydrothermal synthesis and characterization of K x MnO 2 .yH 2 O[END_REF]. However, the most common way to synthesize NaBi is the formation at room temperature and high pH of Na-rich 10Å buserite, which is then partially dehydrated to form NaBi (Giovanoli et al. 1970a;[START_REF] Feng | Synthesis of birnessite-type sodium manganese oxides by solution reaction and hydrothermal methods[END_REF]. [START_REF] Post | Crystal structure determinations of synthetic sodium, magnesium, and potassium birnessite using TEM and the Rietveld method[END_REF] determined for the first time the monoclinic one-layer sub-structure of this phyllomanganate using the Rietveld technique, together with transmission electron microscopy (TEM) and selected area electron diffraction (SAED). [START_REF] Drits | The structure of monoclinic Na-rich birnessite and hexagonal birnessite. Part 1. Results from X-ray diffraction and selected area electron diffraction[END_REF][START_REF] Silvester | The structure of monoclinic Na-rich birnessite and hexagonal birnessite. Part 2. Results from Chemical Studies and EXAFS Spectroscopy[END_REF], using SAED and extended X-ray absorption fine structure (EXAFS) spectroscopy, further studied the structure and crystal chemistry of synthetic NaBi, concentrating on the nature and origin of its super-cell and modulated structures. These authors did not perform any additional XRD analysis and assumed that the substructure determined by [START_REF] Post | Crystal structure determinations of synthetic sodium, magnesium, and potassium birnessite using TEM and the Rietveld method[END_REF] was correct. However, the precision of refined structural parameters (atomic positions and occupancies) was not sufficient to allow a complete and unambiguous determination of all details of NaBi sub-structure, even though its main structural and chemical properties were reasonably known. In particular, they were unable to draw definitive conclusions about the respective contributions of layer Mn 3+ and/or layer vacancies to the layer charge. Accordingly, it was impossible from their results to choose unambiguously between the two structural formulae Na + 0.29 (Mn 4+ 0.71 Mn 3+ 0.29 )O 2 •0.75H 2 O and Na + 0.29 (Mn 4+ 0.93 0.07 )O 2 •0.75H 2 O. Similarly, the position of the interlayer species could not be determined precisely because the O2 site on the difference-Fourier map was very diffuse, and specific sites for interlayer Na and H 2 O could not be determined. This diffuseness was assumed to result from structural disorder [START_REF] Post | Crystal structure determinations of synthetic sodium, magnesium, and potassium birnessite using TEM and the Rietveld method[END_REF]. Finally, the actual structural features responsible for the pronounced anisotropic peak broadening remained unclear. Anisotropic shapes of the coherent scattering domains (CSDs), and structural disorder along one direction, which was consistent with streaking observed in SAED patterns, were the two hypotheses invoked by [START_REF] Post | Crystal structure determinations of synthetic sodium, magnesium, and potassium birnessite using TEM and the Rietveld method[END_REF].

In the present article, this anisotropic broadening is interpreted and, as a result, NaBi is shown to have a one layer triclinic sub-cell. By simulating the XRD pattern of NaBi with this sub-cell, new results concerning its sub-structure were obtained. In particular, interlayer sites for Na and H 2 O have been re-determined and are now consistent with the modulated structure proposed by [START_REF] Drits | The structure of monoclinic Na-rich birnessite and hexagonal birnessite. Part 1. Results from X-ray diffraction and selected area electron diffraction[END_REF] and relating the satellites observed on SAED patterns with a periodic λ = 6b displacement of Na and H 2 O along the b axis. In addition, simulation of the NaBi XRD pattern provides direct evidence that negative layer charge and departure from the hexagonal layer symmetry originate from the Mn 3+ for Mn 4+ substitution in octahedral layers.

EXPERIMENTAL

Experimental methods

Sample NaBi1 was prepared following the procedure of Giovanoli et al. (1970a) as described by [START_REF] Drits | The structure of monoclinic Na-rich birnessite and hexagonal birnessite. Part 1. Results from X-ray diffraction and selected area electron diffraction[END_REF]; its structural formula is Na + 0.31 (Mn 4+ 0.69 Mn 3+ 0.31 )O 2 •nH 2 O [START_REF] Silvester | The structure of monoclinic Na-rich birnessite and hexagonal birnessite. Part 2. Results from Chemical Studies and EXAFS Spectroscopy[END_REF]. One additional NaBi sample (NaBi2) was synthesized following the procedure of [START_REF] Feng | Synthesis of birnessite-type sodium manganese oxides by solution reaction and hydrothermal methods[END_REF]. The initial NaBu suspension was prepared at room temperature using a NaOH/Mn(NO 3 ) 2 ratio of 3.3, and H 2 O 2 as oxidizing agent. This suspension was then hydrothermally treated at 150°C in a 1M NaOH solution for 24 hours to produce a NaBi suspension which was then filtered, washed and dried at room temperature.

