Fragmentation of molecules by virtual photons from remote neighbors
aa r X i v : . [ phy s i c s . a t m - c l u s ] J u l Fragmentation of molecules by virtual photons from remote neighbors
Lorenz S. Cederbaum ∗ Theoretische Chemie, Physikalisch-Chemisches Institut, Universit¨at Heidelberg,Im Neuenheimer Feld 229, Heidelberg D-69120, Germany (Dated: July 17, 2020)It is shown that a molecule can dissociate by the energy transferred from a remote neighbor. Thisneighbor can be an excited neutral or ionic atom or molecule. If it is an atom, the transferred energyis, of course, electronic and in the case of molecules it can also be vibrational. Explicit examplesare given which demonstrate that the transfer can be highly efficient at distances where there is nobonding between the transmitter and the dissociating molecule.
The fragmentation of a molecule through the absorp-tion of light is called photodissociation. Being a funda-mental process in nature, photodissociation attracted avast amount of research, see, for instance, [1–7] and ref-erences therein. As bonds are broken in this process, itsimportance for chemistry, molecular science and astro-physics cannot be denied. As a consequence, nowadays,photodissociation is found to be important for model-ing the chemistry of nearly every type of astrophysicalregion, see [7] and references therein.In general, the molecule undergoing photodissociationis not isolated, and we may ask what is the impact ofa neighbor on this process. We concentrate here on thecase where the molecule and its neighbor are well sep-arated and do not posses a chemical bond such that ifthe photon is absorbed by the molecule, the photodis-sociation process is only little affected. We shall show,however, that if the impinging photon is absorbed by theneighbor, the molecule can still undergo fragmentation.Let us consider the following scenario from the pointof view of the neighbor. The neighbor is excited or ion-ized either by an impinging photon or by the impact ofanother particle, like an electron or ion, and now pos-sesses excess energy. If this excess energy is smaller thanthe ionization potential of the molecule, but larger thanits dissociation energy, which is typically a rather largerange of energy [1–3, 7], then the neighbor can relax andtransfer its excess energy to the molecule which will dis-sociate.The rate of this relaxation process is determined bythe golden rule Γ = 2 π X f |h Ψ i | V | Ψ f i| , (1)where V is the interaction between the molecule and itsneighbor. The wavefunctions Ψ i and Ψ f describe as usualthe initial and final states of the process in the absence ofthis interaction. The initial state is given by the productΨ i = φ Ni φ Mi and the final state by Ψ f = φ Nf φ Mf , where N stands for the neighbor and M for the molecule. Ini-tially, the molecule is in its electronic ground state andvibrationally in any state of interest (usually the groundstate) φ Mi and the neighbor is an excited or ionized state φ Ni as discussed above. After the process, the neighboris in an energetically lower state, usually its ground state(neutral or ionic) φ Nf and the molecule in the energy nor-malized continuum state φ Mf describing the fragmentedmolecule. The sum over the final states also includespossible different states of the fragments.To allow not only for electronic, but also for vibrationalto vibrational and vibrational to electronic and vice versa energy transfer, the interaction V contains the Coulombinteraction among all charged particles, electrons and nu-clei. Let the electronic and nuclear coordinates of theneighbor relative to its center of mass be r i and R k andthose of the molecule relative to its center of mass be r ′ j and R ′ l . Expanding the interaction V in inverse pow-ers of the distance between the two centers of mass R provides the leading contributing term [8] : − u · ˆD N )( u · ˆD M ) + ˆD N · ˆD M R + O ( 1 R ) , (2)where u is the unit vector connecting the two centers ofmass, and ˆD N = − X i r i + X k Z k R k ˆD M = − X j r ′ j + X l Z ′ l R ′ l (3)are the dipole operators of the neighbor and the moleculeincluding all charged particles. The Z indicate nuclearcharges. RESULTS
We now return to the golden rule (1). It is straightfor-ward to express its matrix element needed for the goldenrule. Averaging over the orientations of the molecule andits neighbor leads to Γ = π R P f ′ | D Ni,f | | D Mi,f ′ | , where D Ni,f = h φ Ni | ˆD N | φ Nf i and similarly for the molecule.The dipole matrix elements entering the expression forthe rate are closely related to measurable quantities andcan be conveniently replaced by them. The Einstein co-efficient, i.e., the inverse of the radiative lifetime, A Ni,f ofthe energy E i,f releasing transition of the neighbor reads[9] A Ni,f = 4 E i,f h c | D Ni,f | , (4)where c is the speed of light, and of central importanceto this work, the molecular dipole matrix element deter-mines the photodissociation cross section of the molecule[2]: σ M PD ( E i,f ) = 4 π h E i,f c X f ′ | D Mi,f ′ | . (5)We can view the process discussed above