Auger spectrum of a water molecule after single and double core ionization

Transcription

1 Auge spectum of a wate molecule afte single and double coe ionization L. Inheste, C. F. Bumeiste, G. Goenhof, and H. Gubmülle Citation: J. Chem. Phys. 6, 444 (); doi:.6/.7 View online: View Table of Contents: Published by the Ameican Institute of Physics. Additional infomation on J. Chem. Phys. Jounal Homepage: Jounal Infomation: Top downloads: Infomation fo Authos:

2 THE JOURNAL OF CHEMICAL PHYSICS 6, 444 () Auge spectum of a wate molecule afte single and double coe ionization L. Inheste, a) C. F. Bumeiste, G. Goenhof, and H. Gubmülle Max Planck Institute fo Biophysical Chemisty, Am Faßbeg, 777 Göttingen, Gemany (Received 9 Decembe ; accepted 9 Mach ; published online 9 Apil ) The high intensity of fee electon lases opens up the possibility to pefom single-shot molecule scatteing expeiments. Howeve, even fo small molecules, adiation damage induced by absoption of high intense x-ay adiation is not yet fully undestood. One of the stiking effects which occus unde intense x-ay illumination is the ceation of double coe ionized molecules in consideable quantity. To povide insight into this pocess, we have studied the dynamics of wate molecules in single and double coe ionized states by means of electonic tansition ate calculations and ab initio molecula dynamics (MD) simulations. Fom the MD tajectoies, photoionization and Auge tansition ates wee computed based on electonic continuum wavefunctions obtained by explicit integation of the coupled adial Schödinge equations. These ates seved to solve the maste equations fo the populations of the elevant electonic states. To account fo the nuclea dynamics duing the coe hole lifetime, the calculated electon emission specta fo diffeent molecula geometies wee incoheently accumulated accoding to the obtained time-dependent populations, thus neglecting possible intefeence effects between diffeent decay pathways. We find that, in contast to the single coe ionized wate molecule, the nuclea dynamics fo the double coe ionized wate molecule duing the coe hole lifetime leaves a clea fingepint in the esulting electon emission specta. The lifetime of the double coe ionized wate was found to be significantly shote than half of the single coe hole lifetime. Ameican Institute of Physics. [ I. INTRODUCTION Ulta intense femtosecond fee electon lases (FEL) allow one to study seveal new phenomena in molecules and atoms and hold the pomise to obtain x-ay scatteing infomation fom lage biomolecules such as poteins at the single molecule level., Molecules exposed to intense x-ay pulses ae expected to undego sevee adiation damage. At illumination conditions in the x-ay egime the dominant electonic pocess is photoionization of coe electons into the continuum. These coe ionizations tigge autoionization pocesses, e.g., Auge decay, which cause efilling of the coe hole vacancy while emitting a seconday electon that caies away the excess enegy. Recent theoetical studies have addessed the fomation of multiple coe ionized electonic states by x-ay FEL adiation in atoms. 5 These electonic states mainly esult fom sequential photoionization pocesses whee the second photoionization occus faste than the efilling of the coe shell by Auge decay. Double coe hole states ae of paticula inteest in spectoscopy, as they can povide moe insight into molecula stuctue than conventional single coe spectoscopy. 6 9 In expeiments with intense FEL x-ay pulses at the Linac Coheent Light Souce (LCLS) significant quantities of such multiple coe ionized states in neon atoms and nitogen molecules wee obseved. In addition to the pue electonic adiation damage, a second consequence of the exposue of molecules to intense a) linhest@gwdg.de. x-ay adiation is the fast dissociative motion, the so-called Coulomb explosion, which is tiggeed by the fast chaging of the molecule. This pocess has been studied by molecula dynamics foce field simulations,, 4 in which electonic tansitions ae descibed stochastically, govened by atomic tansition ates. Howeve, because the molecula dynamics stongly depends on the ionization kinetics, accuate molecula photoionization, and Auge decay ates ae desiable. Fom anothe point of view, Auge spectoscopy may povide a means to study the ionization dynamics and might give infomation on the fast nuclea motion. It is theefoe of inteest to elucidate ionization of molecules by intense x-ay adiation and the fomation of the coesponding Auge specta with espect to the seveal ionization steps and the apid nuclea motion. One of the main challenges of calculating molecula Auge decay ates is the appopiate desciption of the continuum electon wavefunction, which cannot be epesented by the commonly used squae-integable (L ) basis functions. Auge tansitions fo small molecules have been studied in seveal appoaches, using (i) Stieltjes imaging, 5 (ii) solving the Lippmann-Schwinge equation on a basis of Gaussiantype functions, 6, 7 (iii) the so-called one-cente appoach using atomic adial Auge integals, 8 and (iv) based on population analysis. 9, The fome two appoaches ely on an asymptotical desciption of the continuum wavefunction with Gaussian basis functions close to the molecule, wheeas the one-cente appoach uses atomic continuum wavefunctions. The method based on population analysis does not include the continuum electon explicitly. -966//6(4)/444//$. 6, 444- Ameican Institute of Physics

3 444- Inheste et al. J. Chem. Phys. 6, 444 () Hee, we calculated molecula ionization and Auge tansition ates using the single cente expansion (SCE) method. In ou appoach the continuum wavefunction is obtained by explicit integation of a set of coupled (static) adial Schödinge equations, wheeas the emaining bound electons ae descibed by usual linea combination of atomic obitals (LCAO). This hybid appoach enabled us to accuately epesent continuum wave functions while taking advantage of efficient L basis sets fo the bound obitals. Using the obtained molecula ionization and Auge decay ates, time-dependent populations of the single and double coe ionized states wee calculated, similaly to pevious appoaches fo atoms. 