Powder XRD patterns were obtained using CuKα radiation with a Siemens D5000 powder diffractometer equipped with a Kevex Si(Li) solid detector. Intensities were measured at a 0.04° 2θ interval with a 30s counting time per step.

The XRD pattern of sample NaBi1 (Fig. 1) is almost identical to that reported by [START_REF] Post | Crystal structure determinations of synthetic sodium, magnesium, and potassium birnessite using TEM and the Rietveld method[END_REF] but does not contain hausmannite. To a first approximation all major diffraction maxima can be indexed with a one-layer monoclinic sub-cell with a = 5.169 Å, b = 2.848 Å, c = 7.321 Å, β = 103.2°. As noted by [START_REF] Post | Crystal structure determinations of synthetic sodium, magnesium, and potassium birnessite using TEM and the Rietveld method[END_REF], there is a strong anisotropic peak broadening, 11l reflections being much broader than 20l reflections for the same l value.

Thermal analysis was carried out with a Netzsch Simultan STA409EP micro-analyzer with a heating rate of 10°C/min up to 1100°C. Thermo-gravimetric analysis (TGA) and differential scanning calorimetry (DSC) were performed on ~20 mg samples to determine the weight loss due to adsorbed and interlayer water. Figure 2 shows that the heating of NaBi1 sample induces two low-temperature weight losses which are likely related to the release of adsorbed (2.1%) and interlayer (7.1%) H 2 O, respectively. To account for the loss of interlayer H 2 O determined from TGA, the structural formula proposed by [START_REF] Silvester | The structure of monoclinic Na-rich birnessite and hexagonal birnessite. Part 2. Results from Chemical Studies and EXAFS Spectroscopy[END_REF] for NaBi1 may be refined as Na + 0.31 (Mn 4+ 0.69 Mn 3+ 0.31 )O 2 •0.40H 2 O.

Simulation of XRD patterns

To build up an initial structure model for NaBi, diffraction effects were calculated for structure models realistic from a crystal chemistry point of view. The agreement between calculated and experimental XRD patterns was first improved using a trial-and-error procedure, as recommended by [START_REF] Drits | X-ray diffraction by disordered lamellar structures: Theory and applications to microdivided silicates and carbons[END_REF] for defective structures. This approach was successfully applied to natural and synthetic one-layer hexagonal birnessites [START_REF] Chukhrov | Crystal structure of birnessite from the Pacific ocean[END_REF][START_REF] Manceau | Structural mechanism of Co 2+ oxidation by the phyllomanganate buserite[END_REF][START_REF] Lanson | Structure of H-exchanged hexagonal birnessite and its mechanism of formation from Na-rich monoclinic buserite at low pH[END_REF], as well as to four-layer monoclinic Ca-exchanged birnessite [START_REF] Drits | Sub-and super-structure of four-layer Ca-exchanged birnessite[END_REF]. Details on the program used to calculate XRD patterns, as well as on the fitting procedure, are given by [START_REF] Drits | Sub-and super-structure of four-layer Ca-exchanged birnessite[END_REF].

The background was assumed to be constant over the angular range considered (30-65°2θ CuKα, 2.95-1.45 Å). The function used to describe the preferred orientation of particles is given by [START_REF] Drits | X-ray diffraction by disordered lamellar structures: Theory and applications to microdivided silicates and carbons[END_REF].

Sub-cell parameters, atomic coordinates, and occupancies obtained from the trial-anderror approach were used to build up an initial structural model for NaBi. Rietveld refinement of this model was subsequently carried out over a larger angular range using the computer program XND [START_REF] Bérar | Xnd code: from X-ray laboratory data to incommensurately modulated phases. Rietveld modeling of complex materials[END_REF]. To de-correlate instrumental broadening from defect broadening, PSF parameters and spectral distribution were first refined using a quartz reference, and then set constant during the refinement of the NaBi sub-structure. As in the previous calculations, the background was held constant over the considered angular range (34-90°2θ CuKα, 2.95-1.09 Å). During initial cycles of the Rietveld refinement, only the scale factor was refined. In a second step, sub-cell parameters were refined prior to atomic coordinates and site occupancy factors. In the final stages of the refinement, preferred orientation was introduced using orientation dependent profiles, and calculating 00l lines as a separate phase having parameters identical to those used to calculate hkl reflections, but with different orientation-dependent profiles.