as follows.The neighbor possessing excess energy can relax by emit-ting a virtual photon which dissociates the molecules. Weremind that the excess energy itself can be deposited by aphoton or by the impact with another particle. At largedistances R , the relaxation rate takes on the appearanceΓ V PD = 3¯ h π (cid:16) cE (cid:17) A N σ M PD R , (6)where indices have been removed for simplicity and thesubscript V PD has been added for later purpose. The rateis faster the faster is the radiative decay of the neigh-bor and the larger is the molecule’s photodissociationcross section. The excess energy and the distance to themolecule influence the rate sensitively. The lifetime ofthe initial state of the neighbor due to the described re-laxation process is τ V PD = ¯ h/ Γ V PD . Of course, one isinterested in cases where this lifetime is faster than thatwithout the presence of the molecule. We shall see thatthis applies in many situations. A schematic picture ofthe process is shown in Figure 1. We would like to callthe process virtual photon dissociation .Above, we illuminated the process at hand from thepoint of view of the neighbor. The molecule under-goes dissociation by the virtual photon emitted fromthe neighbor and we may ask how large is the corre-sponding cross section. For this purpose we have toinclude the excitation process of the neighbor and con-sider the following scenario: The neighbor, for simplic-ity an atom, is excited by a photon of momentum k ph and the formed resonance state can decay either by dis-sociating the molecule or radiatively, the partial widthsbeing Γ V PD and Γ ph = ¯ hA N , respectively. The virtualphotodissociation cross section can be cast in the form[10, 11]: σ M V PD = πk ph g d g i Γ V PD Γ ph ( E ph − E ) + Γ / , (7)where E ph = ¯ hk ph c is the energy of the absorbed photon, E is the excess energy introduced above, and g d and g i FIG. 1. Schematic picture of the dissociation of the moleculeafter excitation of the neighbor. In the upper panel, the neigh-bor is electronically excited. The excitation can be by light,but also by other means, like the impact of an electron orion. If the neighbor is a molecule, the excitation can also bevibrational, see example in the text. The lower panel showsthe decay of the neighbor by the emission of a virtual photonwhich dissociates the molecule. The figure is by courtesy ofTill Jahnke. are the weights of the decaying and initial states. Asusual, the total width of the resonance Γ = Γ
V PD + Γ ph enters the above Breit-Wigner form.To asses the relative size of the virtual and usual crosssections, we make use of the finding that Γ V PD in Eq.(6) contains the latter. The virtual cross section peaksat E ph = E , and at that energy we can convenientlyexpress Eq. (7) to give the desired quotient of both crosssections: σ M V PD σ M PD = 3(2 π ) g d g i (cid:18) Γ ph Γ ph + Γ V PD (cid:19) (cid:18) λ ph R (cid:19) , (8)where λ ph = 2 π/k ph is the wavelength of the photon.This appealing expression has interesting limits. In par-ticular, if Γ ph ≫ Γ V PD , the above quotient is largest anddetermined solely by the geometric factor ( λ ph /R ) . Tobetter understand this, at first sight counterintuitive re-sult, one must keep in mind that the probability for thephoton to be absorbed by the neighbor is determined byΓ ph and without this absorption the virtual photon pro-cess does not take place.There are several mechanisms of photodissociation andsizable cross sections are obtained for photon energiesreaching excited electronic states of the molecule [1–7].The cross section for excitation of a vibrational level ofthe electronic ground state to the dissociation continuumof that state are usually vanishingly small. Can the pres-ence of a neighbor enhance the cross section of the lattercase substantially? Let us discuss an explicit examplewhere all the data needed for the calculation is availablein the literature. There has been much experimental andtheoretical interest in HeH + which is the simplest het-ereonuclear two-electron system made of the two mostabundant elements in the universe, see [12, 13] and ref-erences therein. The cross section σ M PD for the removal ofa proton from the ground state at photon energies belowthe first excited electronic state has been computed andfound to be somewhat smaller than 10 − Mb at the dis-sociation threshold of 1.844 eV, a tiny quantity indeed[13]. As a neighbor we choose a Li atom whose 2s → E = 1 .
85 eV and has an Einstein coefficientof A N = 3 . × s − [14].Using Eq. (6), one readily obtains Γ V PD = 6 . × − cm − at R = 1 nm. Since the radiative widthΓ ph = ¯ hA N = 1 . × − cm − is well larger, the quo-tient of the two cross sections in Eq. (8) is determinedby the geometric factor and takes on the very large value σ M V PD /σ M PD = 4 . × . This leads to σ M V PD of about10 Mb ! Even at the large distance R = 100 ˚A betweenHeH + and Li, the cross section due to the virtual photondissociation is still about 1 Mb. Of course, the enormousenhancement only persists in a narrow Breit-Wigner peakaround E ph = 1 .