5 To also include effects of nuclea motion on the Auge spectum within the coe hole life time, we incoheently summed up instantaneous Auge specta fo diffeent molecula geometies, obtained by classically popagating the nuclei positions with foces calculated on the fly fo the coe ionized electonic states. Such appoach, aleady applied in pevious calculations,, avoids the explicit computation of the many involved potential enegy sufaces but neglects possible intefeence effects on the spectum. Othe appoaches, which addess effects of nuclea motion in a coheent way and, theeby, ae able to addess vibational featues of the spectum, ely on pe-calculated potential enegy sufaces. Fo examples, Eoms et al. 4 used the multi-configuational time-dependent Hatee technique to popagate the nuclea wave packets fo the esonant Auge spectum of wate. Bao et al. 5 pesented a calculation of the nomal Auge spectum of the oxygen molecule based on the Kames-Heisenbeg fomula. As a model system we consideed the Auge spectum of a singly and doubly coe ionized wate molecule. While the single coe Auge spectum (K LL) of wate has been extensively studied, 5,, 6 we ae not awae of any studies of its double coe Auge specta (KK KLL). Ou esults confim that the nuclea motion has little effect on the Auge spectum duing the few femtoseconds of the coe hole lifetime fo single coe ionized wate. Stikingly, howeve, the nuclea motion of double coe ionized wate was found to makedly affect the Auge spectum due to fast dissociation dynamics. The outline of the pape is as follows. Ou appoach to detemine ionization tansition ates is descibed in Sec. II. Section III descibes the computational details of the calculations. Results and conclusion ae pesented in Secs. IV and V. II. CALCULATION OF ELECTRONIC IONIZATION TRANSITIONS RATES Calculation of ionization ates equies the desciption of initial ψ ini and final ψ fin electonic wave functions. To claify the notation, we fist descibe in Subsection II A how the final electonic states ψ fin ae constucted fom a molecula bound pat and a continuum pat. In Subsections IIB and II C we descibe how photoionization coss sections and Auge decay ates ae obtained. A. Constuction of the final electonic state A total final electonic state ψ fin afte ionization is constucted by combining a multi-electon bound pat ψ fin and a single electon pat descibed by the continuum electon wave function φ k,σ () with enegy ɛ = k / and spin σ. Following spin addition ules 7 the state is given by ()S=/ ψ fin,m S =/ = c k,α ψ fin,m S= S =, () ()S=/ ( ψ fin,m S =/ = c k,α ψ fin,m S= S = + c fo doublet states and ψ fin,m S= ( S = = c k,α ψ S=/ fin,m S = / +c k,β k,β ψ fin,m S= S = ), ψ S=/ () fin,m S =/ ), fo singlet states. The additional indices S and M S descibe total spin quantum numbes 8 and c k,σ is the ceation opeato fo a continuum electon with wavefunction φ k,σ (). Fo the evaluation of tansition ates given by fist-ode petubation theoy, matix elements of an opeato O between initial ψini,m S S and final states ψfin,m S S have to be calculated. Assuming O commutes with spin S, expessions fo final states ψ ()S=/ S= fin,m S =/ and ψfin,m S = can be simplified to9 ψ S=/ ini,m S =/ O ψ ()S=/ fin,m S =/ = ψ S=/ O c ψ S= ini,m S =/ k,α ini,m S = O ψ S= fin,m S = = ψ S= O c ini,m S = k,α () ψ S= fin,m S =, (4) ψ S=/ fin,m S = /. (5) The bound pat ψ S fin,m S of the final electonic wavefunction can be epesented by the usual linea combination of atomic obitals (LCAO), while the desciption of the continuum wavefunction φ k,σ equies a continuum epesentation. Hee we epesent φ k,σ in a single cente expansion given by φ k,σ () = P k () Y (θ,φ), (6) whee Y (θ, φ) ae spheical hamonics, and the P k () ae a set of adial wave functions, which solve the set of coupled adial Schödinge equations d d P k () + M,l m ()P l k m () =, (7) l m

4 444- Inheste et al. J. Chem. Phys. 6, 444 () with ( l(l + ) M,l m () := δ l,lδ m,m + l m v l m () ) + ɛ d Y (θ,φ) Y l m (θ,φ)y l m (θ,φ). (8) Hee, d descibes integation ove the solid angle and v () ae the adial pats of the SCE of the potential V(), whee V () = v () Y (θ,φ) = V ne () + V ee () = v ne () + vee () Y (θ,φ). (9) V ne () is the nuclea potential of the molecule and V ee () epesents the inteaction of the continuum electon with the emaining bound electons. Fo the spheical nuclea coodinates R n, θ n, φ n of nucleus n =...N, the adial pats v ne() of the nuclea potential ae given by v ne () = n Z n l < > l+ Y (θ n,φ n ), () with < = min(, R n ) and > = max(, R n ) and Z n being the chage of nucleus n. The electon-electon inteaction V ee ()is detemined by the electostatic potential of the chage density ρ() of the electons in the bound pat ψ fin, J () = d ρ( ), () and the shote anged exchange pat. We used the KSG (Ref. ) exchange potential, ( ) V XC [ ρ()] = π ρ(), () to model the exchange inteaction, which endes Eq. (7) as a homogenous diffeential equation. To futhe simplify the calculations, the non-spheical symmetic pats (l )ofthe electon density ρ() in the exchange potential (Eq. ()) ae appoximated by fist-ode Taylo expansion ( V XC [ ρ()] = π ρ () +O l,m Y ) + l,m ρ ()Y (θ,φ) ρ ()Y ρ ()Y (θ,φ), () ρ ()Y whee ρ () ae the adial pats of the SCE of the electon density ρ() = ρ () Y (θ,φ). (4) The adial pats v ee () of the electon-electon inteactions in Eq. (9) ae finally given by the SCE of Coulomb potential J() and the electon density ρ(), v ee () d Y (θ,φ)j () + Y ( π { fo l = ρ ()/( ρ ()) else. ) ρ () Y (5) Subsequently, the enegy-degeneated solutions of Eq. (7), P k,lm (), ae labeled by the additional index tuple LM. They ae equied to fulfill the bounday conditions P k,lm ( ) =, (6) P k,lm ( ) = πk (δ LM,F l (k) + R LM, G l (k)). (7) F l (k) and G l (k) ae the egula and iegula Coulomb functions and R LM, ae elements of an hemitian matix R detemined by the asymptotic behavio of the solutions. The above mentioned bounday conditions do not povide an enegy-nomalized solution, as equied fo coect tansition ates. Hence, enegy nomalization is achieved by the linea combination P k,lm () = L M + λlm U LM,L M P k,l M, (8) whee the columns of U and λ LM ae eigenvectos and eigenvalues of the Matix R, espectively. Note, that hee the bound electons ae consideed to be not affected by the continuum electon, thus the bound electon pat can be calculated independently. To evaluate electonic tansition ates, electon integals between bound and continuum electons have to be calculated. Fo the pupose of calculating these quantities within the SCE, also the bound obitals ae expanded into the SCE as in Eq. (6). B. Photoionization Following fist-ode petubation theoy, the photoionization coss sections σ ini fin in length gauge is popotional to the dipole matix elements between the initial ψ ini and final ψ fin electonic states, σ ini fin = 4απ ω ψ ini s d ψ fin, (9) whee α /7 is the fine-stuctue constant, ω is the photon enegy, and s is the electic polaization vecto of the electomagnetic wave. The elements of the tansition dipole moment d ae expessed in the SCE by 5 d M = c i c j 4π l m d P i ()Pj () d Y (θ,φ)yl m (θ,φ)y M (θ,φ), () whee P i j () and Pl m () epesent adial pats of the espective bound and continuum electon wave functions and c i, c j