RESULTS

Anisotropic broadening

A careful examination of the experimental XRD patterns (Figs. 1,3a) shows that the broadening of 11l reflections increases with l, whereas 20l reflections are much sharper for the same l value, as described previously by [START_REF] Post | Crystal structure determinations of synthetic sodium, magnesium, and potassium birnessite using TEM and the Rietveld method[END_REF]. However, the intensity of 11l reflections does not decrease strongly as it would if this broadening was related only to structural disorder along the b axis. Because of the very similar breadths of 200 and 020 reflections, this broadening is likely not associated with the anisotropic shape of the CSDs as suggested by [START_REF] Post | Crystal structure determinations of synthetic sodium, magnesium, and potassium birnessite using TEM and the Rietveld method[END_REF]. On the other hand, 11l reflections are clearly split for the NaBi2 sample (Fig. 3b -See especially 11 ± 2bar, 11 ± 3bar and 11 ± 4bar reflections) whereas the overall distribution of diffracted intensity is similar for the two samples. This split proves unambiguously the triclinic character of NaBi; and a triclinic sub-cell with parameters a = 5.174 Å, b = 2.848 Å, c = 7.334 Å, α = 90.5°, β = 103.2°, and γ = 90° indeed permits to explain the observed selective broadening of 11l reflections. The sub-structure refinement was carried out on sample NaBi1 because sample NaBi2 contains a small manganite impurity, and because a model super-structure was available for this particular sample [START_REF] Drits | The structure of monoclinic Na-rich birnessite and hexagonal birnessite. Part 1. Results from X-ray diffraction and selected area electron diffraction[END_REF][START_REF] Silvester | The structure of monoclinic Na-rich birnessite and hexagonal birnessite. Part 2. Results from Chemical Studies and EXAFS Spectroscopy[END_REF]). As a consequence, it was possible to interpret the structure model refined for NaBi1 in the light of available SAED and chemical data as the same sample was used.

SIMULATION OF THE NABI XRD PATTERN

Trial and error approach

The similarity of the monoclinic [START_REF] Post | Crystal structure determinations of synthetic sodium, magnesium, and potassium birnessite using TEM and the Rietveld method[END_REF] and triclinic a, b, c, and α, β, γ sub-cell parameters suggests that the origin of the triclinicity of NaBi is a small layer displacement along the b axis. This shift lowers the space group C2/m [START_REF] Post | Crystal structure determinations of synthetic sodium, magnesium, and potassium birnessite using TEM and the Rietveld method[END_REF] to either C1bar or C1, probably without changing the layer symmetry (2/m). The present sub-structure refinement was performed using the centro-symmetric C1bar space group because no experimental argument in favor of a non-centro-symmetric sub-structure was found. At first, the XRD pattern of NaBi1 was simulated using positions and occupancies of layer and interlayer atomic sites refined by [START_REF] Post | Crystal structure determinations of synthetic sodium, magnesium, and potassium birnessite using TEM and the Rietveld method[END_REF], taking into account the triclinic character of NaBi. In particular, [START_REF] Post | Crystal structure determinations of synthetic sodium, magnesium, and potassium birnessite using TEM and the Rietveld method[END_REF] determined the presence of two interlayer sites, host to Na and H 2 O, both of which were modeled with the scattering factor for oxygen. The predominant site (0.595, 0, 0.5) and the accessory site (0, 0, 0.5) had occupancy factors of 2.0 and 0.3 oxygen atoms per unit cell, respectively. However, the XRD pattern calculated for this model differs from the experimental one (Fig. 4a).

The trial-and-error approach was subsequently used to improve the initial structural model. Because of the modulated super-structures described by [START_REF] Drits | The structure of monoclinic Na-rich birnessite and hexagonal birnessite. Part 1. Results from X-ray diffraction and selected area electron diffraction[END_REF] for sample NaBi1, and because of the diffuseness of the O2 site on the difference Fourier map described by [START_REF] Post | Crystal structure determinations of synthetic sodium, magnesium, and potassium birnessite using TEM and the Rietveld method[END_REF], splitting of the interlayer site was considered; the agreement between the calculated and experimental XRD patterns was significantly improved (Figs. 3,4b) by splitting the predominant interlayer position into four sites related by a center of symmetry (0.551, ±0.135, 0.500) and (0.449, ±0.135, 0.500). Even though at this stage structural sites for Na and H 2 O cannot be separated, the trial-and-error analysis clearly demonstrates that the predominant interlayer site is split (Figs. 4b,4c).

Another conclusion obtained with the trial-and-error method is that NaBi1 contains very few stacking faults or other structural defects which alter its three-dimensional (3D) periodicity. The intensity and profiles of the measured reflections were reproduced assuming that cylinder-like CSDs have a 225 Å radius and contain an average of 20 birnessite layers, the amount of random stacking faults being only 3%. The NaBi sub-structure was further refined using the Rietveld technique.