85 eV.The dissociation of systems containing rare gases hasbeen widely studied. Examples are rare gas dimer andtrimer ions, rare gas complexes with halogen moleculesand with aromatics, see, e.g., [15–17] and referencestherein. The available photodissociation investigationsessentially relate to excited electronic states because thecross sections at photon energies below those states areusually too small to be measured. The binding of raregas atoms in neutral systems is typically weak or evenvery weak. For instance, the binding energy of the NeArdimer is just 40 cm − [18, 19], that of Ar to the aromatic1-naphtol is 474 cm − [17] and it is even small, 637 cm − [20] in the NeAr + ion. How to substantially enhance dis-sociation in the electronic ground state of these systems?Due to the low binding energy, electronic excitations ofthe neighbor are not suitable. We can, however, make useof the fact that Eq. (6) and hence also Eq. (8) are notonly valid for electronic, but also for vibrational energytransfer [8]. In other words, we can use these equationsto describe the energy transfer from a vibrational levelof the neighbor to photoionize the molecule via a virtualphoton. If, for example, we take HCN as a neighbor, itsbending frequency is 712 cm − [21] sufficing to dissoci-ate even the NeAr + ion. Although the radiative width8 × − cm − [21] is small, Γ V PD which also containsthis term as well as the tiny σ M PD , see Eq. (6), can also besmall. There is no data on σ M PD available, but assumingthat it is similar to the value 10 − Mb of HeH + discussedabove, we obtain Γ V PD = 4 . × − cm − at R = 1 nm.According to (8), we now get σ M V PD /σ M PD = 1 . × .An enormous value indeed. After having seen that the typically very small disso-ciation in the electronic ground state can be enhanceddramatically by the presence of a suitable neighbor, wenow consider standard molecules at energies where ph-todissociation is substantial. Our examples are the ni-trogen (N ), water (H O) and methane (CH ) moleculeswhich, being common molecules of interest, have beenmuch studied, see, e.g., [7, 22–30] and references therein.As neighbors we choose rare gas atoms, which are of in-terest by themselves, often serve experiments as matricesto trap and investigate molecules [31], and, importantly,the required data to evaluate Eq. (6) is available forthem.Photodissociation data on small molecules are benefi-cially compiled in [7] and is used here. The photodissci-ation spectrum of CH is continuous from below 140 nmphoton wavelength to above the ionization potential at98 nm (12.65 eV). The low excitation energies of Ar,Kr and Xe fall into this range. The wavelength of the3s ( P / )4s → transition in Ar is 104.82 nmand the Einstein coefficient is 5 . × s − [14]. At thiswavelength σ M PD = 30 Mb. With the aid of Eq. (6) wereadily find Γ V PD = 0 .
16 cm − at a distance of 1 nm be-tween Ar and the center of mass of methane. This impliesthat the lifetime of the isolated excited Ar decreases bymore than two orders of magnitude from τ ph = 1 . τ V PD = 33 ps. If we assume that the Ar was excited bya photon, the dissociation via the virtual photon exceedsthat of σ M PD by five orders of magnitude. The findingswith Kr and Xe as neighbors are similar.H O possesses a continuous photodissociation spec-trum from below 180 nm to above its ionization thresholdat 79.6 nm (15.58 eV) which below 125 nm is accom-panied by many peaks. The 4s ( P / )5s → transition of Kr and 5s ( P / )6s → of Xe havesimilar energies, 128.58 and 129.56 nm, and also similarEinstein coefficients, 3 . × s − and 2 . × s − ,respectively [14]. At these wavelengths σ M PD is approxi-mately 8 Mb. With Xe (Kr) as a neighbor at a distanceof 1 nm we thus obtain Γ V PD = 0 .
