Walter H Watson

Zhenyuan Song

Irina A Kirpich

Ion V Deaciuc

Theresa Chen

Craig J Mcclain

Walter H Watson Ab#

Zhenyuan Song C#

Irina A Kirpich Ab

Ion V Deaciuc Ab

Ethanol exposure modulates hepatic S-adenosylmethionine and Sadenosylhomocysteine levels in the isolated perfused rat liver through changes in the redox state of the NADH/NAD + system

Keywords: perfused rat liver, alcohol, redox state, methionine metabolism, reductive stress

come

Ethanol exposure modulates hepatic S-adenosylmethionine and S-adenosylhomocysteine levels in the isolated perfused rat liver through changes in the redox state of the NADH/NAD system

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Introduction

Ethanol exposure affects a wide spectrum of biochemical and molecular processes in the liver leading to alcoholic liver disease. One prominent feature of the ethanol-exposed liver is altered metabolism of the essential amino acid methionine, manifested as changes in the intracellular concentrations of S-adenosylmethionine (SAM) 1 , S-adenosylhomocysteine (SAH) and other downstream metabolites [START_REF] Mcclain | S-Adenosylmethionine, cytokines, and alcoholic liver disease[END_REF][START_REF] Kharbanda | Alcoholic liver disease and methionine metabolism[END_REF]. SAM is synthesized from methionine and ATP by the enzyme methionine adenosyltransferase (MAT; EC 2.5.1.6), and SAH is formed when the methyl group of SAM is transferred to one of a variety of substrates, including proteins, DNA, RNA and lipids. SAH is a powerful feedback inhibitor of these methyltransferase reactions, and hepatocytes typically keep SAH levels low relative to SAM levels through the action of SAH hydrolase (SAHH; EC 3.3.1.1.) which catalyzes the reversible hydrolysis of SAH to homocysteine and adenosine. Ethanol can alter the expression and activities of these enzymes [START_REF] Lu | Changes in methionine adenosyltransferase and S-adenosylmethionine homeostasis in alcoholic rat liver[END_REF][START_REF] Villanueva | Hepatic transmethylation reactions in micropigs with alcoholic liver disease[END_REF], but the precise mechanisms by which this occurs are still unclear.

Another characteristic response to ethanol exposure is the conversion of NAD + to NADH as a consequence of the metabolism of ethanol to acetaldehyde by the NAD + -dependent enzyme alcohol dehydrogenase (EC 1.1.1.1). The resulting increase in the ratio of NADH/NAD + constitutes a reductive stress [START_REF] Comporti | Ethanol-induced oxidative stress: basic knowledge[END_REF] and has been observed following chronic administration of ethanol to laboratory animals and acute administration to isolated perfused liver [START_REF] Williamson | The redox state of free nicotinamide-adenine dinucleotide in the cytoplasm and mitochondria of rat liver[END_REF][START_REF] Guynn | Dependence on dose of the acute effects of ethanol on liver metabolism in vivo[END_REF][START_REF] Frosander | Influence of Ethanol on the Liver Metabolism of Fed and Starved Rats[END_REF][START_REF] Kreisberg | Effect of alcohol on glucose production and lactate, pyruvate and ketone body metabolism by the isolated perfused rat liver[END_REF][START_REF] Hassinen | Effect of ethanol, thyrxine and fructose on the intracellular redox state of a perfused liver as studied by surface fluorometry[END_REF][START_REF] Bardag-Gorce | The importance of cycling of blood alcohol levels in the pathogenesis of experimental alcoholic liver disease in rats[END_REF][START_REF] Jauhonen | Dependence of ethanol-induced redox shift on hepatic oxygen tensions prevailing in vivo[END_REF]. Ethanol-induced reductive stress has received less attention than ethanol-induced oxidative stress, but interest has been growing in recent years due to accumulating evidence that the NADH/NAD + system has functions beyond serving as a cofactor for oxidation-reduction reactions catalyzed by NAD-dependent dehydrogenases. The discovery of NAD + -consuming enzymes that contribute to epigenetic regulation [START_REF] Houtkooper | The secret life of NAD+: an old metabolite controlling new metabolic signaling pathways[END_REF], and the identification of proteins that respond to changes in the ratio of NADH/NAD + [START_REF] Rutter | Regulation of clock and NPAS2 DNA binding by the redox state of NAD cofactors[END_REF] have broadened our understanding of the ways in which NADH/NAD + contribute to the regulation of cellular processes.

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In this study, we tested the hypothesis that reductive stress plays an important role in the modulation of the intracellular concentrations of SAM and SAH. The experiments we report here were performed on isolated perfused rat livers in which we manipulated the redox state of the NADH/NAD + system by means of administration of ethanol and other compounds that change the NADH/NAD + ratio, and measured the levels of SAM and SAH in liver tissue. Our results indicate for the first time that a short-term shift of the redox state of the NADH/NAD + system rapidly modulates the intracellular concentration of methionine metabolites in the perfused rat liver.

Materials and Methods

Animals and their treatment

Male, Sprague-Dawley rats, weighing 300-350 g, were maintained under standard conditions (

Liver perfusion

The perfusion medium was Krebs-Henseleit bicarbonate buffer continuously gassed with O 2 :CO 2 (95:5%), at 36 o C, in a flow-through system. Gassing of the perfusate was ensured at two sites of the perfusion system: in the pre-heating, pre-gassing, water jacketed vessel and in an in-line rotary disc oxygenator as described by Scholz [START_REF] Scholz | Untersuchungen zur Redox compartmentierung der hemeoglobinfrei perfundierten Leber[END_REF]. The animals were anesthetized with sodium pentobarbital (Nembutal R ; 80 mg•Kg -1 body weight, intraperitoneally), the abdominal cavity opened and the portal vein cannulated at a flow rate of 6 -8 mL•min -1 . Immediately after cannulation and ligation in place of the portal cannula, the infrarenal segment of the inferior vena cava was severed and the flow rate increased to 32 mL•min -1 . This rate was maintained for the entire perfusion time using a peristaltic pump (Cole Parmer Instrument Co., Model 7554-80; Vernon Hills, IL). The portal vein pressure was

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measured with the aid of an in-line pressure gauge (Model 900A Micropressure System (World Precision Instruments, Sarasota, FL) according to manufacturer's instructions, and fluctuated between 12 and 14 mmHg. A second, outflow cannula was inserted through the right atrium in the supradiaphragmatic segment of the inferior vena cava and ligated in place. At this point, the infrarenal segment of the inferior vena cava was ligated and the perfusate diverted entirely through the outflow cannula. The perfusion continued in a flow-through mode for 90 min. At the beginning of the perfusion, which occurred approximately 6 min after opening the abdominal cavity, the caudate lobe of the liver was tied off and immersed in liquid nitrogen. Then, a continuous infusion of a mixture of D-glucose (5 mM) and L-methionine (1 mM; both final concentrations in the perfusate) was started and maintained until the end of perfusion. Glucose and methionine were included in all perfusates to ensure an adequate supply of ATP and methionine, the substrates for SAM synthesis by MAT. Livers were perfused for 30 minutes to establish equilibrium, then concentrated solutions of test compounds (i.e., ethanol, propanol, 4-methylpyrazole and/or pyruvate) were added with the aid of microprocessor-assisted infusion pump (KD Scientific, Model 200, New Hope, PA) into a 2 mL infusion chamber placed in line with the perfusion line at a constant rate of 0.5 mL•min -1 . The stock solutions were prepared such that they would be diluted to the appropriate concentrations in the infusion chamber. Influent and effluent perfusate samples were collected every 10 min throughout the entire perfusion time and immediately processed for lactate and pyruvate measurements by deproteinization in perchloric acid. Thirty minutes after the start of D-glucose + L-methionine infusion, the papilliform lobe of the liver was tied off and immersed in liquid nitrogen. Sampling of the lobes was made such that no leakage of the liver occurred during the perfusion. Immediately after the sampling of the papilliform lobe, infusion of other compounds was started and continued for the remaining 60 min of perfusion. The 90-min time point marked the end of the perfusion when a third part of the liver, the bipartite lobe, was tied off and immersed in liquid nitrogen. A scheme of the liver perfusion is given in

Fig. 1.

Liver biochemical assays

Liver tissue was analyzed for SAM and SAH using a high performance liquid chromatography procedure [START_REF] Song | Alcohol-induced S-adenosylhomocysteine accumulation in the liver sensitizes to TNF hepatotoxicity: possible involvement of mitochondrial S-adenosylmethionine transport[END_REF]. Fifty to 100 mg pieces of frozen liver stored at -85 o C were treated with 10 volumes of

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5% (w/v) metaphosphoric acid on ice, homogenized with a glass-to-glass Potter-Elvehjem homogenizer, centrifuged for 30 min at 15,000 x g, and the supernatant used for metabolite assays. All operations were performed at ice temperature.

Perfusate biochemical assays

The effluent perfusate was deproteinized in perchloric acid, and lactate and pyruvate concentrations were determined using enzymatic methods described in [START_REF] Gutman | Lactat und pyruvat[END_REF]. The deproteinization step was included to prevent interconversion between lactate and pyruvate catalyzed by lactate dehydrogenase that may have been released during perfursion.

Statistical analysis

The statistical analysis was performed using Graph-Pad Prism version 5.01 for Windows. Data are expressed as mean ± standard error of the mean (SEM). One-way Analysis of Variance (ANOVA), and Student's t test were performed to evaluate significant differences between the different experimental conditions. P-values ≤ 0.05 were considered significant.

Results

1.

Ethanol and propanol induced a marked shift of the NADH/NAD + redox couple to a more reduced state in the rat liver.

In the isolated perfused rat liver model used in the present studies, the redox state of the cytoplasmic NADH/NAD + redox couple was determined by measuring the concentrations of lactate and pyruvate in the effluent perfusate [START_REF] Williamson | The redox state of free nicotinamide-adenine dinucleotide in the cytoplasm and mitochondria of rat liver[END_REF][START_REF] Houtkooper | The secret life of NAD+: an old metabolite controlling new metabolic signaling pathways[END_REF]. Because lactate and pyruvate are in equilibrium with free NADH and NAD+ through the reaction catalyzed by lactate dehydrogenase (EC 1.1.1.27; reaction 1), the ratio of lactate/pyruvate is proportional to the ratio of NADH/NAD + . Therefore, we used the lactate/pyruvate ratio in the effluent perfusate as a measure of NADH/NAD + , and by altering the concentrations of lactate and pyruvate in the influent perfusate we could control the intracellular NADH/NAD + ratio.