5 444-4 Inheste et al. J. Chem. Phys. 6, 444 () ae the coesponding ceation/annihilation opeatos, espectively. Aveaging ove all molecula oientations, yields σ ini fin = 4 απ ω ψ ini d M ψ fin. () M=,, The photoionization tansition ate is given by Ɣini fin Photo = σ ini fin F (t), () whee F(t) is the time-dependent photon flux. The total photoionization coss section σ ini eads σ ini = σ ini fin. () fin The above summation involves diffeent continuum solutions (LM) as well as diffeent electonic bound pats ( ψ fin ). C. Auge tansition The tansition ate fo Auge decay Ɣ Auge ini fin fom fistode petubation theoy is given by Ɣ Auge ini fin = π ψ fin H E ini ψ ini. (4) Assuming vanishing state ovelap ψ fin ψ ini =, the Auge tansition ates ae given by matix elements of the electonic Hamiltonian, ψ fin H E ini ψ ini = ψ fin c i c j h ij + c i c j c lc k ij kl ψ ini. ij ij kl (5) The above two- and one-electon integals ae eadily evaluated in the SCE epesentation, h ij = δ σi,σ j dφi ( () ) + V ne() φ j () and = δ σi,σ j ( d dp i () d dp j () d + P i ()v ne,l m ()P j l m () l m l m d Y (θ,φ)y l m (θ,φ)y l m ), (θ,φ) (6) (ik jl) = d = ij kl := δ σi,σ k δ σj,σ l (ik jl), (7) l m l m d φ i ( )φ k ( ) φ j ( )φ l ( ) d y ik j ()Pl m ()P l l m () d Y (θ,φ)y l m (θ,φ)y l m (θ,φ), (8) y ik () := l m l m d 4πPi l m ()P l k m () l + < l > l+ d Y (θ,φ)yl m (θ,φ)y l m (θ,φ). (9) Hee, σ i is the spin of spin obital φ i,σi, < = min(, ) and > = max(, ). Neglecting othe elaxation effects such as fluoescence which is small fo the light nuclei consideed hee the total lifetime τ of the initial state is given by τ = /Ɣ Auge ini = / fin Ɣ Auge ini fin, () whee Ɣ Auge ini is the total tansition ate of the initial state. Again summation index fin descibes the complete elaxation channel given by continuum solution (LM) and bound pat ( ψ fin ). III. COMPUTATIONAL DETAILS A. Single cente expansion Dunning s cc-pvtz basis set 4 was used to epesent bound molecula obitals (MOs) in all computations. Fo each ionization and Auge decay step, molecula obitals wee calculated by a esticted (open shell) Hatee Fock pocedue (R(O)HF) optimized fo the initial electonic state. Thus, we calculated MOs fo the neutal and double coe ionized state by RHF, espectively, and MOs fo the single coe ionized state with ROHF. The self-consistent field optimization fo coe ionized states was caied using a modified PSI quantum package 5 as descibed elsewhee. 6 This pocedue is known to account fo most of the coe electon vacancy induced obital elaxation effects. 7 Fom the esulting MOs, the SCE of each obital φ i () (Eq. (6)) has the adial pats P i () = d φ i () Y (θ,φ). () These adial pats as well as the adial pats of thei electostatic potential and density wee numeically calculated using Gaussian-Legende integation with integation points in angle space. Because all of these quantities ae eal, eal valued tesseal spheical hamonics 8 instead of the usual complex valued spheical hamonics wee used to educe the computational cost. Angula integals ove thee spheical hamonics, also known as Gaunt coefficients, ae eadily calculated by evaluation of Wigne j symbols. 9 Fo the adial coodinate non-equidistant adial gid points wee used implicitly detemined by ρ() = α+ β ln + n actan R n. () γ Hee R n ae the distances of atom n to the cente of the expansion, and vaiable ρ is discetized on an equidistant gid. The numbe of adial gid points was 5, the lagest adial gid point was set to = a.u., and the cente fo the expansion was chosen at the position of the oxygen atom, which allowed the SCE to limit to angula quantum numbes l 5,

6 444-5 Inheste et al. J. Chem. Phys. 6, 444 () esulting in 6 diffeent () tuples. These paametes tuned out to epesent the molecula obitals and elevant continuum wavefunctions of the wate molecule sufficiently accuate and thus have been used fo all subsequent calculations d OH =.96 a θ=.5 B. Configuational mixing We applied the fozen obital appoximation, i.e., initial and final electonic states wee epesented by the same obital set. In paticula, obitals optimized fo the initial state by the R(O)HF calculation mentioned above wee used. The molecula obital integals, obtained fom the PSI quantum package, 5 wee used to pefom spin adapted multi-efeence configuation inteaction (MRCI) calculations fo the final bound electonic states ψ S fin,m S and single efeence configuation inteaction (CI) calculations 4 fo the initial states ψ S ini,m S. As efeences fo the final state we have chosen the initial state efeence with all possible combinations of one additional vacancy (fo photoionization) o two valence electons emoved and a e-occupied coe obital (fo Auge decay). Fom the efeence occupations configuation state functions (CSFs) wee built consideing single excitations within the full MOs space and double excitations up to the th MO. This tuncation of the CI space was used fo both single and the multi-efeence calculations, leading to a numbe of CSFs fo the initial states (single efeence CI) and 85 8 CSFs fo the final state (MRCI). Fom these calculations only solution vectos with significant contibutions in the efeences (nom of pojection into efeence subspace >.) wee used fo subsequent calculations. Fo these solution vectos the SCE of the electostatic potential and the electon density of the espective final electonic state was obtained as a linea combination of the electostatic potential and electon density of the MOs. C. Integation of the continuum wavefunction Fo the elevant CI vectos, the 6 solutions fo the continuum wavefunctions in the potential of the molecule of the specific final electonic state ψ S fin,m S wee integated accoding to Eq. (7) fo the given bounday conditions (Eq. (7))usingthevecto sweep integation method adopted fom Ref.. The continuum nomalization was caied out by diagonalizing the obtained R matix as descibed by Eq. (8). D. Tansitions Photoionization coss sections and Auge decay tansition ates wee calculated by evaluating Eqs. () and (5) using Simpson s ule. In paticula, fo the two electon integals a system of coupled diffeential equations, 4 was solved fo y ik () and then Eq. (8) was integated by Simpson s ule, as had been descibed fo atoms. 