046 cm − (0.05 cm − )which corresponds to a lifetime τ V PD = 116 ps (100 ps)due to the virtual photon dissociation.For CH and H O we see that the dissociation by thevirtual photon emitted by the neighbor is efficient at theexcess energy deposited in the neighbor. In our last ex-ample we follow a different scenario. Here, we wish toshow that one can avoid having an enhancement of dis-sociation at a specific incoming photon energy only. Acartoon describing the scenario is shown in Figure 2. Tobe specific, we consider N with Ar as a neighbor. Thethreshold for 3s ionization of Ar is 29.24 eV [32] and thusany photon of wavelength larger than 42.4 nm may pro-duce an Ar ∗ + ion in the state 3s which has an excessenergy of 13.48 eV [14], well below the ionization poten-tial (15.65 eV) of N . The photodissociation spectrumof N is dense between about 95 and 85 nm with manyintense peaks rising up to 8 × Mb and at the transi-tion 3s → ( P / ), i.e., at 91.976 nm, the crosssection is larger than 150 Mb. The Einstein coefficientfor this transition is 1 . × s − [14]. FIG. 2. Schematic picture of the dissociation of the moleculeafter ionization of the neighbor. In the upper panel, the neigh-bor is ionized. The ionization can be by light, but also byother means, like the impact of an electron or ion. The lowerpanel shows the decay of the neighbor by the emission of a vir-tual photon which dissociates the molecule. If the ionizationis by light, note that the virtual photon dissociation processcontinuously takes place as a function of photon energy oncethe frequency of the light is above the required ionization en-ergy. The figure is by courtesy of Till Jahnke.
With the above data we can employ Eq. (6) and ob-tain Γ
V PD = 0 .
122 cm − at a distance of 1 nm betweenAr and the center of mass of N implying that insteadof decaying radiatively in τ ph = 7 . ∗ + decays much faster in τ V PD = 43 . . At R = 5 ˚A, whichis still a rather large distance where no bonding betweenN and Ar occurs, τ V PD reduces further and becomes subps (0.68 ps).The fast relaxation of the Ar ∗ + suggests an experi-ment. Ar-N is a well studied cluster with equilibriumdistance 3.77 ˚A, see [33] and references therein. By us-ing modern techniques, one can measure in coincidencethe momenta of the charged particles, see, e.g., [34, 35].In the present case, this implies that one can measurethe photoelectrons identifying the creation of Ar ∗ + witha hole in 3s in coincidence with the momentum of therelaxed Ar + in its ground state 3s . The dissocia-tion of N should be reflected in this momentum dis-tribution and can be well distinguished from the relax-ation of Ar ∗ + -N by photon emission forming Ar + -N .The possibility of measuring in coincidence photons, ions and electrons in decay processes has been demonstrated[36, 37]. So, at least in principle, one could also measureadditionally in coincidence the possible photons emittedfrom the atomic nitrogen fragments formed by the vir-tual photodissociation. For completeness, we just men-tion that there are several methods of N-atom productdetection by fluorescence and other means, see [38–40]and references therein. DISCUSSION
We have seen that fragmentation of molecules by vir-tual photons emitted from remote neighbors can be ef-ficient and fast. The involved energy transferred can beelectronic, but it can also be vibrational. Two scenar-ios have been discussed: excitation or ionization of theneighbor. If the excess energy is deposited by a photon,the enhancement of the photodissociation cross sectioncan be dramatic at the exciting photon energy. In thecase of ionization, the dependence on the ionizing pho-ton energy is continuous. The exciting/ionizing particledoes not have to be a photon, it can be an electron or anion.We notice that there is an analogy to another process.Interatomic Coulombic decay (ICD) is an efficient de-cay channel in excited/ionized systems, such as van derWaals and hydrogen bonded clusters and solutions. Inthe ICD process the de-excitation of a excited/ionizedatom or molecule via energy transfer to the environmentcauses ionization of the environment through long rangeelectronic correlation. Since its prediction [41], ICDhas been widely studied (see, e.g., [42] and referencestherein), found to be ultrafast (typically on the fem-tosecond timescale), and in most cases to be fast enoughto quench concurrent electronic and nuclear mechanisms[43–46]. Although ICD can be purely electronic, e.g., beoperative between atoms, and there is no need for nuclearmotion, one can learn much from ICD on virtual photondissociation discussed here. In ICD, retardation tendsto enhance the decay rate [47, 48] and we may assumethat this is also the case here. Excitation of the neighborby a photon may influence considerably the photoioniza-tion cross section of another atom [49] and we have seenthat this also applies here for the photodissociation ofthe molecule. ICD becomes particularly fast when theintermolecular (interatomic) distances are small and thenumber of involved species is large [50–53]. Similarly, ifseveral molecules which can be dissociated at the wave-length at hand are available, the excited/ionized neighbordecays faster as the number of decay channels grows. Ifthere are more neighbors, the probability for the imping-ing particle to excite/ionize one of them, trivially growsand hence also that to dissociate the molecule. We con-clude by stressing that virtual photon dissociation andICD complement each other. If the excess energy sufficesto ionize the environment, ICD can take place, and if not,virtual photon dissociation can be operative. ∗ E-mail: [email protected][1] R. Schinke,
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The author thanks K. Gokhberg, T. Jahnke, A. I.Kuleff and A. Saenz for valuable contributions. Finan- cial support by the European Research Council (ERC)(Advanced Investigator Grant No. 692657) is gratefullyacknowledged