The Rietveld approach

Atomic coordinates obtained with the trial-and-error approach in space group C1bar were transformed to space group P1bar. The Rietveld refinement of the sub-cell parameters gave values of a P = 2.9513(4) Å, b P = 2.9547(4) Å, c P = 7.334(1) Å, α P = 78.72(2)°, β P = 101.79(1)°a nd γ P = 122.33(1)°. The refined positions and occupancies of structural sites for this sub-cell led to the fit shown in Figure 5. Final R WP and R B are respectively 10.7 and 6.2%. In spite of these high values, the refinement was stopped because the main discrepancies between experimental and calculated patterns arise from the presence of super-cell reflections in the XRD pattern of NaBi1. Furthermore, the discrepancy observed for the 200 reflection (Fig. 5) may be explained by the small amount of random stacking faults in this sample. Indeed, calculations performed using the trial-and-error approach show that the intensity calculated for this reflection strongly depends on the amount of such defects, and that the addition of random stacking faults (3-4%) to the refined model significantly improves the agreement between calculated and experimental distributions of intensities. As a consequence no significant modification was expected for the sub-cell parameters. Refined structural parameters and selected inter-atomic distances of NaBi are found in Tables 1 and2.

Refinement of the layer Mn site occupancy factor for NaBi (range: 1.00 to 1.02) indicated that NaBi layers do not contain vacant octahedral sites. In addition, because the refined position of layer oxygen atoms (O layer ) is slightly off the long diagonal of the P sub-cell, the octahedral layer has no mirror plane, and its symmetry is lower than 2/m. Figure 6 shows the positions of four refined interlayer sites and the distances from these sites to the closest O layer atoms. The analysis of these bond lengths allows determination of the nature of the interlayer species in these sites. For example, the two symmetrically related interlayer sites at 2.46 Å from the nearest O layer are likely occupied by Na cations, whereas the other two symmetrically related sites, at ~2.70 Å from the nearest O layer are likely occupied by H 2 O (Fig. 6). Using this assumption, refined site occupancy factors for both Na (0.18 ±0.02) and H 2 O (0.27 ±0.02) are similar to the values derived from the chemical formula (0.145, and 0.20, respectively) and support the hypothesized site allocation.

One may note that the same quality of fit may be obtained for a configuration of interlayer sites which, with respect to the first model, is almost symmetrically reflected by a plane passing through the c * axis and the long diagonal of the P sub-cell. However, this alternative model was rejected because distances from each of the interlayer sites to the nearest O layer atoms are significantly shorter than 2.5 Å, which is short for interlayer H 2 O -O layer bond lengths.

DISCUSSION

Presence and azimutal orientation of Mn 3+ octahedra in the layer

One of the most notable features of the NaBi layer is the strong elongation of individual octahedra along the a axis. Within a MnO 6 octahedron, the two Mn-O distances oriented in the ac * plane are much longer (2.003 Å) than the other four Mn-O distances (1.945, and 1.925 Å -Table 2). This elongation originates from the displacement along the a axis of O layer atoms lowering the ideal symmetry of the octahedron. Additionally, this elongation is responsible for the departure from the hexagonal symmetry of the layer (a/b = 1.817 = 30 .

3 ). This distortion likely results from the presence of a significant amount of Mn 3+ cations in NaBi;

two Mn 3+ -O distances in Mn 3+ octahedra are commonly much longer than the other four because of the Jahn-Teller effect. In crednerite for example, Mn 3+ octahedra contain two 2.260 Å and four 1.929 Å Mn 3+ -O bond lengths, with an average <Mn-O> distance of 2.04 Å (Töpfer et al. 1995). Similar distortions of Mn 3+ octahedra and <Mn 3+ -O> have been reported

by [START_REF] Shannon | Effect of octahedral distortion on mean Mn 3+ -O distances[END_REF] cations, rather than the existence of vacant layer sites. Furthermore, the distribution of Mn-O distances indicates that all Mn 3+ octahedra have the same azimutal orientation with their long Mn-O distance in the ac * plane. This unique azimutal orientation is responsible for the departure from the hexagonal layer symmetry.

Structure of the interlayer region

As compared with the model proposed by [START_REF] Post | Crystal structure determinations of synthetic sodium, magnesium, and potassium birnessite using TEM and the Rietveld method[END_REF], the main difference in our refined model of the NaBi interlayer region is the determination of specific sites for Na and H 2 O. [START_REF] Post | Crystal structure determinations of synthetic sodium, magnesium, and potassium birnessite using TEM and the Rietveld method[END_REF] did not determine the respective positions of these interlayer species because the maximum observed in their difference Fourier map was very diffuse, and because of the very similar scattering powers of these two species. To account for this diffuseness, these authors invoked positional disorder in the distribution of interlayer species possibly occupying "different sites within a unit-cell and a range of positions in different unit cells". The results of our refinement support this hypothesis, as within the subcell Na and H 2 O occupy different sites related by a center of symmetry (Figs. 6,7 & 8). Such split of Na and H 2 O sites is likely responsible for the diffuse character of the electron density described by [START_REF] Post | Crystal structure determinations of synthetic sodium, magnesium, and potassium birnessite using TEM and the Rietveld method[END_REF].