4 As diffeent angula continuum channels wee not distinguished hee, tansition ates fo diffeent continuum solutions (LM) wee finally summed up. E in Hatee θ in (U)HF R(O)HF/CI d OH in a FIG.. Cuts though the potential enegy sufaces fo single (top) and double coe (bottom) ionized wate. (Left) Both hydogen atoms ae at equilibium distance d OH =.96a to the oxygen fo diffeent HOH angles. (Right) One hydogen atom is fixed at d OH =.96a and the othe is at vaiable distances to the oxygen atom, while is at equilibium value of.5. E. Molecula dynamics calculations All calculations of Auge decay ates wee pefomed fo a set of molecula geometies, obtained by ab initio molecula dynamics (MD) simulation. The nuclei wee popagated using the Beeman integation scheme 4 with a. fs time step in the single and double coe ionized state. Enegy gadients wee calculated with GAUSSIAN 9 (Ref. 44) fom unesticted Hatee Fock (UHF) calculations fo the single coe ionized state and esticted Hatee Fock (RHF) calculations fo the double coe ionized state. Convegence of the selfconsistent field pocedue to the desied open coe shell states was achieved by choosing an initial guess based on neutal optimized obitals with single o unoccupied coe obital, espectively. To assess the accuacy of the (U)HF method used to geneate the tajectoies, we compaed two sections of the potential enegy suface obtained by the single deteminant (U)HF method with that obtained by the CI method descibed in Sec. II B. As seen in Fig., vey simila cuves ae obtained, apat fom a nealy constant offset. Theefoe, we conside the gadients at (U)HF level to be sufficiently accuate to descibe the nuclea dynamics afte coe ionizations. F. Initial conditions The Auge spectum of wate is dominated by Fanck- Condon boadening due to the vey steep potential enegy sufaces of the final electonic states. To estimate this boadening (as descibed futhe below), multiple MD tajectoies wee calculated with initial conditions at the σ standad deviation of the gound state Wigne distibution. To that aim, neutal gound state optimization and vibational mode calculations 4 using hamonic appoximation wee pefomed with GAUSSIAN 9 (Ref. 44) on the MP level. Fom the optimized gound state geomety six diffeent initial conditions with zeo velocities wee geneated by vaying the geomety in positive and negative diections along each of the thee nomal modes by the standad deviation σ of the vibational gound state distibution. Six futhe initial conditions wee

7 444-6 Inheste et al. J. Chem. Phys. 6, 444 () geneated fom the optimized geomety with initial velocities in positive and negative diections along the vibational nomal modes. These velocities wee chosen to match the standad deviation σ of the vibational gound state velocity distibution. Togethe with the optimized geomety with zeo velocity, a total set of sets of initial conditions wee thus obtained. vibational goundstate distibution time dependent photon flux F (t) time t τ t τ t neutal single coe double coe G. Spectum Fo each of the initial conditions, molecula dynamics simulations in the single coe ionized state wee stated. Rathe than estimating the Fanck Condon boadening of the Auge lines fom a weighted aveage ove many Wignedistibuted tajectoies (which would equie a consideable numbe of tajectoies to achieve sufficient sampling), the vaiance of the assumed Gaussian line pofile of the dominant Auge tansitions was estimated fom the diffeences ɛ(t)in the Auge tansition enegies between the cental tajectoy (stated fom the optimized geomety with zeo initial velocities) and the satellite tajectoies (stated fom alteed initial conditions) as σ (t) = σ exp + σ lifetime + ɛi,+x (t) + ɛ i, x (t) + ɛ i,+v (t) + ɛ i, v (t). () This computationally moe efficient estimate ests on the assumption that the Gaussian shape of the nuclei wave packet is appoximately maintained duing the shot simulation time. Moe pecisely, we assume that the peak of the wave-packet emains sufficiently close to the cental tajectoy, and the satellite tajectoies emain on aveage sufficiently close to the suounding σ hypesuface, such that the width of the wave packet can be estimated fom thei aveage distance to the cental tajectoy. Additionally, we assume that within the phase space egion coveed by the wave packet, the tansition enegy is sufficiently linea in the atomic coodinates and the individual tansition ates ae constant. Visual inspection of the tajectoies showed that these conditions ae satisfied. This allowed us to estict the computation of Auge tansition ates to the cental tajectoy, while fo the satellite tajectoies only tansition enegies needed to be calculated. In Eq. () ɛ i,±x (t) denotes the diffeence of the tansition enegy between the cental and the satellite tajectoy, stated with geometies modified along the vibational mode i. Similaly, ɛ i,±v (t) denotes the diffeence of the tansition enegy between the cental and the satellite tajectoy, stated with velocities modified along vibational mode i. Additionally, the effect due to limited expeimental esolution and due to line boadening was included, with σ exp =.7 ev (.4 ev FWHM (Ref. 45)), and σ lifetime estimated fom the decay ates calculated in Sec. IV B. To follow the evolution of double coe ionized wate afte a peiod of nuclea dynamics in the single coe ionized state, additional simulations of the double coe ionized state i= Auge photoionization FIG.. Illustation of diffeent ionization pathways. The total Auge spectum is obtained fom a supeposition of specta esulting fom diffeent tajectoies along the neutal, single coe ionized, and double coe ionized states. Hee, as an example, 4 pathways contibuting to the single and the double coe Auge spectum ae illustated. wee stated fom selected snapshots of the single coe ionized state tajectoies ( cental and satellite ) at,,...9fs, thus esulting in a total of double coe tajectoies. The simulation time fo each of the concatenated single and double coe tajectoies was limited to fs, and the simulation time in the double coe ionized state was limited to 7 fs. Figue illustates how the specta wee composed of