Lactate + NAD + ↔ Pyruvate + NADH (Reaction 1)

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When a mixture of lactate and pyruvate at physiological concentrations (1 mM and 0.1 mM, respectively) and at a ratio of 10:1 was infused, the ratio of lactate/pyruvate in the effluent was maintained close to 10:1 during the entire perfusion (Fig. 2). Infusion of either ethanol or propanol induced a marked increase of the lactate to pyruvate ratio in the effluent, reaching 45.9 ± 3.0 and 25.1±2.0, respectively, at the end of perfusion. The effect of both alcohols was markedly blocked by co-infusion of 4-methylpyrazole, an inhibitor of alcohol dehydrogenase [START_REF] Blomstrand | Pyrazoles as inhibitors of alcohol oxidation and as important tools in alcohol research: an approach to therapy against methanol poisoning[END_REF].

3. 2. Ethanol and propanol decreased SAM content of the liver.

In the absence of alcohols, a relatively stable concentration of hepatic SAM was reached within 30 min after the start of perfusion and was maintained until the end of the perfusion. As described in Materials and Methods, all perfusion media contained glucose and methionine to ensure that neither ATP nor methionine, the two substrates for SAM synthesis, became limiting over the course of the experiments. Upon infusion of ethanol or propanol, SAM content in the liver at the end of perfusion was decreased (Fig. 3). Thus, ethanol produced a decrease from 176.7±22.1 nmoles•g -1 liver to 73.4±7.0 nmoles•g -1 liver (p<0.05), while propanol produced a smaller decrease, i.e., to 120.0±6.7 nmoles•g -1 liver (p<0.05). The effect of ethanol was blocked by 4-methylpyrazole. The inhibitor alone did not affect the intracellular content of SAM or SAH (not shown).

Inhibition of alcohol dehydrogenase by 4-methylpyrazole provides evidence that the effects of ethanol and propanol are due to the oxidation of both alcohols by alcohol dehydrogenase. We also used a physiological approach to counteract the effect of ethanol by infusing pyruvate, a metabolite which would rapidly consume NADH through the reverse of reaction 1, thus shifting the redox state of the NADH/NAD + system to a more oxidized state. The pyruvate concentration in the influent perfusate was increased from 0.1 mM to 10 mM, resulting in a lactate/pyruvate ratio of 0.1. Under these conditions, the decrease in SAM concentrations induced by ethanol was blunted; hepatic SAM levels were 1.8-fold higher in the presence of 10 mM pyruvate than with ethanol alone (Fig. 3). A similar pattern was obtained with propanol and pyruvate (not shown).

3. 3. Ethanol and propanol increased SAH content of the perfused liver.

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Like SAM, hepatic SAH reached a stable concentration within 30 min of the start of glucose + methionine infusion. Both ethanol and propanol increased the content of SAH in the liver, and the effect was more pronounced with ethanol (Fig. 4). Ethanol increased the SAH level from 12.8±1.1 nmoles•g -1 liver to 26.6±1.2 nmoles•g -1 liver (p<0.05), while propanol increased the level to 18.6±1.4 nmoles•g -1 liver (p<0.05). Infusion of either 4-methylpyrazole or 10 mM pyruvate completely blocked the effect of ethanol and propanol on the SAH content.

3. 4. NADH/NAD + redox shift induced changes in SAM to SAH ratio.

Changes in the SAM and SAH content in the liver have an impact on the SAM/SAH ratio. This ratio is taken as a marker of cellular methylation potential because SAM serves as a donor of methyl group while SAH inhibits methyl transfer reactions. Ethanol and propanol significantly (p<0.05) decreased the SAM/SAH ratio compared to the basal experimental conditions: 2.7+0.6, 6.8+0.7 and 14.1+1.9, respectively. Pyruvate and 4-methylpyrazole counteracted the effects of alcohols (Fig. 5).

Discussion

The goal of this study was to test the hypothesis that ethanol-induced dysregulation of liver methionine metabolism may be ascribed, at least in part, to ethanol-induced changes in the redox state of the NADH/NAD + couple. To determine this, we used isolated perfused rat livers in which we manipulated the free NADH/NAD + redox couple by infusing substrates of NADH-and NAD + -dependent dehydrogenases and measuring the tissue levels of SAM and SAH. These experiments confirmed earlier findings that ethanol infusion increased the ratio of NADH/NAD + [START_REF] Frosander | Influence of Ethanol on the Liver Metabolism of Fed and Starved Rats[END_REF][START_REF] Bardag-Gorce | The importance of cycling of blood alcohol levels in the pathogenesis of experimental alcoholic liver disease in rats[END_REF] and, through the use of 4-methylpyrazole, demonstrated that the increase in NADH was largely due to the activity of alcohol dehydrogenase. Consistent with this finding, a second alcohol dehydrogenase substrate, propanol, had a similar effect on the NADH/NAD + ratio. In support of our hypothesis, conditions that increased the NADH/NAD + ratio (ethanol and propanol infusion) led to a decrease in SAM, an increase in SAH, and a decrease in the SAM/SAH ratio, whereas conditions that normalized the NADH/NAD + ratio (4-methylpyrazole and pyruvate) blocked the perturbations in methionine metabolism. The observation that 4-methylpyrazole blocked the effects of ethanol and propanol

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10 demonstrated that formation of NADH by the NAD + -dependent alcohol dehydrogenase is necessary for the changes in SAM and SAH content rather than other factors related to the molecular properties of the two alcohols such as their effects on membrane fluidity.

It is not clear whether a change in the NADH/NAD + ratio is sufficient to induce changes in SAM and SAH. In addition to NADH, aldehydes are also products of the alcohol dehydrogenase-catalyzed oxidation of ethanol and propanol, and acetaldehyde (ethanal) and propionaladehyde (propanal) may have contributed to the changes in methionine metabolism. We consistently observed that the effects of propanol on NADH/NAD + and SAM/SAH were weaker than the effects of ethanol even though rat liver alcohol dehydrogenase has the same K m for both ethanol and propanol, and propanol is oxidized by the perfused rat liver at the same velocity as ethanol [START_REF] Carlson | Comparison of the metabolism of alcohols by rat hepatic and pulmonary alcohol dehydrogenase[END_REF]. One potential explanation for the observed differences between ethanol and propanol may be that subsequent metabolic steps occur at different rates. The second enzyme in the metabolism of both ethanol and propanol is aldehyde dehydrogenase

(EC 1.2.1.
3), which is also NAD + -dependent. If propionaldehyde is oxidized at a lower rate than acetaldehyde, then less NADH would be produced. When excess pyruvate was infused, the rates of metabolism of both the parent alcohol and the product aldehyde by alcohol dehydrogenase and aldehyde dehydrogenase, respectively, would be expected to increase due to increased re-oxidation of NADH to NAD + . Others have shown that when glycerate was used to increase the rate of NADH re-oxidiation in ethanol-fed rats, the clearance of ethanol was increased and there was a trend toward increased acetaldehyde levels [START_REF] Eriksson | Acceleration of ethanol and acetaldehyde oxidation by D-glycerate in rats[END_REF], suggesting that oxidation of ethanol to acetaldehyde was faster than oxidation of acetaldehyde to acetate. We did not measure aldehyde concentrations as part of this study, but it is unlikely that aldehyde concentrations decreased significantly upon infusion of pyruvate.

Therefore, the effects of pyruvate on ethanol-and propanol-induced changes in SAM and SAH levels are probably not the result of decreased aldehyde levels.

Earlier studies showed that chronic exposure to ethanol decreased SAM and increased SAH [3-4, 16, 21-24]. We reproduced these effects with an acute exposure to ethanol as well as propanol. It

is not clear whether similar mechanisms are responsible for both the acute and chronic effects. Little is known about the effects of ethanol on SAM and SAH at very early time points; it is possible that the SAM/SAH ratio decreases transiently as an early response to ethanol exposure, coincident with an early shift in the NADH/NAD + ratio [START_REF] Guynn | Dependence on dose of the acute effects of ethanol on liver metabolism in vivo[END_REF]. The shift in the NADH/NAD + ratio tends to be less

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pronounced in chronic ethanol exposure models than in acute models [START_REF] Salaspuro | Ethanol and galactose metabolism as influenced by 4-methylpyrazole in alcoholics with and without nutritional deficiencies. Preliminary report of a new approach to pathogenesis and treatment in alcoholic liver disease[END_REF][START_REF] Salaspuro | Attenuation of the ethanol-induced hepatic redox change after chronic alcohol consumption in baboons: metabolic consequences in vivo and in vitro[END_REF] as a result of increased re-oxidation of NADH and/or increased metabolism of ethanol through enzymes other than alcohol dehydrogenase [START_REF] Salaspuro | Effect of 4-methylpyrazole on ethanol elimination rate and hepatic redox changes in alcoholics with adequate or inadequate nutrition and in nonalcoholic controls[END_REF]. While acute ethanol exposure can produce more dramatic changes in the NADH/NAD + ratio, changes similar to those described here have been observed in chronic ethanol models. Direct measurement of the NADH/NAD + ratio in rats fed ethanol intragastrically for 1 month showed a 3-to 10-fold increase in the ratio, depending on whether the measurements were made when blood alcohol levels were at their highest or lowest points [START_REF] Bardag-Gorce | The importance of cycling of blood alcohol levels in the pathogenesis of experimental alcoholic liver disease in rats[END_REF]. Similarly, Baraona et al., reported that although the lactate/pyruvate ratio was only 30% higher in chronic alcohol-fed baboons than in pair-fed controls, the ratio increased 15-fold in response to acute exposure to ethanol in both groups [START_REF] Baraona | Zonal redox changes as a cause of selective perivenular hepatotoxicity of alcohol[END_REF].

Therefore, changes in the NADH/NAD + ratio similar to or greater than the 2.5-to 5-fold increase we observed in our liver perfusion model are achievable in chronic ethanol exposure models, and may be sufficient to sustain prolonged changes in methionine metabolism.

The enzymes that directly contribute to SAM concentrations are MAT (SAM synthesis), methyltransferases (substrate methylation), and SAM decarboxylase (polyamine synthesis). It is not clear which of these enzymes are responsible for the decrease in SAM levels that we observed following infusion of ethanol or propanol. Decreased SAM synthesis by MAT is certainly a possibility.