The existence of two distinct H 2 O sites is favored by the equal opportunity for the formation of strong hydrogen bonds between interlayer H 2 O and layer oxygen atoms offered by each of these positions (Fig. 7). According to our refinement, the two H typical for the formation of H-bonds (Fig. 8). Furthermore, the 145° angle between O 1 , O 2 , and H 2 O i (i = 1, 2 -Fig. 7) is also suitable for the formation of strong H-bonds which are responsible for most of the cohesion between layers.

The two Na sites are also shifted along the c * axis, one towards the lower layer (Na 1 ) and the other towards the upper layer (Na 2 ; Fig. 8), leading to typical Na i -O i distances (2.46 Å).

Origin of the super-cell and modulated structures

The results of our refinement on NaBi1 confirm unambiguously the assumption of Post and Veblen (1990), Manceau et al. (1992), and [START_REF] Drits | The structure of monoclinic Na-rich birnessite and hexagonal birnessite. Part 1. Results from X-ray diffraction and selected area electron diffraction[END_REF] that super-cell reflections and satellites observed in the SAED patterns of NaBi micro-crystals result from a regular distribution of interlayer species rather than vacancies in the octahedral layer.

For NaBi micro-crystals, two types of SAED patterns have been reported [START_REF] Drits | The structure of monoclinic Na-rich birnessite and hexagonal birnessite. Part 1. Results from X-ray diffraction and selected area electron diffraction[END_REF]. et al. (1997), the (0, 0) and the (0.5, 0.5) sites should be fully occupied whereas other sites are occupied by Na with a lower probability to keep its base-centered character to the interlayer Na distribution (Fig. 9b).

In both NaBi varieties, the super-periodicity A = 3a arises from the ordered distribution of This distribution presents several remarkable features. First, interlayer cohesion is insured by a set of chains, elongated along the a axis, in which Na and H 2 O are distributed in a Na i - 12, one may note that successive chains are not linked to each other by inter-chain interlayer species. Second, the distance between successive chains changes periodically along the b axis with a 6b super-periodicity because the displacements of Na cations along the b axis induce the same periodicity in the distribution of associated H 2 O. As shown on Figure 13a, the shift of interlayer Na along the c axis and in the ab plane likely results from the Mn 3+ -Mn 3+ -Mn 3+ -Mn 3+ -Mn 2+ -Mn 4+ … distribution of heterovalent Mn layer in Mn 3+ -rich rows described by Drits et la. (1997), interlayer Na being shifted towards Mn 2+ cations. To account for the observed B = 3b super-cell periodicity, the respective positions of heterovalent Mn layer sequences in adjacent layers have to be considered. If Mn 3+ -rich rows from adjacent layers are shifted by -a/3 with respect to each other, sequences of heterovalent Mn layer cations in these Mn 3+ -rich rows should be shifted The distance between these chains along the b axis changes with a λ = 6b period, accounting for the satellites observed in SAED patterns of NaBi type I crystals [START_REF] Drits | The structure of monoclinic Na-rich birnessite and hexagonal birnessite. Part 1. Results from X-ray diffraction and selected area electron diffraction[END_REF]). An alternative origin for these satellites [START_REF] Drits | The origin of satellites in the reciprocal lattice of cymrite[END_REF]) is a periodic displacement of interlayer species along the a axis with a 6b period along the b axis (Fig. 15a). In these chains there are two equally probable configuration for H 2 O-H 2 O pairs. As a consequence, propagation of these chains along the a axis may be random (Fig. 15b A trial-and-error approach was used to determine an appropriate initial model which was further refined using the Rietveld technique. The NaBi structure consists of vacancy-free layers whose negative charge arises from the substitution of Mn 3+ for Mn 4+ . The Jahn-Teller distortion of Mn 3+ octahedra, which are systematically elongated along the a axis, leads to the departure from the hexagonal symmetry of layers. In the NaBi interlayer two positions were determined for both Na and H 2 O. The presence of such split positions, which are related by a center of symmetry, for interlayer Na and H 2 O allows to propose 2D distributions of interlayer species responsible for the super-cell and modulated structures observed for NaBi type-I and -II micro-crystals [START_REF] Drits | The structure of monoclinic Na-rich birnessite and hexagonal birnessite. Part 1. Results from X-ray diffraction and selected area electron diffraction[END_REF]. and (0, 0, 0.5), and the occupancy factors, 2.0 and 0.3, refined by [START_REF] Post | Crystal structure determinations of synthetic sodium, magnesium, and potassium birnessite using TEM and the Rietveld method[END_REF].