diffeent tajectoies using diffeent pathways though the single and double coe ionized states. The accumulated single and double coe hole Auge specta, S and S, wee calculated consideing all possible pathways by S = dt t dt p (t,t ) s (t t ), (4) t t S = dt dt dt p (t,t,t ) s (t t,t t t ), (5) whee s (τ ) is the time-dependent single coe Auge spectum obtained as instantaneous spectum fom the geomety esulting fom popagating nuclei in the single coe ionized state fo a time inteval τ. Similaly, s (τ, τ ) denotes the instantaneous double coe Auge spectum esulting afte a time inteval τ of nuclea dynamics in the single coe ionized state and subsequent nuclea dynamics in the double coe ionized state fo an inteval τ. The instantaneous single coe Auge specta s (τ ) wee weighted hee with the joint pobability p (t, t ) dt of finding the molecule at time t in the single coe ionized state afte ionization at time t. Similaly the double coe Auge specta s (τ, τ ) wee weighted with the joint pobability p (t, t, t ) dt dt of finding the molecule in double coe ionized state given that the fist and second ionizations have occued at times t and t, espectively. The joint pobability densities p (t, t ) and p (t, t, t ) ae expessed in tems of conditional pobabilities p (t t ) and p (t t, t )by p (t,t ) = p (t t ) p (t )σ F (t ), (6) p (t,t,t ) = p (t t,t ) p (t,t )σ F (t ), (7) whee p (t) is the pobability of the neutal electonic state at time t, F(t) is the photon flux, and σ i j ae the patial

8 444-7 Inheste et al. J. Chem. Phys. 6, 444 () photoionization coss sections. Hee the indices,, and denote the neutal, single coe ionized, and double coe ionized states, espectively. The conditional pobabilities wee obtained fom the numeical solution of the maste equations dp (t) = p (t)σ F (t); p (t ) =, (8) dt dp (t t ) dt dp (t t,t ) dt = p (t t ) ( Ɣ Auge + σ F (t) ) ; p (t t ) =, (9) = p (t, t,t )Ɣ Auge ; p (t, t,t ) =. (4) Hee we neglected any geomety (i.e., time) dependence of the total Auge decay ates Ɣ Auge and Ɣ Auge as well as of the espective photoionization coss sections σ i j, because they all wee found to vay by less than % fo all obtained geometies. To gain moe insight in the ionization pocess, also the total populations of the single and double coe ionized state at time t wee consideed, given by N (t) = N (t) = t t dt t H. Illumination conditions dt p (t,t ) (4) dt p (t,t,t ). (4) We assumed a Gaussian x-ay pulse with fs FWHM. Soft x-ay photon beams at photon enegies of kev wee consideed, with peak intensities of photons/fs Å ˆ.6 6 W/cm, photons/fs Å ˆ.6 7 W/cm, and photons/fs Å ˆ.6 8 W/cm, espectively. These paametes agee with the egime offeed by the Atomic, Molecula and Optical science instument at the LCLS (Ref. 46) and povide a high ionization ate such that a consideable amount of double coe ionizations is eached. Fo visible light one might ague that such high fluxes give ise to instantaneous multi-photon ionization o tunnel ionization events. These effects ae elevant if the pondeomotive enegy U p = 8π I exceeds the ionization enegy. 7 4ω Howeve, fo the x-ay pulses consideed hee, U p.4 ev, which is fa below the ionization enegy. Despite the high intensities, these effects can theefoe assumed to be negligible, and thus the petubative appoach of sequential photoionization is justified. IV. RESULTS AND DISCUSSION A. Photoionization coss sections We fist tested ou appoach by compaing calculated total photoionization coss sections fo neon with values fom McMastes compilation of x-ay coss sections 47 fo selected TABLE I. Calculated total photoionization coss sections fo atomic neon compaed to values fom Ref. 47. Photon This wok Fom McMaste et al. 47 enegy in in (kev) a.u. a.u photon enegies. As shown in Table I, ou calculated ionization coss sections agee well with the tabulated values. Table II lists the obtained total and patial coss sections of wate fo a photon enegy of ω = kev. Fo wate, the atio of the patial coss sections to the single coe ionized state to the total coss section σ /σ is about 8%. The emaining % mostly involve coe ionization with additional shake-up tansitions of valence electons. A simila atio is seen fo the second ionization step. B. Auge decay ates To also validate Auge decay ate calculations, we compaed in Table III calculations fo neon with calculations fom Koloenč and Avebukh, 48 Bhalla et al., 49 Yazhemsky and Sgamellotti 5 (single coe), Kelly 5 (single coe), Pelicon et al. 5 (double coe), and Chen 5 (double coe). The values epoted in these studies vay by about % fo the single coe and by up to % fo the double coe Auge tansition. As can be seen in Table III, ou values fall within these anges. Table IVcompaes the calculated single coe Auge tansition ates fo wate with pevious calculations of absolute 5 and elative values. 6 We have adjusted the calculated enegies to the expeimental spectum (see Fig. 5) by subtacting an oveall offset of. ev. This offset may esult fom neglecting elativistic effects in ou calculation, tuncation of the CI space, o incomplete basis sets. As can be seen, the elative ates (nomalized to the dominant S peak) compae well in the highe enegy egime, whee final states consist of two oute valence holes. In the lowe enegy ange, somewhat lage deviations ae seen, which can be explained by the stonge influence of shake-up contibutions. Notably, ou values tend to be lage than the Auge decay ates obtained by Stieltjes imaging calculations by Caavetta and Ågen 5 and Koloenč and Avebukh, 48 with a total tansition ate of 6. a.u. compaed to TABLE II. Calculated total and patial photoionization coss sections fo wate at kev. The indices,, and denote the neutal, single coe ionized, and double coe ionized state, espectively. Tansition Coss section in a.u. σ.84 σ.8 σ.6 σ.7