The MAT substrates, methionine and ATP, are probably present in sufficient concentrations (methionine was infused throughout the experiments, and ATP depletion was not observed in an earlier study using similar conditions [START_REF] Strubelt | The toxic and metabolic effects of 23 aliphatic alcohols in the isolated perfused rat liver[END_REF]), but MAT activity is subject to transcriptional and allosteric regulation [START_REF] Lu | Changes in methionine adenosyltransferase and S-adenosylmethionine homeostasis in alcoholic rat liver[END_REF][START_REF] Pajares | Modulation of rat liver S-adenosylmethionine synthetase activity by glutathione[END_REF]. In light of this, it will be interesting to test the hypothesis that the NADH/NAD + ratio modulates MAT activity. Oxidation or alkylation of nucleophilic cysteine residues within the MAT enzyme results in loss of activity [START_REF] Pajares | Modulation of rat liver S-adenosylmethionine synthetase activity by glutathione[END_REF][START_REF] Corrales | Inactivation and dissociation of S-adenosylmethionine synthetase by modification of sulfhydryl groups and its possible occurrence in cirrhosis[END_REF][START_REF] Martinez-Chantar | Role of thioltransferases on the modulation of rat liver S-adenosylmethionine synthetase activity by glutathione[END_REF]. Therefore, it will be important to determine whether NADH/NAD + -dependent alterations in activity are mediated by intracellular reactive oxygen species or electrophilic aldehydes such as acetaldehyde or lipid peroxidation byproducts. An increase in the use of SAM as a methyl donor in methyltransferase reactions would also account for the decrease of SAM and the increase of SAH. Another possibility is that SAM is used in other reactions such as polyamine formation which would divert a part of its pool from SAH formation, but this pathway has not been tested in ethanol exposed liver.

SAH is a product of SAM-dependent methyltransferases (EC 2.1.1.X) and is a feedback inhibitor of

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ACCEPTED MANUSCRIPT 12 many of these enzymes [START_REF] Finkelstein | Metabolic regulatory properties of S-adenosylmethionine and S-adenosylhomocysteine[END_REF]. Using the average total water content of 0.74 mL•g -1 liver measured at the beginning and end of the perfusion (not shown), we calculated the intracellular concentration of SAH after 90 minutes infusion of ethanol to be approximately 43 μM. This concentration is within the range at which SAH inhibits several methyltransferase reactions, including RNA methyltransferases (EC 2.1.1.X) and glycine methyltransferase (EC 2.1.1.20) [START_REF] Kerr | Competing methyltransferase systems[END_REF][START_REF] James | Elevation in S-adenosylhomocysteine and DNA hypomethylation: potential epigenetic mechanism for homocysteine-related pathology[END_REF]. An increase in SAH relative to SAM could therefore lead to decreased methylation of various cellular components, including proteins, nucleic acids and lipids. In addition, SAH is the main precursor of circulating homocysteine whose increased levels are implicated as a risk factor in a number of diseases, particularly of the cardiovascular system [START_REF] Asfar | Homocysteine levels and peripheral arterial occlusive disease: a prospective cohort study and review of the literature[END_REF]. Therefore, an increase in SAH may have far reaching effects on cell function and disease progression.

The precise mechanism(s) underlying the increase in intracellular concentrations of SAH in the liver of alcohol exposed animals are not fully understood. The intracellular concentration of SAH reflects the balance between the rate of formation and the rate of elimination by SAHH-mediated hydrolysis to homocysteine and adenosine. SAH can be formed in two ways: by SAM-dependent methyltransferases, as discussed above, and by SAHH-mediated synthesis from homocysteine and adenosine (the reverse of the hydrolysis reaction). A potential mechanism for the increase in SAH that we have observed may be a decrease in SAH hydrolysis catalyzed by SAHH. Decreased activity of SAHH has been reported in the livers of micropigs fed alcohol, but this was most likely due to a decrease in expression levels of the enzyme [START_REF] Villanueva | Hepatic transmethylation reactions in micropigs with alcoholic liver disease[END_REF]. An intriguing possibility with respect to the current studies is that SAHH activity may be sensitive to changes in the redox state of the NADH/NAD + system. In SAHH, NAD + is tightly bound to the enzyme and acts catalytically in both the forward and reverse reactions [START_REF] Palmer | The mechanism of action of S-adenosylhomocysteinase[END_REF][START_REF] Yang | Catalytic strategy of S-adenosyl-L-homocysteine hydrolase: transition-state stabilization and the avoidance of abortive reactions[END_REF]. Replacing NAD + with NADH in the active site inhibits SAHH hydrolysis activity by increasing its affinity for adenosine [START_REF] Kloor | Adenosine binding sites at S-adenosylhomocysteine hydrolase are controlled by the NAD+/NADH ratio of the enzyme[END_REF]. The active site of SAHH is not in rapid equilibrium with the pool of free NADH/NAD + , but under conditions of chronic reductive stress this regulatory mechanism may become more important. Newly synthesized SAHH must bind NAD + in order to become catalytically active. During this time, therefore, the redox state of the free NADH/NAD + pool may have an impact on the redox state of the enzyme bound NADH/NAD + . Such a scenario may be encountered in long-term alcohol exposure of the liver as happens in human ethanol abusers and in animals chronically exposed to alcohol.

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Conclusions

The data presented in this study suggest that the shift in the redox state of the NADH/NAD + system to a more reduced state causes rapid changes in liver methionine metabolism, reflected by decreased SAM and increased SAH content. These observations provide a totally new mechanism for the metabolically important changes in SAM and SAH observed in alcoholic liver disease.

Furthermore, the data add a novel metabolic pathway to an already large list of processes that are regulated by the redox state of the NADH/NAD + system [START_REF] Houtkooper | The secret life of NAD+: an old metabolite controlling new metabolic signaling pathways[END_REF][START_REF] Ying | NAD+ and NADH in cellular functions and cell death[END_REF].

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  12 hour dark/light cycle, 20 -22 o C, constant humidity, and Purina rat chow and water ad libitum). The animals were taken for experimentation between 8.00 and 11.00 a. m. without overnight fasting. The animals were treated in accordance with the Guide for Care and Use of Laboratory Animals. The experimental protocol of the study was approved by the Institutional Animal Care and Use Committee of the University of Louisville.

Fig. 1 .Fig. 2 .

 12 Fig. 1. Scheme of the rat liver perfusion experiment. The 90 minute time line represents the duration of the perfusion in flow-through mode. The position of each box indicates the time period over which each component was infused (final concentrations in the influent perfusate are indicated in parentheses). Liver samples were taken by tying off lobules at the beginning of the experiment (0 time point) and after 30 and 90 minutes of perfusion, as indicated by arrows. n = 6 animals per group.

Fig. 3 .

 3 Fig. 3. Infusion of ethanol or propanol decreased hepatic SAM concentration. Isolated rat liver was infused with ethanol (EtOH, 20 mM) or propanol (Prop, 20 mM) for 60 minutes, and S-adenosylmethionine (SAM) content was measured in liver lobules. Ethanol or propanol infusion resulted in a significant decrease of the liver SAM. The effect of ethanol was markedly attenuated by co-infusion of 4-methylpyrazole (4-MP, 4 mM), an inhibitor of alcohol dehydrogenase, and pyruvate (Pyr, 10 mM). Groups that do not share a common letter are significantly different (p<0.05).

Fig. 4 .

 4 Fig. 4. Ethanol and Propanol increased hepatic SAH concentrations. The effect of ethanol (EtOH, 20 mM) and propanol (Prop, 20 mM) on the S-adenosylhomocysteine (SAH) content was assessed in the perfused rat liver. Ethanol and propanol resulted in a significant increase of liver SAH. The effect of ethanol was markedly attenuated by co-infusion of 4-methylpyrazole (4-MP, 4 mM) and pyruvate (Pyr, 10 mM). Groups that do not share a common letter are significantly different (p<0.05).

Fig. 5 .

 5 Fig. 5. The SAM/SAH ratio decreased in rat liver perfused with ethanol (EtOH, 20 mM) or propanol (Prop, 20 mM). The SAM/SAH ratio was calculated from the hepatic concentrations of SAM and SAH. The ratio was restored by infusion of 4-methylpyrazole (4-MP, 4 mM) or pyruvate (10 mM). Groups that do not share a common letter are significantly different (p<0.05).

Abbreviations: SAM, S-adenosylmethionine; SAH, S-adenosylhomocysteine; MAT, methionine adenosyltransferase; SAHH, S-adenosylhomocysteine hydrolase; G+M, glucose + methionine; 4-MP, 4-methylpyrazole; Pyr, pyruvate; EtOH, ethanol; Prop, propanol.

Acknowledgements. The work presented in this study was supported by NIH grants R21 AA014611 and R21 AA015611 (IVD), K01 AA 015344 (ZS), R01 AA010486, R37 AA010762, R01 AA0015970, P01 AA017103, and the Department of Veterans Affairs (CJM).


M Peters

T T Nguyen-Dang

E Charron

A Keller

O Atabek

Laser induced electron diffraction: a tool for molecular orbital imaging

We explore the laser-induced ionization dynamics of N2 and CO2 molecules subjected to a fewcycle, linearly polarized, 800 nm laser pulse using effective two-dimensional single active electron time-dependent quantum simulations. We show that the electron recollision process taking place after an initial tunnel ionization stage results in quantum interference patterns in the energy resolved photo-electron signals. If the molecule is initially aligned perpendicular to the field polarization, the position and relative heights of the associated fringes can be related to the molecular geometrical and orbital structure, using a simple inversion algorithm which takes into account the symmetry of the initial molecular orbital from which the ionized electron is produced. We show that it is possible to extract inter-atomic distances in the molecule from an averaged photon-electron signal with an accuracy of a few percents.

I. INTRODUCTION

The proposal and experimental realizations of new ultra-fast molecular imaging techniques based on electron dynamics in intense ultra-short laser pulses have been the subject of intense research activity in the last decade [1]. Due to the ability they offer to visualize, at the atomic scale, the ultrafast molecular dynamics taking place in a chemical reaction or during an internal rearrangement process, these techniques obviously will find many applications in biological, chemical and material sciences. Conventional methods used to achieve atomic resolution, for example X-ray or electron diffraction, are much more limited in time resolution than ultrafast electron-dynamics based imaging techniques. One of those new imaging approaches consists of probing a molecule by its own electrons, strongly driven and ionized by an intense laser pulse [2].

This idea originates from the so-called three-step electron rescattering mechanism [3][4][5]. It is now widely accepted that, during strong-field atomic or molecular ionization by an optical field, a free electronic wave packet is formed each time the laser field passes its maximum value. These wave packets, initially accelerated by the field, have a large probability to return to the vicinity of the parent ion when the electric field reverses its sign, half a cycle later. This creates a series of recollision processes, taking place with very high electron kinetic energies. The consequences of these electron-ion rescattering events are diverse and can lead to several different physical processes: elastic scattering of the returning electron, electronic excitation of the parent ion, non-sequential double ionization (NSDI), but also recombination of the recolliding electron with the orbital from which it was extracted, thus producing high order harmonics (HHG).