(H 2 O) i -Na i -(H 2 O) i … sequence (i = 1, 2

ACKNOWLEDGMENTS

Other structural parameters as in the text and the caption of In Mn 3+rich rows, Mn cations are distributed as Mn 3+ -Mn 3+ -Mn 3+ -Mn 3+ -Mn 2+ -Mn 4+ … (Drits et al. 1997). The shift of interlayer Na induced by the presence of Mn 2+ is shown by the arrow. b) Distribution of heterovalent Mn cations in the Mn 3+ -rich rows of two adjacent layers. Mn layer from the upper layer are shifted by -a/3 with respect to those of the lower layer. Shift of interlayer Na towards Mn 2+ cations is indicated by arrows.

For

  clarity and to be consistent with previously published data, all structural features of NaBi are hereafter discussed in terms of the base-centered sub-cell with parameters a = 5.174 Å, b = 2.848 Å, c = 7.334 Å, α = 90.53°, β = 103.20°, γ = 90.07°, which is equivalent to the refined primitive sub-cell with a P = 2.951 Å, b P = 2.955 Å, c P = 7.334 Å, α P = 78.72°, β P = 101.79°, γ P = 122.33°.

  The distribution of super-cell reflections in NaBi type I corresponds to a base-centered layer super-cell with A = 3a = 15.52 Å, B = b = 2.848 Å, and γ = 90°. To account for these super-cell reflections,[START_REF] Drits | The structure of monoclinic Na-rich birnessite and hexagonal birnessite. Part 1. Results from X-ray diffraction and selected area electron diffraction[END_REF] proposed the arrangement of interlayer Na cations shown in Figure9a. In addition, any sub-and super-cell reflection is surrounded by satellites at ±b * /6 along the [01] * direction. According to[START_REF] Drits | The structure of monoclinic Na-rich birnessite and hexagonal birnessite. Part 1. Results from X-ray diffraction and selected area electron diffraction[END_REF] these satellites arise from a periodic variation of distances between planes parallel to (010) containing interlayer species (Fig.10). The distribution of super-cell reflections reported for NaBi type II corresponds to a base-centered super-cell with A = 3a = 15.52 Å, B = 3b = 8.55 Å, and γ = 90°. A possible distribution of Na within the interlayer super-cell is shown in Figure9b. As discussed by Drits

  Mn 3+ -rich rows parallel to[010] and separated from each other along [100] by two Mn 4+ rows[START_REF] Drits | The structure of monoclinic Na-rich birnessite and hexagonal birnessite. Part 1. Results from X-ray diffraction and selected area electron diffraction[END_REF]. Along with sub-and super-cell reflections, SAED patterns of NaBi type II contain two types of satellites: 1) the first type are similar to those observed for NaBi type I, and 2) the second type are elongated along [11] * and [11bar] * . Satellites elongated along [11bar] * are located in the middle of two nearest point reflections located along [11] * . A possible arrangement of interlayer species which accounts for the main diffraction features of NaBi type II is shown in Figure 11. A periodic displacement of Na and H 2 O along the b axis with B S = 6b accounts for the origin of the satellites of the first type. A periodic modulation along [13] and [1bar3] of Na and H 2 O species located along (110) and (11bar0), respectively, gives rise to the satellites of the second type. As can be seen in Figure 11, waves parallel to (110) located in two neighboring rhomb-shaped unit cells (I and II) do not scatter X-rays strictly in phase because of opposite displacements of Na and H 2 O along the b axis. This phase difference is likely responsible for the elongation of the satellites (Drits et al. 1997). Results from the present refinement may be combined with the above interpretation of SAED patterns to propose consistent 2D distributions of interlayer Na and H 2 O responsible for the presence of super-cell reflections and satellites for both NaBi type I and II crystals. NaBi type II. In agreement with SAED data, Na cations should be distributed with a 3b period along the b axis to a super-cell with A = 3a, B = 3b, γ = 90° (Fig. 9b). To comply with the two positions determined for Na, these cations are likely shifted along the b axis alternately towards the upper and lower layer surfaces to provide homogeneous compensation of the negative layer charge. As a result, Na cations form a Na 1 -Na 2 -Na 1 … sequence inducing a 6b period along the b axis consistent with the satellite periodicity. In addition to O layer atoms, Na cations are likely coordinated by H 2 O molecules at 2.60-2.65 Å, and H 2 O molecules coordinating Na i cations are located in the H 2 O i sites. The combination of the above conditions leads to the 2D distribution of interlayer Na and H 2 O shown in Figure 12.