9 444-8 Inheste et al. J. Chem. Phys. 6, 444 () TABLE III. Compaison of calculated Auge decay ates fo singly and doubly coe ionized neon in a.u. Total single coe Auge decay ate in a.u. Koloenč Yazhemsky This wok and Avebukh 48 and Sgamellotti 5 Kelly 5 Bhalla et al Total double coe Auge decay ate in a.u. Koloenč This wok and Avebukh 48 Pelicon et al. 5 Chen 5 Bhalla et al a.u. (Ref. 5) and 5.4 a.u. (Ref. 48). Howeve, ou value fo the total Auge decay ate is simila to the single coe hole decay ate measued by Sankai et al., ±. a.u. Table V shows tansition ates obtained fo the double coe Auge spectum of wate. As fo the single coe Auge tansitions, enegies have been shifted by. ev. We note, howeve, that elativistic contibutions not taken into account in ou calculations may contibute about ev moe in the double than in the single coe hole case. 55 The obtained total double coe decay ate (8. a.u.) is about thee times lage than the single coe decay ate. It is also significantly lage than the value epoted by Koloenč and Avebukh 48 who used Stieltjes imaging method (.4 a.u.). These authos 48 estimated thei value to be % too low due to insufficient inclusion of initial state obital elaxation effects. This estimation was based on the discepancy between thei esults and othe calculations fo atomic neon (see Table III). Wheeas Koloenč and Avebukh 48 used neutal state opti- TABLE V. Total Ɣ Auge and patial Ɣ Auge f Auge tansition ates of wate (double coe) fo the main tansition channels (MP optimized equilibium geomety). Channel Enegy in ev Ɣ Auge ini fin in 4 a.u a a a a 5..4 a a a a Ɣ Auge 8. mized obitals and cove obital elaxation effects in initial and final states with the ADC()x (Ref. 56) method, ou calculation is based on initial state optimized obitals and incopoates final state obital elaxation by configuational inteaction. We theefoe assume that ou calculation does not suffe fom these poblems. C. Population Figue shows the populations of the neutal, single, and double coe ionized states obtained fom Eqs. (4) and (4) fo diffeent beam intensities. The decease ate of the neutal population inceases with pulse intensity, wheeas the tansition ate to single and double coe ionized populations TABLE IV. Total Ɣ Auge and patial Ɣ Auge fin Auge tansition ates of wate (single coe) fo the main tansition channels (MP optimized geomety) compaed to calculations fom Caavetta and Ågen 5 and Siegbahn et al. 6 Enegy in ev Ɣ Auge ini fin in 4 a.u. Relative Ɣ Auge ini fin Channel This wok This wok Fom Ref. 5 This wok Fom Ref. 5 Fom Ref. 6 a T S a S T S a S a T a S S a T a a T a T a S a a S a S a S Ɣ Auge

10 444-9 Inheste et al. J. Chem. Phys. 6, 444 () neutal.6* W/cm.6* W/cm.6* W/cm single coe ionized double coe ionized time in fs FIG.. Integated population of the neutal N (t), single coe ionized N (t), and double coe ionized states N (t). A Gaussian shaped x-ay pulse centeed at time t = fs, width of fs FWHM and a photon enegy of kev was assumed. incease, such that thei peak positions ae shifted to ealie times. As can be seen in Table VI, the pobability fo double coe ionization is about times smalle than fo single coe ionization at.6 6 W/cm, and eaches a atio of.465 at.6 8 W/cm.At.6 8 W/cm the fist ionization step is satuated, and the pobability of fist coe ionization agees with the atio σ /σ =.8. Note that the missing % fom shake-up contibutions and valence ionizations ae not consideed hee and thus ae missing in ou simulations. Afte Auge decay the molecule may undego futhe coe ionizations, as has been obseved fo Neon. Note that these futhe ionization steps, which involve a lage numbe of diffeent channels, ae not included in ou simulation. D. Single and double coe ionized Auge specta Figue 4 illustates tajectoies stating fom zeo velocities and equilibium geomety by snapshots of the electon densities in the molecula plane, calculated fom the CI wavefunctions. The evolution of the OH bond length is shown in the uppe left panel, that of the HOH bond angle in the supplementay mateial. 4 As can be seen, in the single coe ionized state the nuclei motion is mainly a bending motion, wheeas in the double coe ionized state potons ae apidly expelled fom the molecule within a few femtoseconds. Duing that pocess, and with futhe ionization, the electon density becomes inceasingly isotopic. TABLE VI. Total pobability of single and double coe ionization fo diffeent flux intensities. Intensity (W/cm ) st coe ionization nd coe ionization Ratio nd/st y/a d OH /a single coe double coe t in fs t= fs t= fs x/a t=5 fs t= fs t= fs t=4 fs FIG. 4. Dynamics of a wate molecule afte single (middle ow) and double (bottom ow) coe ionization. The uppe left plot shows the evolution of the OH-bond length d OH fo the single and double coe ionized state. The othe plots show cuts though the electon density in the molecula plane (contou lines) at thee selected times fo the neutal, single coe ionized, and double coe ionized state. Cosses denote the positions of the nuclei; tiangles mak the neutal equilibium positions of the nuclei. All plots efe to the tajectoy stating fom equilibium geomety with zeo initial velocities. Figue 5 compaes the calculated single coe spectum at peak intensity.6 6 W/cm with the spectum measued by Moddeman et al. 45 Also shown fo compaison ae the calculated specta obtained with fozen nuclei. As can be seen, the calculated spectum captues most of the featues of the expeimental spectum vey well. Only small deviations to the spectum with fozen nuclei ae obseved. Obviously, fo single coe ionization the influence of the nuclea motion Intensity / abitay units calculated spectum calculated spectum without nuclei dynamics expeimental spectum a S a S a a S a S ε in ev a T a a T a T S a S S a T T a S S neutal single coe ionized double coe ionized a S a T FIG. 5. Single coe Auge spectum. Compaison of expeimental spectum 45 (dashed line) and calculated spectum fo peak intensity I =.6 6 W/cm with (ed line) and without (geen line) nuclei dynamics. The position of the peaks wee labeled accoding to thei dominant hole configuations.

11 444- Inheste et al. J. Chem. Phys. 6, 444 () t= fs t= fs t= 4 fs t= fs t= fs t= fs.5.6* W/cm.6* W/cm.6* W/cm. Intensity / abitay units Intensity / abitay units Electons / ev ε in ev ε in ev FIG. 6. (Left) Instantaneous single coe Auge specta afte fs, fs, and 4 fs of the single coe ionized state. (Right) Instantaneous double coe Auge specta afte fs, fs, and fs of the double coe ionized state. Fo the double coe Auge spectum, nuclea motion was calculated only fo the double coe ionized state. The fast dissociation in the double coe ionized state is eflected by the fast shift of the Auge spectum to highe enegies. in the Auge spectum is athe small, which can also be seen in Fig. 6 (left), whee a set of instantaneous Auge specta of selected snapshots of the single coe ionized state tajectoies ae shown. Indeed, fo diffeent times these Auge specta ae vey simila, with the notable exception at about ev, whee final states ae associated with vacancies in oute valence obitals and a, which ae