Using a few-cycle laser pulse, the measurement of the outcome of the electron-ion rescattering events, which take place only half an optical cycle after ionization, can be seen as an ultra-fast probe of the molecule by its own electrons. It has for instance been realized that the high order harmonic spectrum carries very precise informations on the molecular structure and dynamics [6], with exceptional temporal and spatial resolutions. The harmonic spectrum is indeed related to the transition dipole between the ground and continuum electronic states. Assuming a simple form for the continuum wave functions allows one to reconstruct the molecular ground state from a complete measurement of the harmonic phases and amplitudes. This method was proposed and used for the first time in 2004 by the group of P. B. Corkum for the accurate experimental reconstruction of the highest occupied molecular orbital (HOMO) of N 2 [6]. Even though the assumptions made in such a reconstruction method are still a matter of debate, this experimental breakthrough has generated a wealth of promising new theoretical and experimental studies [7][8][9][10][11][12].

For instance, the intensity modulation of the harmonic spectra generated by aligned linear molecules was shown to be related to a multi-center molecular interference which can be used to measure the internuclear separation in the molecule [7][8][9]. It should however be mentioned that the measured harmonic spectrum may be rather complex since it carries informations not only on the HOMO orbital, but also on lower lying orbitals (HOMO-n, n = 1, 2, . . .), which, depending on the molecule itself and on its orientation, may contribute significantly or not to the HHG spectrum [10][11][12].

Elastically scattered electrons also carry structural informations on the molecule, since their momentum distribution can be influenced by interferences between the amplitudes arising from the different scattering centers constituted by the nuclei. This imaging technique, first proposed in 1996 by T. Zuo, A. D. Bandrauk and P. B. Corkum, was called "Laser-induced electron diffraction" (LIED) [13]. This proposal was investigated theoretically by different groups [14][15][16][17][18]. It was first shown in restricted two-dimensional quantum simulations that clear signatures of two-center interferences could be dis-tinguished in the re-scattered electron angular distributions [14]. Other investigations analyzed how these interferences could be used to image the structure of small molecules [15]. This question was then addressed in full three-dimensional quantum simulations, and a relatively simple analytic transformations was proposed to extract the internuclear separation of a diatomic molecule from the full three-dimensional electron momentum distribution [16,17]. The first successful experimental demonstration of LIED was finally reported in 2008 from aligned O 2 and N 2 diatomic gases [19].

Contrarily to conventional electron diffraction, the ability to extract structural information from the outcome of a rescattering event, be it harmonic generation or laser-induced electron diffraction, relies heavily on the ability to align the molecule in the laboratory frame. Assuming a realistic molecular alignment, we have recently demonstrated an accurate, simple and robust method to extract the molecular structure from the photo-electron spectra of a laser-driven linear, symmetric, polyatomic molecule [20]. We have also shown that the detailed structure of the diffraction image reflects the symmetry of the molecular orbital from which the recolliding electron emanates.

The present paper gives a more detailed theoretical analysis of this effect, illustrated first on the diatomic N 2 molecule then on the triatomic CO 2 molecule, one of the simplest linear polyatomic system. It is organized as follows. The physical process of laser-induced electron diffraction is described in Sec. II. Our twodimensional effective quantum model and the optimization of the model parameters are presented in Sec. III. This section also presents results of our simulations on N 2 , taken as an illustrative example of a model diatomic molecule, together with a number of assumptions leading to a simplified treatment of the electronic diffraction dynamics. In Sec. IV, we present our theoretical predictions and analysis of the electron momentum distribution for the triatomic molecule CO 2 . The spectra obtained with the three highest molecular orbitals are successively discussed, and the influence of various model assumptions are then presented. We conclude the work with a summary of our findings in section V.

II. LIED AND THE THREE-STEP MECHANISM

We start with a brief review of the so-called threestep mechanism, the principal idea from which the above dynamical molecular imaging schemes originate. Figure 1, inspired from the work of P. B. Corkum and F. Krauss [21], depicts the electron motion occurring in the combined oscillating force of an external laser field and that deriving from its binding potential, the latter being chosen as the double-well Coulomb potential of a laser driven diatomic molecule, N 2 for instance.

The coordinate system (x, y) refers to the plane defined by the molecular axis and the polarization vector ε of the linearly polarized electric field. The molecule is assumed to be initially aligned along the laboratory y-axis, while the intense laser pulse, which drives the electron dynamics, is polarized along the laboratory x-axis, as illustrated in the first row of Fig. 1. Due to the form of the radiative interaction, the subsequent electronic dynamics takes place essentially in the (x, y)-plane. Also illustrated in the first row of Fig. 1 is a single-cycle laser pulse defined as E (t) = E sin(ω L t) ε for t T , where T = 2π/ω L is the period of the field. Here, ω L = 0.06 a.u., corresponding to the wavelength λ = 800 nm of a Ti-Sapphire laser, and E = 0.15 a.u., corresponding to a peak intensity I = 8×10 14 W/cm 2 . The middle panels of Fig. 1 display, for specific times within the optical cycle, the total double-well Coulomb potential distorted by the radiative interaction potential r • E (t), written in the length gauge, r being an electron coordinate. In addition to the potential surfaces, the electron motion is schematically indicated by classical trajectory-style representations. Pan-els (a) to (d) refer to the different times indicated by the labeled dots placed on the electric-field waveform on the first row of the figure.

Panel (a) is for t = 0 and corresponds to the field-free Coulomb potential. Panel (b) is for t = 0.2 T , a time close to the field peak intensity. The total Coulomb plus field interaction potential is then characterized by a maximum positive slope. The molecule is ionized around that time, when the electric field amplitude is sufficiently high to lower the potential barrier allowing tunnel ionization. The positive electric field then accelerates the electron in the "forward" direction, x < 0. At t = T /2, the electric field changes its sign and so does the force on the electron, which is accelerated from then on in the opposite direction x > 0. The potential slope is then negative and the electron trajectory is classically accelerated back to the double-well region. The wave packet still extends in the forward direction until some time later, t = 0.7 T , panel (c) when the absolute value of the slope of the potential reaches its maximum. This is the classical turning point of the electron trajectory. Panel (d), for t = 0.9 T , illustrates the subsequent recollision process of the electron with the nuclei. The trajectories starting around t = T /4 + T /20 lead to the most energetic recollision events [2]. The system has thus produced its own electron gun. In addition to this, at times close to 3T /4, the field is again large enough to significantly lower the potential energy barrier through which tunnel ionization could occur. There is then a possibility for a second electronic wave packet to be launched and accelerated in the backward direction. Finally, it is important to note the asymmetry of the forward and backward scattering. The electron wave packet launched at a time close to T /4 is accelerated in the forward direction from the time of its birth and until it starts to return to the core. This oscillating wave packet interferes with a second wave packet produced around t = 3T /4. This explains the observation of interference patterns in the backward direction (x > 0) in the lower rows of Fig. 1.

A wealth of complex phenomena are induced by the recollision process in this three-step mechanism [3][4][5] depicted in Fig. 2. Indeed, the recollision may resultthrough the temporary recapture of the electron -in the formation of a transient excited species [M + . . . e -] * leading to several possible decay channels. One of them corresponds to the definite recapture of the electron with the emission of high order harmonics (HHG) currently exploited in the generation of attosecond pulses [21]. Another possible process is non-sequential double-ionization (NSDI) [22,23]. Yet another possibility, which is the subject of this paper is laser induced electron diffraction (LIED) [19], yielding a re-scattered electron with high momentum k. It has also been observed that the recollision process may even leave the electron in highly excited Rydberg states, as the ionized electron, being decelerated over many laser cycles, may be recaptured when the field is over. This last process has been called "frustrated tunnel ionization" [24,25]. In the following, we wish to demonstrate how photoelectron momentum distributions of a laser-driven (and ionized) linear molecule reflect the geometrical and orbital structures of the molecule. The aim is to show how LIED can be used as a nuclear geometry imaging technique, and to establish a reliable procedure to retrieve geometrical informations from the photoelectron spectrum. We will refer to this procedure as an inversion algorithm.

III. THEORETICAL MODEL

The laser-driven electron dynamics and the photoelectron spectra are calculated using a number of simplifying assumptions. These fall roughly into two classes: those pertaining to the model of the physical system itself, and those relevant to questions of experimental feasibility.

The two assumptions that may be of most experimental concern are the hypothesis of a perfect molecular alignment and that of a few cycle pulse excitation. As will be shown later, the default to alignment is a major limitation for the quality requirements of the imaging technique. Indeed, different alignment angles with respect to the polarization vector lead to different diffraction patterns, and we will have to assess the robustness of the results of our LIED analysis with respect to misalignment of the molecule. Another experimental challenge is the production of few-cycle laser pulses. We started out with observations made with a single-cycle pulse, for which the LIED is most transparent. The inclusion of several cycles will induce complicated diffraction patterns due to several back and forth oscillations of the electronic wave packet along with multiple recollision events. One can expect, in a near future, more sophisticated molecular alignment techniques, on one hand, and the production of intense near single-cycle pulses, on the other hand. How the contrast of LIED images is affected by the lack of perfect alignment and the effect of multi-cycle pulses on the diffraction spectra will however be discussed in detail in the second part of this article and illustrated explicitly for the case of the CO 2 molecule.

As to the model used for the physical (molecular) systems, the major limitation is the restricted number of degrees of freedom for the description at two levels: First, concerning nuclear degrees of freedom, a fixed nuclei approximation is used and the nuclear vibrational and rotational dynamics are completely neglected. This is justified by the ultrafast time scale of the electronic LIED motion (attosecond range) as compared to the vibrational and rotational time scales (femtosecond to picosecond range). Second, a single active electron (SAE) model with a soft-Coulomb pseudo-potential is used to describe the electron dynamics, and the associated timedependent Schrödinger equation (TDSE) is solved in twodimensions.