  along the b axis as shown on Figure13bto induce the observed B = 3b super-cell periodicity.As a result, Na 1 and Na 2 cations are shifted in opposite directions along the c axis and in the ab plane leading to the modulated distance between successive chains of interlayer species, which in turn is responsible for satellites of the first type observed in SAED patterns of NaBi type II crystals.The third feature of the interlayer species distribution shown on Figure12is the fluctuation, along[13]and [13bar], of interlayer Na and H 2 O atomic positions. As an illustration, a rhomb-shaped unit-cell with 2(A + B) and 2(B -A), or 12a P and 12b P , may be chosen (A, B, a P , and b P are parameters of the base-centered super-cell and of the primitive sub-cell, respectively -Fig.14). One may note that interlayer Na and H 2 O (not shown) are distributed periodically as waves parallel to (110), or b p . The amplitude of these waves varies along[13] with a 8d (310) (or 12a p ) period.[START_REF] Drits | The origin of satellites in the reciprocal lattice of cymrite[END_REF] showed that a periodic displacement of atoms along the a axis with a λ = nb period along the b axis induces, in reciprocal space, satellites which are located along the b * axis and separated from the main nodes by b * /n. Similarly, the periodic displacement of interlayer species (Na, and H 2 O) along [13] with a 8d (310) (or 12a p ) period along [11] should induce satellites distributed along [11bar] * and separated from super-cell reflections by a p * /12 (or B * /4sinγ). Similar displacements of interlayer species along [13bar] (not shown) induce additional satellites located along [11] * at the same distance from super-cell reflections as the previous group of satellites. Such satellite distribution has been described as satellites of the second type in SAED patterns of NaBi crystals type II (Drits et al. 1997). NaBi type I. As mentioned, this variety has a A = 3a = 15.52 Å, B = b, γ = 90° layer super-cell (Fig. 9a), whereas the presence of satellites indicates the existence of a modulated super-cell with a 6b period. As a consequence, the 2D distribution of interlayer Na and H 2 O in NaBi type I crystals should satisfy several conditions. First, Na cations should form rows parallel to the b axis and separated from each other by A/2 along the a axis. In addition, successive Na should be separated from each other by distances of 2b along the b axis to avoid Na-Na pairs. Second, to generate the modulated structure responsible for the satellites, Na cations should be specifically distributed along these rows with a 6b period, as in Na 1 -Na 1 -Na 2 -Na 1 …, and Na 2 -Na 2 -Na 1 -Na 2 … sequences. In addition, one may assume that predominant Na-O layer distances are 2.61-2.64 Å, inducing Na i -O i -O i -Na i … (i = 1, 2) sequences of interlayer species along the a axis. Third, to induce an average b period for the base-centered super-cell, the rows separated by A, as well as rows separated by A/2, should be shifted at random with respect to each other by ±b. A 2D distribution of interlayer species in NaBi type I satisfying these conditions is shown in Figure 15. As for NaBi type II, each Na has a tetrahedral coordination with the same bond lengths and each H 2 O is bound to the two nearest O layer atoms and to two Na cations. Figure 15a shows that interlayer Na and H 2 O form chains elongated along the a axis (solid lines).

  -dashed lines), inducing the average B = b periodicity. SUMMARY A synthetic analogue of sodium birnessite (NaBi) prepared along the protocol of Giovanoli et al. (1970a) has a one-layer triclinic base-centered sub-cell with parameters a = 5.174 Å, b = 2.848 Å, c = 7.334 Å, α = 90.53°, β = 103.20° and γ = 90.07°, which is equivalent to the refined primitive sub-cell with a P = 2.951 Å, b P = 2.955 Å, c P = 7.334 Å, α P = 78.72°, β P = 101.79° and γ P = 122.33°.

Figure Captions Figure 1 .

 Captions1 Figure Captions

Figure 2 .

 2 Figure 2. Experimental DTA-TG (a) and DSC (b) profiles obtained for synthetic NaBi1 and

Figure 3 .

 3 Figure 3. X-ray diffraction patterns of synthetic NaBi1 (a) and NaBi2 (b) samples. The

Figure 4 .

 4 Figure 4. Influence of structural parameters on calculated XRD patterns as compared with the

Figure 3 .

 3 (b) Optimum fit obtained by splitting the predominant interlayer position into four sites related by a center of symmetry, (0.551, ±0.135, 0.500) and (0.449, ±0.135, 0.500), with 0.39 Na or H 2 O per site. (C) Influence of the predominant interlayer position. Interlayer species are located in a unique non-split interlayer site (0.551(5), 0, 0.5) with 1.56 Na or H 2 O per unit cell. All other structural parameters are constant. R wp were calculated for restricted 34-62 °2θ range as for Figure 3.