mostly affected by the hydogen bending movement and thus sensitive to small geomety changes. The valence obital is only weakly affected by geomety changes as its nodal plane is identical with the molecula plane. Figue 7 shows the calculated double coe spectum at peak intensity.6 8 W/cm and the calculated spectum with fozen nuclei. Hee the nuclea dynamics causes a long tail to highe enegies fo each peak in the spectum. The instantaneous specta at diffeent time steps (Fig. 6, ight) confim that the specta indeed shift to highe enegies as the Intensity / abitay units calculated spectum calculated spectum without nuclei dynamics a a a ε in ev FIG. 7. Double coe Auge spectum. Calculated spectum fo peak intensity I =.6 8 W/cm with (ed line) and without (geen line) nuclei dynamics. The positions of the peaks ae labeled accoding to thei dominant hole configuations. a a a ε in ev FIG. 8. Single and double coe Auge spectum fo diffeent peak intensities. At peak intensity.6 8 W/cm the two pats of the Auge spectum ae at compaable intensity. double coe ionized state evolves. This is a esult of the stong dissociative motion in the double coe ionized state (see Fig. 4, bottom). As both potons ae epelled, positive chage is emoved fom the molecule. As a esult, the subsequent Auge ecombination involves lage enegy diffeences. The combined single and double coe Auge specta, composed accoding to Eqs. (4) and (5), ae shown in Fig. 8 fo the same flux paametes as in Fig.. Fo.6 6 W/cm the single coe Auge spectum clealy dominates, while at.6 7 W/cm the double coe spectum has aleady significant contibution. At peak intensity.6 8 W/cm, the double coe and single coe specta each the same intensity. Hee, about 5% of the population is double coe ionized, such that the single coe Auge spectum becomes even smalle than fo.6 7 W/cm. V. CONCLUSION We have developed a pocedue fo calculating ab initio tansition ates fo photoionization and molecula Auge decay, which was validated against pevious calculations and expeimental data fo neon. Ou test calculations demonstates that the single cente expansion method in combination with LCAO fo the bound MOs povides eliable coss sections and tansition ates fo wate. It was demonstated that fo descibing ionization dynamics the second coe ionization pocess must be consideed fo intensities above 7 W/cm. Auge specta wee computed fo a single and double coe ionized wate molecule. Fo these calculations, the nuclea dynamics duing the coe hole lifetime wee descibed by an MD appoach based on the coe ionized UHF/RHF wavefunction. The obtained total Auge decay ates as well as the specta agee well with pevious expeimental data. Stikingly, the Auge decay ate of double coe ionized wate tuned out to be thee times lage than that of single coe ionized wate. Only small effects of the nuclea motion on the single coe Auge spectum wee seen. In contast, fo the double coe ionized wate molecule fast dissociation dynamics is seen, which

12 444- Inheste et al. J. Chem. Phys. 6, 444 () stongly affects the espective Auge specta. A paticula signatue of nuclea motion, which should be seen in futue FEL expeiments, ae maked tails at the highe enegy side of most peaks. This signatue, theefoe, might povide an independent pobe fo detecting fast nuclea motion on femtosecond time scales. Futue wok should addess possible vibational intefeence effects fo the nuclea motion, which in ou incoheent accumulation of specta have been neglected. Although only small intefeence effects ae expected fo the single coe hole Auge spectum of the wate monome, moe ponounced fingepints might be visible in the double coe hole Auge spectum, and in paticula in the specta of the wate dime as 57, 58 has been demonstated fo the x-ay emission spectum. ACKNOWLEDGMENTS This wok has been suppoted by the DFG, Gant No. SFB 755. R. Neutze, R. Wouts, D. van de Spoel, E. Wecket, and J. Hajdu, Potential fo biomolecula imaging with femtosecond X-ay pulses, Natue (London) 46, 75 (). K. J. Gaffney and H. N. Chapman, Imaging atomic stuctue and dynamics with ultafast X-ay scatteing, Science 6, 444 (7). M. Makis, P. Lambopoulos, and A. Mihelič, Theoy of multiphoton multielecton ionization of xenon unde stong 9-eV adiation, Phys. Rev. Lett., (9). 4 N. Rohinge and R. Santa, X-ay nonlinea optical pocesses using a self-amplified spontaneous emission fee-electon lase, Phys. Rev. A 76, 46 (7). 5 S. Son, L. Young, and R. Santa, Impact of hollow-atom fomation on coheent x-ay scatteing at high intensity, Phys. Rev. A8, 4 (). 6 R. Santa, N. V. Kyzhevoi, and L. S. Cedebaum, X-ay two-photon photoelecton spectoscopy: a theoetical study of inne-shell specta of the oganic paa-aminophenol molecule, Phys. Rev. Lett., (9). 7 M. Tashio, M. Ehaa, H. Fukuzawa, K. Ueda, C. Buth, N. V. Kyzhevoi, and L. S. Cedebaum, Molecula double coe hole electon spectoscopy fo chemical analysis, J. Chem. Phys., 84 (). 8 L. S. Cedebaum, F. Taantelli, A. Sgamellotti, and J. Schime, On double vacancies in the coe, J. Chem. Phys. 85, 65 (986). 9 J. H. D. Eland, M. Tashio, P. Linusson, M. Ehaa, K. Ueda, and R. Feifel, Double coe hole ceation and subsequent auge decay in nh and ch 4 molecules, Phys. Rev. Lett. 5, 5 (). L. Young, E. P. Kante, B. Kaessig, Y. Li, A. M. Mach, S. T. Patt, R. Santa, S. H. Southwoth, N. Rohinge, L. F. DiMauo, G. Doumy, C. A. Roedig, N. Beah, L. Fang, M. Hoene, P. H. Bucksbaum, J. P. Cyan, S. Ghimie, J. M. Glownia, D. A. Reis, J. D. Bozek, C. Bostedt, and M. Messeschmidt, Femtosecond electonic esponse of atoms to ultaintense X-ays, Natue (London) 466, 56 (). L. Fang, M. Hoene, O. Gessne, F. Taantelli, S. T. Patt, O. Konilov, C. Buth, M. Gueh, E. P. Kante, C. Bostedt, J. D. Bozek, P. H. Bucksbaum, M. Chen, R. Coffee, J. Cyan, M. Glownia, E. Kukk, S. R. Leone, and N. Beah, Double coe-hole poduction in N-: beating the Auge clock, Phys. Rev. Lett. 5, 85 (). C. Gnodtke, U. Saaann, and J. M. Rost, Ionization and chage migation though stong intenal fields in clustes exposed to intense x-ay pulses, Phys. Rev. A 79, 4 (9). Z. Juek and G. Faigel, The effect of tampe laye on the explosion dynamics of atom clustes, Eu.Phys.J.D5, 5 (8). 4 Z. Juek and G. Faigel, The effect of inhomogenities on single-molecule imaging by had XFEL pulses, EPL 86, 68 (9). 5 V. Caavetta and H. Ågen, Stieltjes imaging method fo molecula auge tansition ates - application to the auge spectum of wate, Phys. Rev. A 5, (987). 