A. SAE Hamiltonian

Electronic wave packets are generated from an initial wave function Φ( r, t 0 ) by writing

Φ( r, t) = U H (t, t 0 )Φ( r, t 0 ) , (1)
U H being the unitary time evolution operator satisfying the TDSE

i ∂ ∂t U H (t, t 0 ) = H( r, t)U H (t, t 0 ), U H (t 0 , t 0 ) = 1 (2)
with a SAE-Hamiltonian of the form

H( r, t) = p 2 2 + 3 α=1 -Z α ( r α ) | r α | 2 + a 2 α + r • E (t) , (3)
p being the electron momentum operator. Atomic units are used throughout the paper unless stated otherwise. α = 1, ..., N labels the nuclei at fixed positions ρ α and r α = r -ρ α where r ≡ (x, y) denotes the electron position in the two-dimensional xy-plane. The positiondependent effective charge Z α ( r) is chosen of the following analytical form

Z α ( r) = Z ∞ α + Z 0 α -Z ∞ α exp - | r -ρ α | 2 σ 2 α , (4)
where Z ∞ α denotes the effective nuclear charge of the nucleus α as seen by an electron at infinite distance. Z 0 α is the bare charge of nucleus α and σ α is a parameter which characterizes the decrease of the effective charge of that nucleus with distance, and is introduced to account for distance-dependent electron-electron screening effects. The value of Z ∞ α is derived from a Mulliken analysis carried out in an ab-initio study performed on the parent ion. As a result, the sum α Z ∞ α , denoting the total charge of the parent molecular ion, is equal to 1.

The SAE potential defined in Eqs. ( 3) and ( 4) thus represents the force field seen by the active electron. It takes into account the interaction with the different nuclei of the molecule and with the N -1 other electrons, which screen the nuclei.

B. Electron wave packets

Wave packet propagation algorithm

The third-order split-operator technique [26] is used to solve the TDSE together with a Volkov-type asymptotic analysis [27]. Following Refs. [28] and [29], one divides the electronic space into two regions, an asymptotic region (A) where the Coulomb potential energy is neglected and an internal region (I) where all interaction potentials act. The wave function ( 1) is split accordingly as follows

Φ(x, y, t) = Φ I (x, y, t) + Φ A (x, y, t) , (5)
where Φ I (x, y, t) is non-zero in the internal region only and Φ A (x, y, t) is non-zero in the asymptotic region only. This is done through some smooth function f (x, y) which is equal to 1 in the interaction region and equal to 0 in the asymptotic one, such that

Φ I (x, y, t) = f (x, y) × Φ(x, y, t) (6a) Φ A (x, y, t) = [1 -f (x, y)] × Φ(x, y, t) . ( 6b
)
The linearity of the Schrödinger equation allows one to propagate the wave functions Φ I (x, y, t) and Φ A (x, y, t) separately, using different methods as suited to each region. In the internal region, the electronic wave function Φ(x, y, t) is then replaced by its internal part Φ I (x, y, t), and its propagation is done using the third-order splitoperator algorithm. Any portion of this propagated function that acquires a significant amplitude in the asymptotic region is collected as the asymptotic component Φ A (x, y, t). Each newly removed asymptotic component is then propagated separately using the analytical Volkov solution [29] and added to the previously cut-out and propagated asymptotic components. The final time of propagation is chosen such that the accumulation of the outgoing components is converged. Typically, the temporal propagation lasts a few tens of femtoseconds after the end of the pulse. As for the function f (x, y), an infinite number of choices exists, with only one limitation: that its spatial variation must be slow enough to avoid unphysical quantum reflexions but fast enough to separate efficiently the internal region from the asymptotic domain [29].

Initial state

The initial state Φ(x, y, 0) of Eq.( 1) is typically taken as one of the HOMO-n (n = 0, 1, 2) orbitals of the molecule. In the two-dimensional SAE model with a soft-Coulomb pseudo-potential, as shown in Fig. 3(a) for N 2 , this initial state is obtained by integrating the time-dependent Schrödinger equation in imaginary time (t → -it) using the split-operator technique [26]. Starting with an arbitrary trial initial wave function of appropriate symmetry and requiring this wave function to remain normalized, the propagated wave function will converge to the lowest energy state of the same symmetry, the excited components dying off exponentially [30,31]. Excited states of this symmetry can also be obtained by propagating different initial trial wave functions of the correct symmetry with an additional Gram-Schmidt orthogonalization to lower energy states after each time step.

In the case of N 2 , only the HOMO orbital is considered as the initial state. In a complete N = 14-electron description, it is the third orbital, (in order of increasing energy), of σ g symmetry, and is noted 3σ g . This HOMO orbital is characterized by a symmetry of revolution about the molecular axis and by two nodal planes orthogonal to this axis. The HOMO calculated, by imaginary-time wave packet propagation, in this SAE 2D model, has indeed two nodes on the y-axis, as shown in Fig. 3(b). The energy of this HOMO orbital depends on what is chosen for the parameters of the soft-Coulomb potential. Here, the value of σ α is optimized in such a way that the energy of the HOMO approaches the experimental ionization potential of this particular orbital, determined by electron impact [START_REF] Lias | Ionization Energy Evaluation[END_REF] while all other parameters remain fixed. For the case of N 2 , since only one molecular orbital will be considered, the exact energy value could be obtained.

In the case of the CO 2 molecule, three different molecular orbitals, the HOMO, HOMO-n, with n = 1, 2, were considered separately as possible initial states of the SAE dynamics. The HOMO, of π g symmetry, is doubly degenerate in 3D, and, in the crudest LCAO scheme, can be expressed as the antisymmetric combination of the Oxygens' 2p orbitals that are oriented perpendicular to the molecular axis. The HOMO is characterized by two nodal planes: one containing the molecular axis and another one perpendicular to and bisecting the molecular axis. The HOMO-1 is of π u symmetry, and is also doubly degenerate. It can be expressed as the symmetric linear combination of the Oxygen and Carbon 2p orbitals that are perpendicular to the molecular axis (for example the 2p x atomic orbitals). These orbitals are characterized by a nodal plane containing the molecular axis. The HOMO-2, of σ u symmetry, is non degenerate and can be expressed as a linear combination of the Carbon 2p orbital oriented along the direction of the molecular axis with an anti-symmetric combination of the Oxygens' 2s and a symmetric one of the Oxygens' 2p orbitals oriented along the molecular axis. It has a nodal plane perpendicular to and bisecting the molecular axis.

Enforcement of these symmetry properties, together with imaginary-time propagation, yield the three desired molecular orbitals depicted in Fig. 5. Table I lists the potential parameters optimized for this case. This optimization of the soft-Coulomb potential's parameters was done such that the ionization potentials calculated with it agree satisfactorily and simultaneously for all three orbitals with the experimental measurement [START_REF] Lias | Ionization Energy Evaluation[END_REF]. Therefore, with the optimal values given in Table I, one finds 13.6 eV, 17.8 eV and 22.3 eV for the ionization potentials of the HOMO, HOMO-1 and HOMO-2 respectively, as compared to their respective measurement of 13.8 eV, 19.7 eV and 20.3 eV, as reported in [START_REF] Lias | Ionization Energy Evaluation[END_REF]. The maximum relative deviation from the experimental data remains within 10% in this case. In principle, the effective potential seen by the active electron depends on the molecular orbital under scrutiny, as in Hartree-Fock theory, where exchange and Coulomb parts of the Fock operator depend on the unknown orbitals. The simpler picture adopted here is thus expected to have some limitations in the simultaneous description of many orbital states [START_REF] Saugout | [END_REF]. For the purpose of this work, where the feasibility of LIED as a molecular imaging scheme is assessed, this description must be sufficiently accurate to capture the essentials of the electron dynamics.

Spatial wave packet: Diffraction fringe structures

Using the propagation algorithm described above, electronic wave packets were calculated (for both N 2 and CO 2 ) on a spatial grid of 800 points on each axis, spanning -170 a 0 to 170 a 0 . The split-operator algorithm for the time-evolution operator uses the time interval δt = 0.05 a.u. Snapshots of the wave packet obtained at the different times of a single-cycle pulse as indicated on the waveform of the field shown on the first row of Fig. 1 are given in the lower panels of this figure, for N 2 and CO 2 . The series of spatial wave packet contour plots does reflect the sequence of classical events that was discussed in detail in section II, corresponding to different phases of the three-step mechanism. Thus at time t 0.2 T , the electronic wave packet starts its forward motion (toward x < 0), while at t = 0.7 T it reached its maximum extension in the forward (x < 0) direction. Of note is that at t = T an interference pattern is observed between the direct and rescattered components of the ionized wave packet.

The most important observations to be drawn from these snapshots describing the LIED process are: (i) the fringes obtained in the forward direction [x < 0 in the third and fourth rows of Fig. 1(c)] as a signature of the lobes of the corresponding HOMO orbitals acting as electron ejection sources, on one hand, and (ii) the rich pattern of interference fringes in the backward direction [x > 0 in the third and fourth rows of Fig. 1(d)] resulting from the electron diffraction analogous to Young's slit experiments, on the other hand. In this analogy, the slits are replaced by atomic diffusion centers. In the case of N 2 , the HOMO shows three lobes along y-axis leading to three main fringes in the diffraction pattern (see Fig. 1, third row, right panel). On the contrary, the HOMO of CO 2 displays only two lobes along the y-axis and its diffraction pattern shows two families of fringes with a nodal structure along y = 0 (see Fig. 1, fourth row, right panel).

C. Electron momentum distribution and its qualitative analysis

The experimentally accessible observable, through time-of-flight electron velocity mapping is the diffraction pattern in the reciprocal momentum space (k x , k y ). Theoretically, this is observed in the electronic asymptotic momentum distribution, defined as the Fourier transform ΦA (k x , k y ) of the asymptotic electron wave packet Φ A (x, y, t f ) at the end of the propagation (t = t f ) ΦA (k x , k y ) ∝ Φ A (x, y, t f )e -ikxx e -ikyy dxdy (7) In the following, the tilde (˜) symbol will denote a Fourier transform. In practice, the final time t f needs not be infinite, but is taken sufficiently large to ensure convergence. The upper panels of Fig. 4 It was shown that the overall three-step recollision process leads to a maximum collision energy given as 3.17 U p [34][START_REF] Kulander | NATO Advanced Study Institute, Series B: Physics[END_REF][START_REF] Huillier | [END_REF], where U p = E 2 /4ω 2 L is the electron ponderomotive energy. Assuming elastic collisions distributed over all angles of the (x, y)-plane, this defines a circle C (k x , k y ) in the reciprocal (k x , k y )-plane, with a well defined finite radius, such that all classical trajectories with an energy smaller than or equal to 3.17 U p remain within the boundaries of this circle. The ponderomotive motion circle associated to the maximum recollision energy for the Ti-Sapphire laser of peak intensity 8 × 10 14 W/cm 2 is drawn on all the upper panels of Fig. 4.

Two observations are in order: (i ) The asymptotic electron momentum distribution basically fill up the ponderomotive motion circle, but lead also to the observation of asymmetrical high energy spectral components in the backward (k x > 0) scattering region going beyond a classical trajectory-type interpretation frame. This region of the spectra is filled by constructive and destructive interference patterns between backward scattering and direct ionization in the x < 0 direction. In addition, the larger the internuclear separation, the more pronounced this asymmetric feature will be. (ii ) Even more important is the observation of fringes in the momentum distribution, similar to what is observed in the spatial wave packet at the end of the pulse. These fringes are markedly well resolved in the forward (k x < 0) direction. Moreover the number of fringes increases with the internuclear distance. As a more quantitative analysis will make it clear, this last information is precisely the key point of the LIED imaging technique for geometrical structure determination.