Figure 5 .

 5 Figure 5. Final experimental (crosses), calculated , and difference powder XRD patterns for

Figure 6 .

 6 Figure 6. Schematic location of interlayer Na and H 2 O sites with respect to layer O and Mn atoms in projection on the ab plane. Inter-atomic distances are given in Å. Positions and

Figure 7 .

 7 Figure 7. Schematic location of interlayer H 2 O sites with respect to layer O atoms in projection along the a axis. Inter-atomic distances are given in Å. Positions and distances

Figure 8 .

 8 Figure 8. Schematic location of interlayer Na sites with respect to layer O atoms in projection

Figure 9 .

 9 Figure 9. Super-cells for NaBi types I and II with A = 3a and B = b (a), and A = 3a and B =

Figure 10 .

 10 Figure 10. Idealized super-structure model for NaBi interlayers of type I. Grey triangles

Figure 11 .

 11 Figure 11. Idealized super-structure model for NaBi interlayers of type II. Large and small

Figure 12 .

 12 Figure 12. Schematic distribution of interlayer species in projection on the ab plane for NaBi

Figure 13 .

 13 Figure 13. Schematic distribution of interlayer species in projection on the ab plane for NaBi

Figure 14 .

 14 Figure 14. Schematic distribution of interlayer species in projection on the ab plane for NaBi

Figure 15 .

 15 Figure 15. Schematic distribution of interlayer species in projection on the ab plane for NaBi

β

  P = 101.79(1)°, γ P = 122.33(1)°. Positions and site occupancy factors (Occ.) are given for space group P1bar. Un-refined isotropic B factors are 0.5 for Mn, 1 for O, and 2 for interlayer species. Layer and Int subscripts refer to the location of the atoms in the layer and in the interlayer, respectively.

  for α-MnOOH(2.041 Å -Glasser and Ingram 1968), γ-MnOOH(2.037 Å -Dachs 1963), and α-Mn 2 O 3(2.039-2.045 Å -Norrestam 1967).If one assumes that <Mn-O> is a statistically-weighted sum of the mean Mn 3+ -O distance determined for crednerite and for Mn oxy-hydroxides (~2.04 Å), and of the Mn 4+ -O distance, determined for λ-MnO 2 (1.912 Å -[START_REF] Thackeray | Synthesis and structural characterisation of defect spinels in the lithium-manganese-oxide system[END_REF], the mean Mn-O distance for the 0.69:0.31 cation composition of NaBi1 should be 1.952 Å. This value is consistent with the observed one (1.958 Å). For the same average layer cation composition (Mn 4+ 0.69 Mn 3+ 0.31 ), the long Mn-O distance calculated as the weighted sum of long Mn-O distances determined for Mn 4+ and Mn 3+ octahedra for λ-MnO 2 (1.912 Å) and crednerite(2.26 Å), respectively, equals 2.003 Å and coincides with the experimentally determined value (2.002 Å). As a consequence, the origin of the negative layer charge is undoubtedly the presence of Mn 3+

  2 O sites differ from each other by their z-coordinate; one site is shifted towards the lower octahedral layer (z = 0.495 -(H 2 O) 1 ) whereas the other is shifted towards the upper layer (z = 0.505 -(H 2 O) 2 ). If the nearest oxygen atoms of adjacent lower and upper layers are respectively labeled O 1 and O 2 , bond lengths (H 2 O) 1 -O 1 , (H 2 O) 2 -O 2 (2.69 Å), and (H 2 O) 2 -O 1 and (H 2 O) 1 -O 2 (2.71 Å) are

Table 1 .

 1 Atomic positions, and site occupancy factors determined in the triclinic P sub-cell from the refined model shown in Figure 5. Note: a P = 2.9513(4) Å, b P = 2.9547(4) Å, c P = 7.334(1) Å, α P = 78.72(2)°,

	Atom	x	y	Z	Occ.
	Mn layer	0	0	0	0 . 5
	O layer	0.3886(26)	-0.3733(26)	0.1396(6)	1
	Na Int	0.628(11)	0.476(13)	0.481(3)	0.182(14)
	H 2 O Int	0.290(8)	-0.819(15)	0.496(3)	0.272(16)

The authors thank Ewen Silvester for supplying the NaBi 1 birnessite sample, Pr. Alain Plançon and Jean-François Bérar for their programs. VAD is grateful to the Russian Science Foundation for financial support. BL acknowledges financial support from INSU/Géomatériaux and CNRS/PICS709 programs. Michel Garais (University of Poitiers) is thanked for the thermal analysis of NaBi samples. Constructive remarks and comments by AE David Bish and two anonymous reviewers helped to improved the original manuscript.