6 B. Schimmelpfennig, B. Nestmann, and S. Peyeimhoff, Ab initio calculation of tansition ates fo autoionization: the Auge specta of HF and F, J. Electon Spectosc. Relat. Phenom. 74, 7 (995). 7 R. Colle and S. Simonucci, Method fo calculating Auge decay-ates in molecules, Phys. Rev. A 9, 647 (989). 8 E. Chelkowska and F. Lakins, Auge-Spectoscopy fo molecules tables of matix-elements fo tansition-ate calculations coesponding to an s- type, p-type, o d-type initial hole, At. Data Nucl. Data Tables 49, (99). 9 M. Mitani, O. Takahashi, K. Saito, and S. Iwata, Theoetical molecula Auge specta with electon population analysis, J. Electon Spectosc. Relat. Phenom. 8, (). M. Tashio, K. Ueda, and M. Ehaa, Auge decay of molecula double coe-hole state, J. Chem. Phys. 5, 547 (). P. Demekhin, A. Ehesmann, and V. Sukhoukov, Single cente method: a computational tool fo ionization and electonic excitation studies of molecules, J. Chem. Phys. 4, 4 (). M. Odelius, Molecula dynamics simulations of fine stuctue in oxygen K-edge x-ay emission specta of liquid wate and ice, Phys. Rev. B 79, 444 (9). O. Takahashi, M. Odelius, D. Nodlund, A. Nilsson, H. Bluhm, and L. Pettesson, Auge decay calculations with coe-hole excited-state molecula-dynamics simulations of wate, J. Chem. Phys. 4, 647 (6). 4 M. Eoms, O. Vendell, M. Jungen, H. Meye, and L. Cedebaum, Nuclea dynamics duing the esonant Auge decay of wate molecules, J. Chem. Phys., 547 (9). 5 Z. Bao, R. Fink, O. Tavnikova, D. Ceolin, S. Svensson, and M. Piancastelli, Detailed theoetical and expeimental desciption of nomal Auge decay in O, J. Phys. B 4, 5 (8). 6 H. Siegbahn, L. Asplund, and P. Kelfve, The Auge electon spectum of wate vapou, Chem. Phys. Lett. 5, (975). 7 R. Pauncz, Spin Eigenfunctions: Constuction and Use (Plenum, New Yok, 979). 8 In cases whee the total spin quantum numbes ae ielevant these additional indices ae skipped in the following. 9 M. Deleuze, B. T. Pickup, and J. Delhalle, Plane wave and othogonalized plane wave many-body geen s function calculations of photoionization intensities, Mol. Phys. 8, 655 (994). R. Gaspa, Übe eine appoximation des Hateefockschen potentials duch eine univeselle potentialfunktion, Acta Phys. Acad. Sci. Hung., 6 (954). P. Demekhin, D. Omelyanenko, B. Lagutin, V. Sukhoukov, L. Wene, A. Ehesmann, K.-H. Schatne, and H. Schmoanze, Investigation of photoionization and photodissociation of an oxygen molecule by the method of coupled diffeential equations, Opt. Spectosc., 8 (7). J. Sakuai and S. F. Tuan, Moden Quantum Mechanics (Benjamin/Cummings, 985), Vol.. R. Manne and H. Ågen, Auge tansition amplitudes fom geneal manyelecton wavefunctions, Chem. Phys. 9, (985). 4 T. H. Dunning, Gaussian basis sets fo use in coelated molecula calculations. I. The atoms boon though neon and hydogen, J. Chem. Phys. 9, 7 (989). 5 T. D. Cawfod, C. D. Sheill, E. F. Valeev, J. T. Femann, R. A. King, M. L. Leininge, S. T. Bown, C. L. Janssen, E. T. Seidl, J. P. Kenny, and W. D. Allen, PSI: an open-souce ab initio electonic stuctue package, J. Comp. Chem. 8, 6 (7). 6 P. S. Bagus, Self-consistent-field wave functions fo hole states of some Ne-like and A-like ions, Phys. Rev. 9, A69 (965). 7 N. V. Kyzhevoi, R. Santa, and L. S. Cedebaum, Inne-shell single and double ionization potentials of aminophenol isomes, J. Chem. Phys. 5, 84 (). 8 E. Whittake and G. Watson, A Couse of Moden Analysis (Cambidge Univesity Pess, 95). 9 L. Biedenhan, J. Louck, and P. Cauthes, Angula Momentum in Quantum Physics: Theoy and Application, Encyclopedia of Mathematics and its Applications Seies, Vol. 8 (Addison-Wesley, Reading, 98). 4 See supplementay mateial at fo detailed esults fo initial state enegies, the obtained gound state optimized geomety and vibational mode fequencies, and the HOH bond angle evolution. 4 L. Chenysheva, Computation of Atomic Pocesses: A Handbook fo the ATOM Pogams, Bd. (Institute of Physics, 997). 4 J. McDougall, The calculation of the tems of the optical spectum of an atom with one seies electon, Poc. R. Soc. Lond. Se. A 8, 55 (9), see 55.full.pdf+html. 4 D. Fenkel and B. Smit, Undestanding Molecula Simulation (Academic, San Diego, CA, 996).

13 444- Inheste et al. J. Chem. Phys. 6, 444 () 44 M. J. Fisch, G. W. Tucks, H. B. Schlegel et al., GAUSSIAN 9, Revision A., Gaussian, Inc., Wallingfod, CT, W. Moddeman, T. Calson, M. Kause, B. Pullen, W. Bull, and G. K. Schweitz, Detemination of K-LL auge specta of N,O,CO, NO, H O, and CO, J. Chem. Phys. 55, 7 (97). 46 SLAC National Acceleato Laboatoy, Atomic, molecula & optical science,, see 47 W. McMaste, N. del Gande, J. Mallett, and J. Hubbell, Compilation of X-ay coss sections, Lawence Radiation Laboatoy Repot No. UCRL- 574, Sec. II, Rev. (969). 48 P. Koloenč and V. Avebukh, K-shell Auge lifetime vaiation in doubly ionized Ne and fist ow hydides, J. Chem. Phys. 5, 44 (). 49 C. P. Bhalla, N. O. Folland, and M. A. Hein, Theoetical K-shell Auge ates, tansition enegies, and fluoescence yields fo multiply ionized neon, Phys.Rev.A8, 649 (97). 5 V. Yazhemsky and A. Sgamellotti, Auge ates of second-ow atoms calculated by many-body petubation theoy, J. Electon Spectosc. Relat. Phenom. 5, (). 5 H. P. Kelly, K Auge ates calculated fo Ne +, Phys. Rev. A., 556 (975). 5 P. Pelicon, I. Čadež, M. Žitnik, Ž. Šmit, S. Dolenc, A. Mühleisen, and R. I. Hall, Fomation of the hollow s o S state of Ne + by electon impact: obsevation by means of an Auge hypesatellite, Phys. Rev. A 6, 74 (). 5 M. H. Chen, Auge tansition ates and fluoescence yields fo the double- K -hole state, Phys.Rev.A44, 9 (99). 54 R. Sankai, M. Ehaa, H. Nakatsuji, Y. Senba, K. Hosokawa, H. Yoshida, A. D. Fanis, Y. Tamenoi, S. Aksela, and K. Ueda, Vibationally esolved O s photoelecton spectum of wate, Chem. Phys. Lett. 8, 647 (). 55 J. Niskanen, P. Noman, H. Aksela, and H. Agen, Relativistic contibutions to single and double coe electon ionization enegies of noble gases, J. Chem. Phys. 5, 54 (). 56 P. Koloenč, V. Avebukh, K. Gokhbeg, and L. S. Cedebaum, Ab initio calculation of inteatomic decay ates of excited doubly ionized states in clustes, J. Chem. Phys. 9, 44 (8). 57 M. P. Ljungbeg, A. Nilsson, and L. G. M. Pettesson, Semiclassical desciption of nuclea dynamics in x-ay emission of wate, Phys. Rev. B 8, 455 (). 58 M. P. Ljungbeg, L. G. M. Pettesson, and A. Nilsson, Vibational intefeence effects in x-ay emission of a model wate dime: implications fo the intepetation of the liquid spectum, J. Chem. Phys. 4, 445 ().