Such an analysis is conducted by first averaging over k x the momentum distribution to give

S(k y ) = | ΦA (k x , k y )| 2 dk x (8)
The corresponding spectra, a diffraction pattern, are displayed, on a logarithmic scale, on the lower panels of Fig. 4. One can relate the number of fringes observed in this diffraction pattern S(k y ) with the internuclear distance: using the four tick mark rulers displayed in the lower panels of Fig. 4, it turns out that, within a typical interval of 0 < k y < 1.5 a -1 0 , there are twice as many fringes for R = 2.2 Å as there are, (in the same k y interval) for R = 1.1 Å, and again twice as many fringes for R = 4.4 Å as there are for R = 2.2 Å. This important observation can be summarized as

R ∝ 1/∆k y (9)
where ∆k y denotes the momentum separation between the fringes. A complete quantitative analysis requires the determination of the proportionality factor. This will be done in the next Section. It is however worthwhile to note that, even at this level of analysis, the diffraction spectra S(k y ) may already serve as a tool for molecular imaging once the averaged diffraction spectrum has been measured at the known equilibrium internuclear distance. A possible limitation of this technique seems to be the small numbers of fringes in the spectra, which may render more difficult the measurement of the geometrical structure for short internuclear separations. This limitation can however be overcome by changing the laser parameters, such as to increase the ponderomotive energy U p and consequently the radius of the ponderomotive motion circle C (k x , k y ) to encircle a larger area covered by recolliding trajectories. For a given frequency, the issue is to increase the peak intensity. As an indication, the laser parameters used in obtaining the spectra of Fig. 4 fulfill the requirement for the appearance of at least two fringes in C (k x , k y ).

IV. LIED ANALYSIS AND ROBUSTNESS

We now give a complete proof of the feasibility of imaging molecular geometrical and orbital structures of linear molecules. We shall first take a closer look at the diffraction pattern defined by the k x -averaged asymptotic electron distribution S(k y ), Eq.( 8), and illustrated in Fig. 4 for the case of the N 2 molecule. We shall demonstrate how the geometrical and orbital structures of the molecule can be inferred from a comprehensive reading of S(k y ), in what constitutes to us an inversion algorithm for LIED. Emphasis will then be placed on results obtained for the CO 2 molecule, in the simple yet relatively realistic model (for the purpose of the present paper) described in the preceding section. This molecule constitutes an ideal example of a polyatomic linear symmetric molecule. The choice of this system is motivated by different considerations. With three nuclei it has enough internal degrees of freedom to make its molecular structure determination challenging. It remains however sim-ple enough when referring to its alignment properties, due to its linear and symmetric equilibrium geometry. We will then explore and discuss the robustness of the LIED analysis when three main approximations are relaxed, at least partially. These are, in order of conceptual importance, the SAE, the perfect alignment and the single-cycle pulse excitation assumptions.

A. Inversion algorithm

The LIED inversion algorithm used here is basically related to structural properties of the initial state that are well conserved during the electron wave packet propagation, i.e. during the time-evolution of the system under the combined effect of the Coulomb forces and of the laser field polarized perpendicularly to the molecule.

For centrosymmetric linear molecule, such as CO 2 and N 2 , the addition of the time-dependent interaction with the linearly polarized laser electric field lowers the symmetry of the molecular force field from D ∞h to C 2v , where the C 2 symmetry axis corresponds to the direction x of the field polarization. The symmetry operators of this subgroup (E, C 2 , σ v and σ v ) commute with the Hamiltonian H( r, t) and with the associated timeevolution operator U H . As a consequence, if the initial state is one of the eigenstates of the field-free Hamiltonian, which necessarily transform under the various symmetry operations according to one of the four irreducible representations of the C 2v group, the time-dependent state that evolves from this will also transform accordingly. It is thus relevant to look at symmetry properties of the initial states considered here for the N 2 and CO 2 molecules.

N2 case

For N 2 , the HOMO, a σ g orbital in D ∞h , is of symmetry a 1 in C 2v . Its nodal structure, as seen in Fig. 3, is that of a symmetry-adapted (SA) LCAO form

Φ 3σg ( r) ∝ f 0 ( r -R/2) + [S f 0 ]( r + R/2) ( 10
)
where f 0 = a φ 2s + b φ 2p + . . . denotes a Nitrogen hybrid atomic orbital and S a symmetry operation of C 2v which exchanges the two N atoms. At any time t, the wave function can be expressed similarly, as

Φ 3σg ( r, t) ∝ f t ( r -R/2) + [S f t ]( r + R/2) (11)
From this, we infer that the electron momentum wave function is

Φ3σg ( k, t) ∝ e +i k. R/2 ft ( k) + e -i k. R/2 [S ft ]( k) (12)
f t can also be written as a sum of a-and b-symmetry functions as and then

f t = f a t + f b t and S f t = f a t -f b t (13)
Φ3σg ( k, t) ∝ cos(k y R/2) f a t ( k)+i sin(k y R/2) f b t ( k) (14)
The modulus square of this function, integrated over k x , yields an oscillatory pattern that depends on the widths of the functions f a t , f b t and on the interplay between the cosine and sine terms, which have a common period of

∆k y = 4π/R, ( 15
)
Without delving into the details of the actual variations of S(k y ) as defined by this, we can at least infer from the period of the (modulated) oscillations in its fringe pattern the value of the internuclear distance R, as announced in Eq.( 9) of the previous section.

CO2 case

For CO 2 , we considered ionization out of either the HOMO, or the HOMO-n (n = 1, 2). These orbitals are sufficiently close in energy to each other in the field-free molecule to play an important role in the laser-driven ionization process for the choice of laser parameters under consideration [37][38][39]. Their well-known LCAO structures have been recalled in the preceding section. They are shown in Fig. 5. The differences in symmetry and nodal properties between these three molecular orbitals (MO) are obvious from this figure. The most important feature in this respect and for the purpose of the present discussion is that the HOMO and HOMO-2 are anti-symmetric with respect to the mirror plane orthogonal to the molecular axis and containing the C atom, while the HOMO-1 is symmetric. This is at the origin of the different fringe structures found in both the 2D photoelectron momentum distribution and k x -averaged spectra, i.e. diffraction patterns S(k y ), as shown in Fig. 6, lower panels.

The HOMO and the HOMO-2 (panels (a) and (c) of Fig. 5) are both of symmetry b 2 in C 2v , and have a general LCAO composition of the form where f O ( r) will generally be a hybrid Oxygen atomic orbital, and f b2 C ( r) denotes some function of symmetry b 2 centered on the C atom. R here denotes the C-O internuclear distance. Similarly, one can write

Φ( r) ∝ f O ( r -R) + f b2 C ( r) -[S f O ]( r + R) (16)
Φ( k, t) ∝ e i k. R fO,t ( k)+ f b2 C,t ( k)-e -i k. R [S fO,t ]( k) (17)
such that, writing once again

f O,t = f a O,t +f b O,t and S f O,t = f a O,t -f b O,t (18)
the momentum wave function now reads

Φ( k, t) ∝ 2 cos(k y R) f b O,t ( k) + f b2 C,t ( k) +2i sin(k y R) f a O,t ( k) (19)
As shown in Fig. 7(a) and (c), in the cases of the HOMO and HOMO-2, the diffraction patterns S(k y ) have fringes at large k y that are mainly governed by a sin 2 (k y R) term, with zeroes at k y = nπ/R. The curves in black (solid) lines represent this pure sin 2 (k y R) diffraction pattern. The calculated patterns shown in red (dashed) lines follow closely this behavior and exhibit zeroes at the predicted positions. The last term of Eq.( 19) is thus seen to dominate the calculated spectrum in this range of k y . At smaller values of |k y |, the zeroes of this sin 2 (k y R) part are masked by the contributions of the first terms of Eq.( 19), which die out rapidly as |k y | increases. Indeed, f b O,t ( k) and f b2 C,t ( k) are essentially the Fourier transforms of the contributions arising from the diffuse 2p y atomic orbitals centered on O and C, which are narrower in k-space than the contribution f a O,t ( k) arising from the 2s atomic orbital. The HOMO-1 is a π u bonding orbital and is of symmetry a 1 in C 2v . Its analysis thus proceeds as done above for the HOMO of N 2 , except that here the LCAO pattern will be a 3-center one. In fact, of the n = 2 atomic orbitals of O and C, only the 2p x orbital is simultaneously of π symmetry in D ∞h and a 1 in C 2v . The HOMO-1 can thus be well described by the combination

Φ πu ( r) ∝ f O ( r -R) + f C ( r) + f O ( r + R) (20)
Assuming that this structure is conserved during the propagation in the presence of the field, the above reasoning leads to

Φπu ( k, t) ∝ 2 cos(k y R) fO,t ( k) + fC,t ( k) (21)
If we assume that f C ( r) f O ( r) = g( r), i.e. that O and C have the same 2p x orbitals, we get Φ πu (x, y, t) g t (x, y -R) + g t (x, y) + g t (x, y + R) (22) and the diffraction pattern S(k y ) should exhibit a double peak structure just as [1 + 2 cos(k y R)] 2 with zeroes for Φ πu (x, y, t) = g t (x, y -R)+γ g t (x, y)+g t (x, y +R) (24) and in which case

k y = n + 1 3 2π R and k y = n + 2 3 2π R (
| Φπu (k x , k y , t)| 2 ∝ [2 + γ 2 + 2 cos(2Rk y ) + 4γ cos(Rk y )] |g t (k x , k y )| 2 (25)
One can then show that the ratio, σ, of the amplitudes of two successive peaks in the diffraction pattern resulting from this is related to the ratio γ of the LCAO coefficients of the 2p x atomic orbitals of C and O in this molecular orbital. Indeed, we obtain

σ (γ -2) 2 /(γ +2) 2 or, equiv- alently, γ = (2 -2 √ σ)/(1 + √ σ).
The intensity ratio of the double peak structure of the calculated diffraction pattern for the HOMO-1 orbital, shown in panel (b) of Fig. 7 in red (dashed) line, is σ 1/11.8 which gives a parameter γ 1.10, in support of the simpler expansion of Eq.( 22) and in agreement with the pictorial representation of the HOMO-1 orbital in Fig. 5(b).

Generalization

A number of general features of the above analysis, permitting the retrieval of the geometrical and orbital structure of the molecule from observable photoelectron momentum distributions, and constituting the LIED inversion algorithm, is worth noting here.

We first note that, at large values of |k y |, S(k y ) is dominated by a single oscillatory term, associated with that atomic component of the initial molecular orbital that is widest in the momentum space. Thus, disregarding the modulation of the interference pattern in S(k y ), in particular its decay as k y increases, we can concentrate on this oscillatory pattern at large |k y | to retrieve informations on the bond-length(s) of a symmetric linear molecule. More precisely the positions of the zeroes and peaks of the calculated spectra are accurately predicted by a simple periodic function of k y R, the central element of the inversion algorithm, so that one can recover the sought-for geometrical information through relations of the type R = nπ/∆k y where ∆k y denotes the k y -spacing separating two consecutive maxima. This is the quantitative aspect of the inversion algorithm.

We have been able to reproduce, by using just the simple formula evoked above, the regular oscillation patterns found in the photoelectron parallel momentum spectrum, S(k y ), in all cases and trace these patterns back to the symmetry and nodal, or rather LCAO structure of the initial molecular orbital: for example, the simple structure of the spectrum in the case of the HOMO [Fig. 7(a)] as opposed to that of the HOMO-2 [Fig. 7(c)], reflects the simple LCAO structure of this orbital. The double peak structure of the HOMO-1 [Fig. 7(b)] reflects also the 3centered LCAO structure of this orbital. This qualitative aspect of the inversion algorithm distinguishes it from previous, existing LIED analysis approaches, and is the feature that depends most on the alignment condition as it is deeply rooted in the structure and symmetry of the molecular system.

We turn next to the role and importance of the model assumptions, and assess the robustness of this LIED analysis, as an imaging technique for the retrieval of geometrical information, with respect to the variations in the model parameters. This is conducted by examining separately those concerning molecular degrees of freedom and the ones in relation with the laser field.

B. Robustness with respect to molecular degrees of freedom

It is useful to make here a distinction between internal and external molecular degrees of freedom when referring to the various assumptions made above. Assumptions on internal degrees of freedom are those made in describing electronic and vibrational dynamics. Assumptions on external degrees of freedom are those concerning the rotational dynamics. Let us examine the consequences of these approximations successively.

Internal degrees of freedom: electronic dynamics

Within the framework of the SAE approximation, we have considered the ionization and the subsequent electronic dynamics to start from one of the highest occupied orbitals of the CO 2 molecule. The resulting spectra for the HOMO-n (n = 0, 1, 2) gathered in Fig. 7 show that: (i ) Very sharp minima at the momenta k y are predicted by the inversion algorithm from which the value of the C-O bond length R can be obtained. More precisely, from the spectra shown in the figure, one obtains R = 4.92 Å with an error of less than 3% over 8 peaks, when compared with the input parameter R = 4.80 Å. (ii ) The observation of a periodic sequence of minima is clearest in the high energy region of the spectra, which incidentally corresponds mainly to backward scattering.

Without relaxing yet the SAE approximation, we have also shown in panel (d) of Fig. 7 how an incoherent superposition of the three HOMOs with equal probability still results in a spectrum with sharply defined minima from which the geometrical information can be extracted with a comparable accuracy.

We now argue that, provided the molecule is perfectly aligned as assumed, going beyond the SAE should presumably not affect the readability of the diffraction pattern. Indeed, the symmetry character of the quantum state holds for the N -electron, fully correlated, timedependent wave function, as well as for the orbitals (one electron wave functions). Thus, all what was written above concerning the SAE wave function should also apply to the N -electron wave function. Now, the photoelectron momentum distribution can certainly be analyzed in terms of properties of the oneelectron (time-dependent) density matrix, γ 1 (t), derived from the exact N -electron state |Ψ N (t) , hence from its eigenfunctions, the natural orbitals, ϕ i ( r, t) [41].

For the purpose of this demonstration, let us first define the N -electron density matrix

ρ N (t) ≡ |Ψ N (t) Ψ N (t)| (26)
The one-electron density matrix γ 1 (t), which can be obtained from the partial trace

γ 1 (t) = N Tr N-1 ρ N (t) (27)
over the remaining (N -1) electrons, defines the set of natural orbitals using

γ 1 (t) = i n i |ϕ i (t) ϕ i (t)| . (28)
The exact time-dependent electron momentum density can then be written as

ρ1 ( k, t) = i n i | φi ( k, t)| 2 (29)
For symmetric linear molecules aligned perpendicular to the laser polarization, whose common reduced point group is C 2v , the natural orbitals can only be of a 1(2) or b 1(2) symmetry, and can always be expanded in a complete atomic basis. Their asymptotic momentum distribution should be either of the form given in Eq.( 19), for a general a 1(2) orbital, or of the form of Eq.( 21), for a general b 1(2) orbital. It follows then that the exact photoelectron momentum distribution should be the incoherent sum, defined by Eq.( 29), of the natural orbitals' asymptotic densities of the form

| Φ( k, t)| 2 ∝ a cos( k. R) f1 ( k) + b sin( k. R) f2 ( k) + c f3 ( k) 2 (30)
where f1 (2,3) , are some functions of the free electron momentum k (here in 3D), whose explicit forms depend on the natural orbitals.

The same type of periodic fringe pattern as shown in panel (d) of Fig. 7, with readable periodic zeroes and maxima at large momenta, should thus characterize this photoelectron momentum distribution. Work is in progress in our group in order to study in detail this process and the influence of electron correlations in the LIED process.

Internal degrees of freedom: vibrational dynamics

The structural reading made above of the diffraction patterns refers to a molecule in a fixed nuclear geometry configuration, perfectly aligned perpendicular to the linear polarization vector ε. In reality, molecules are subject to vibrations and rotations, and these geometrical parameters are usually distributed over some range depending on the vibrational and rotational state in which it is initially prepared. The possible blurring due to molecular vibrations in the observed diffraction spectra can be taken into account through an incoherent average of the diffraction signal over independent calculations performed for different internuclear distances. Here we choose a range of R going from (R 0 -1 Å) to (R 0 + 1 Å) both with R 0 = 1.2 Å and 4.8 Å, under the same excitation conditions as in Fig. 4. The weight associated with each value of the internuclear distance R is chosen as a Gaussian distribution D(R) normalized over the finite

interval R ∈ [R 0 -1 Å, R 0 + 1 Å] D(R) = N exp - [R -R 0 ] 2 2ς 2 (31)
where the standard deviation ς, fixed at 0.2 Å, is much larger than that of the vibrational ground state associated with the symmetric stretch mode of CO 2 . The incoherently averaged diffraction patterns are displayed in Fig. 8. They were obtained by retaining the higher energy signal only. We have chosen here |k| > 3.15 a.u., which results in a significant increase of the contrast between constructive and destructive interference fringes. As can be seen from Fig. 8, the diffraction patterns are almost insensitive to the initial distribution of R. At both the equilibrium and stretched geometries, the contrast that remains after the vibrational motion has been taken into account statistically in this way, allows for an accurate determination of the position of the fringes, which are not much affected by the bond length dispersion. A more quantitative estimation of this sensitivity can be reached by analyzing the Fourier transform of the asymptotic momentum amplitude associated with the HOMO, which reads

Φ( k, t) sin( k. R) g( k), (32)
Replacing R by R 0 +∆ R, setting k.( R 0 +∆ R) = k y .(R 0 + ∆R), assuming perfect alignment, and considering the symmetric variations of both CO bond length, corresponding to a symmetric stretching mode, as done in the calculations shown in Fig. 8, one gets

Φ( k, t) [sin(k y R 0 ) + k y ∆R cos(k y R 0 )] g( k) (33)
A strong sensitivity with respect to ∆R therefore appears only for |k y | 1/∆R. In other words, for small bond length dispersions, only high energy fringes are significantly affected. This is confirmed in Fig. 8. The same analysis also applies to a non symmetric variation of the two C-O bond lengths, corresponding to a small amplitude vibrational motion in the anti-symmetric stretch mode. Likewise, a bending motion would give the same effect as the symmetric variations of the C-O bond lengths considered above. We thus expect diffraction patterns that are no less analyzable than the one shown in Fig. 8, and from which the C-O equilibrium bond length can be extracted unambiguously.

External degrees of freedom

Finally, we examine the validity of the model assumptions with respect to the introduction of the external degree of freedom, that is the effect of angular dispersion. In that respect, it is useful to imagine that a first laser pulse, an alignment pulse, has created a rotational wave packet which periodically re-phases, yielding thus, at the revival times, a strong alignment of the CO 2 molecule perpendicular to the polarization direction of the subsequent ionizing laser pulse. As previously done for the vibrational motion, the effect of molecular rotation is simulated through similar statistical incoherent averaging over various molecular alignment angles θ. The respective weight of each angle is determined by an analog of Eq.(31), D(θ) now denoting a Gaussian angular distribution taken in the interval θ ∈ [0, π]. The standard deviation ς is chosen such that 90% of the molecules are contained within a cone of aperture 20 • , corresponding to a strong alignment scenario that has recently been achieved experimentally for CO 2 [40]. We apply to the scattered electron momentum spectrum a filter that retains only the high-energy electrons. The resulting sig- nal, displayed in Fig. 9, is nicely contrasted and reproduces all the fringes of the aligned molecule. The only caveat is the weakness of the measured signal where the fringes are contrasted, but this could be overcome by a high repetition data recording. A quantitative analysis of the sensitivity with respect to ∆θ leads to

k. R = 1 + k x k y tan(∆θ) k y R cos(∆θ) (34)
which, for small ∆θ, results into:

Φ( k, t) sin[k y R(1 + (k x /k y )∆θ)] g( k) (35)
giving a higher sensitivity to ∆θ for |k y /k x | 1. The fringes are actually slightly less contrasted for small k y values in Fig. 9(b). But we still expect diffraction patterns that are as analyzable as the ones shown in Fig. 9, and from which the C-O equilibrium and stretched bond lengths can be again unambiguously extracted.

C. External field parameters

In order to remove the limitations of an unrealistic single-cycle 800 nm laser pulse, we present results of calculations made with 8 optical cycles at the same carrier frequency. We will compare the results obtained with a rectangular pulse shape [upper row of Fig. 10(a)] with those obtained with a more realistic 10 fs Full Width at Half Maximum (FWHM) sine-square pulsed excitation [upper row of Fig. 10(b)]. Both pulses cover about 8 optical cycles. The lower row of Fig. 10 shows the diffraction pattern calculated for R = 4.8 Å assuming perfect alignment at fixed internuclear distance with this few-cycle excitation. With these long pulses, at least two electron wave packets are driven by the field in opposite directions before being both returned to the parent ion. Scattering signals arising from these different "pathways" interfere with each other, blurring out the diffraction pattern. Let us examine in some more details the role of the corresponding classical trajectories. The first row of Fig. 10 depicts the time variation of the electric field, in red, and shows two classical trajectories corresponding to the maximum recollision energy, one (in blue) starting at a time at which the electric field is positive, and the other (in green) when the electric field is negative.

A recollision event arising from any of these trajectories occurs when the associated colored line intersects the x = 0 axis, where the incident (returning) electron momentum corresponds to a kinetic energy of 3.17 U p . The electron associated with the first trajectory (blue line) scatters in every direction and continues to drift in the field, but since the recollision time is approximately 0.95 T , the residual shift in velocity is negligible (0.12 a.u.). This gives rise to a signal mostly contained within a circle of radius |k| 3.15 a.u. centered about (k x = 0.12 a.u., k y = 0). This is not the case for the second (green) trajectory where the electron continues to drift, after recollision, for some half integer multiple of the optical period T , giving rise to a large residual momentum drift of 4.87 a.u. The associated signal is thus contained within a circle of same radius but centered about (k x = 4.87 a.u., k y = 0).

A different situation occurs with the sine-square pulsed laser excitation. If one performs the same calculation as described above for the maximal recollision energies starting at the times associated with the opposite maxima of the electric field, one finds slightly different recollision energies for the "blue" and "green" trajectories, respectively 2.8 U p and 3.1 U p . These are also characterized by different residual drift momentum, respectively -2.12 a.u. and 2.29 a.u. The recollision energy is therefore reduced compared to the one obtained with a rectangular pulse shape for two reasons: (i ) the electric field is smaller in amplitude yielding different values of U p , and (ii ) the electric field driving the electron back to the ion is constantly decreasing in amplitude, as opposed to the case of a rectangular pulse shape.

With these long laser pulses, interferences between the different electron wave packets emitted in the forward and backward directions decrease the contrast obtained in the fringe spectrum. This difficulty can however be circumvented by applying a filter which restricts the S(k y ) signal to high-energy electrons as shown in Fig. 11 displaying this distributions in the high energy domain, with an average [Eq.( 8)] performed over |k x | > 3.15 a.u. only. (solid) lines display the similar result for a single-cycle excitation, retaining again only high energy electrons to enhance the contrast. Once again, the contrast is perfectly sufficient to extract the corresponding C-O bond lengths.

V. CONCLUSION

In this paper, we have presented a detailed theoretical analysis of the ionization dynamics of the N 2 and CO 2 molecules in intense ultra-short linearly polarized laser fields at 800 nm. Using two-dimensional single active electron effective potentials, we have calculated the momentum distribution of the emitted photo-electrons by solving the time-dependent Schrödinger equation for the electronic motion.

We have shown that, if the molecule is initially aligned perpendicular to the field polarization, a simple averaging and an inversion procedure can be used to determine the molecular bond length with an accuracy of a few percents. We have also shown that the photo-electron momentum distribution carries information on the structure and symmetry of the ionized molecular orbital. The robustness of the structure determination with respect to vibrational and rotational motions has also been demonstrated, and the inaccuracies introduced by such perturbations have been assessed and interpreted. These results throw some light on how to image such geometric and orbital information for a linear polyatomic molecule on an attosecond timescale by laser induced electron diffraction. Finally, taking into account the very short time acquisition of these molecular images, we also claim that a stroboscopical animation of the vibrational motion is achievable using the LIED strategy.

FIG. 1 .

 1 FIG. 1. (Color online) Schematic view of the three-step mechanism as inspired from [21]. The upper part displays the positioning of the molecule and the electric field polarization vector in the laboratory frame, together with the time variation of its amplitude (ωL = 0.06 a.u., E = 0.15 a.u.). Panels (a) to (d) illustrate the time evolution of the field-dressed twocenter Coulomb potentials corresponding to different snapshots within the optical cycle, as indicated by the dots on the field evolution. Numerical results for the square modulus of the electron wave packet are shown on the bottom part, starting initially from the HOMO of N2 (upper part) and of CO2 (lower part). See text for details.

M

  FIG.2. Three-step electron rescattering mechanism.

FIG. 3 .

 3 FIG. 3. (Color online) (a) Soft-Coulomb potential associated with N2. (b) Two-dimensional representation of the HOMO probability amplitude distribution of N2. The change of color denotes a change of sign of the orbital wave function.

  display the square modulus of ΦA (k x , k y ), for three different internuclear distances R of N 2 : (a) for the molecule in its equilibrium geometry R = 1.1 Å ; (b) and (c) for a stretched N 2 with R = 2.2 Å and 4.4 Å, respectively.

FIG. 4 .

 4 FIG. 4. (Color online) Electron diffraction patterns | ΦA(kx, ky)| 2 for N2 in the momentum space representation (upper panels) and for three different inter-atomic distances R from equilibrium to stretched geometries and under the same excitation conditions as Fig. 1. Dashed circles indicate points with final energy equal to the maximum classical recollision energy. The lower panels display the averaged diffraction spectra S(ky) on a logarithmic scale normalized to the maximal value. Tick mark rulers are given to facilitate the counting of peaks within a given ky-interval.

FIG. 5 .

 5 FIG. 5. (Color online) Two-dimensional representation of the (a) HOMO, (b) HOMO-1 and (c) HOMO-2 probability amplitude distributions of CO2. The change of color denotes a change of sign of the orbital wave function.

FIG. 6 .

 6 FIG. 6. (Color online) Electron diffraction patterns of a symmetrically stretched CO2 (R = 4.8 Å) in the momentum space (upper panels) for the HOMO (a) and HOMO-1 (b) orbitals under the same excitation conditions as in Fig. 4. The lower panels display the averaged diffraction spectra S(ky) on a logarithmic scale normalized to the maximum value.

  23) and maxima at k y = nπ/R. The profile of the pure [1+2 cos(R k y )] 2 diffraction pattern is shown in the panel (b) of Fig. 7 in black (solid) line with the numerical result shown in (red) dotted line. This comparison confirms again the validity of the proposed inversion algorithm.A less stringent assumption would be that f C ( r) = γ g( r) while f O ( r) = g( r), i.e.

FIG. 7 .

 7 FIG. 7. (Color online) Averaged diffraction patterns S(ky) on a logarithmic scale normalized to the maximum value calculated from the three different initial states of a symmetrically stretched CO2 molecule with a C-O distance of R = 4.8 Å under the same excitation conditions as in Fig. 1. Panels (a) to (c) correspond respectively to the HOMO, HOMO-1 and HOMO-2. Black solid lines show the theoretical reconstruction through the inversion algorithm and the red dotted lines show the numerical results. Panel (d) displays the same observable for an incoherent superposition of the three orbitals with equal weights. Gray dotted lines indicate the expected positions ky = nπ/R of the minima of the function sin 2 (kyR).

FIG. 8 .

 8 FIG. 8. (Color online) High energy (|k| > 3.15 a.u) averaged diffraction patterns S(ky) from the HOMO of CO2 in its equilibrium (R0 = 1.2 Å, panel a) and symmetrically stretched (R0 = 4.8 Å, panel b) geometries, under the same excitation conditions as in Fig. 1. Black solid lines show the results in the absence of vibration (fixed R) and the red dotted lines take into account a vibrational dispersion of ∆R = ±1 Å. The gray dotted lines indicate the expected positions of the minima of the function sin 2 (R0 ky), for the corresponding values of the internuclear distance. See text for details.

FIG. 9 .

 9 FIG. 9. (Color online) High energy (|k| > 3.15 a.u) averaged diffraction patterns S(ky) from the HOMO of CO2 in its equilibrium (R = 1.2 Å, panel a) and symmetrically stretched (R = 4.8 Å, panel b) geometries, under the same excitation conditions as in Fig. 1. The black solid lines show the results corresponding to perfect alignment at a fixed internuclear distance and the red dotted lines take into account an angular dispersion of 20 • with the same internuclear distance. The gray dotted lines indicate the expected positions of the minima of the function sin 2 (R ky), for the corresponding values of the internuclear distance. See text for details.

Fig. 11

 11 shows the averaged diffraction patterns obtained for R = 1.2 Å in panel (a), and for R = 4.8 Å in panel (b), with a sine-square pulsed excitation of FWHM 10 fs, represented in red dotted lines. In comparison, the black

FIG. 10

 10 FIG. 10. (Color online) Electron diffraction patterns of the HOMO of CO2 with R = 4.8 Å) in the momentum space representation (lower panels) for a rectangular pulse shape (panel a) and a sine-square pulse shape (panel b). The upper panels display the time variations in arbitrary units of the corresponding electric fields (in red dotted lines) and the two classical trajectories discussed in the text, one starting at positive electric field, in blue, and the other at negative electric field, in green.

FIG. 11 .

 11 FIG. 11. (Color online) High energy (|kx| > 3.15 a.u.) averaged diffraction patterns S(ky) from the HOMO of CO2 in its equilibrium (R = 1.2 Å, panel a) and symmetrically stretched (R = 4.8 Å, panel b) geometries, under the same multi-cycle excitation conditions as in Fig. 10. Black solid lines show the results corresponding to a single-cycle excitation. The gray dotted lines indicate the expected positions of the minima of the function sin 2 (kyR), for the corresponding values of the internuclear distance.

TABLE I .

 I Table I summarizes the values of all parameters defining the SAE potential for the case of the N 2 and CO 2 molecules. Values of all parameters used for the single active electron effective potentials of N2 and CO2.

	Molecule	N2			CO2
	Atom	N	N	O	C	O
	aα (a.u.)	1.2	1.2	1.0	1.0	1.0
	σα (a.u.) 0.700 0.700 0.577 0.750 0.577
	Z 0 α	7	7	8	6	8
	Z ∞ α	0.500 0.500 0.173 0.654 0.173

VI. ACKNOWLEDGEMENTS

The authors would also like to acknowledge useful and stimulating discussions with Dr. Christian Cornaggia of the CEA IRAMIS in Saclay, France. M.P. and T.T.N.D. acknowledge the Natural Sciences and Engineering Research Council of Canada (NSERC) for financial supports. The authors also acknowledge supports from CFQCU (contract number 2010-19), from ANR (contracts ImageFemto ANR-07-BLAN-0162 and Attowave ANR-09-BLAN-0031-01), from FCS Digiteo -Triangle de la Physique (project 2010-078T -High Rep Image), and from the EU (Project ITN-2010-264951, CORINF), and the joint NSF (USA) -ANR (France) FRAMOLSENT project.