Photofragmentation of the H molecule, including Jahn-Teller 3 coupling effects 4 Viatcheslav Kokoouline and Chris H. Greene 0 ∗ 0 Department of Physics and JILA, University of Colorado, 2 n Boulder, Colorado 80309-0440, USA a J (Dated: February 2, 2008) 6 ] Abstract h p We have developed a theoretical method for interpretation of photoionization experiments with - m theH molecule. Inthepresentstudywegiveadetailed descriptionofthemethod,whichcombines o 3 t a multichannel quantum defect theory, the adiabatic hyperspherical approach, and the techniques of . s c outgoing Siegert pseudostates. The present method accounts for vibrational and rotation excita- i s y tionsofthemolecule,dealswithallsymmetryrestrictionsimposedbythegeometryofthemolecule, h p includingvibrational, rotational, electronic andnuclear spinsymmetries. Themethodwas recently [ 1 applied to treat dissociative recombination of the H+3 ion. Since H+3 dissociative recombination has v 4 been a controversial problem, the present study also allows us to test the method on the process 1 0 of photoionization, which is understood better than dissociative recombination. Good agreement 1 0 with two photoionization experiments is obtained. 4 0 / s PACS numbers: 33.80.Eh,33.80.-b, 33.20.Wr, 33.20.Vq c i s y h p : v i X r a ∗ Present address: Department of Physics, University of Central Florida, Orlando, FL 32816-2385,USA 1 I. INTRODUCTION The simplest polyatomic molecules, H and H+, have been intensively studied during the 3 3 last decades. Interest in these molecules is motivated by the fact that the H+ ion plays an 3 important role in the chain of chemical reactions in interstellar space, acting as protonator in chemical reactions with almost all atoms. In particular, dissociation recombination (DR) of H+ with an electron leads via several intermediate steps to the production of water in 3 interstellar space. Many successful models in interstellar chemistry are based on H+ DR. 3 In addition, H+ attracts theorists as a benchmark for high accuracy calculations with small 3 molecules: Theoretical ab initio methods can be tested against existing experimental H+ 3 spectroscopydata. TheinterestintheneutralH moleculeiscloselyrelatedtotheproblemof 3 H+ DR.ButH alsopresents greatinterest fromanother pointofview. Experimental studies 3 3 [1, 2, 3, 4, 5] of the metastable H molecule and several later theoretical studies [4, 5, 6, 7] 3 revealed non-Born-Oppenheimer effects of coupling between its electronic, vibrational, and rotational degrees of freedom. As was shown recently [8, 9], these non-Born-Oppenheimer effectsplayanimportantroleinH+ DRaswell. Thepresent studyisdevotedtoatheoretical 3 treatment of H photoionization. 3 There are three main reasons for this study. The first one is to explore a new theo- retical method for the treatment of polyatomic photoionization. Our method is based on multi-channel quantum defect theory (MQDT) [10, 11, 12, 13], the adiabatic hyperspherical approach to vibrational dynamics of three nuclei, the formalism of outgoing wave Siegert states [14, 15, 16], and inclusion of a non-Born-Oppenheimer coupling—Jahn-Teller effect. The second reason is related to a recent study of H+ DR [8, 9], where the reported method 3 was very successful in treatment of H+ DR, giving good agreement between theoretical 3 calculations and experimental results from storage rings [17, 18, 19]. However, since that method is new, it is desirable to test it in greater detail. An application to the interpretation of H photoionization experiments [4, 5] is such a test. These photoionization experiments 3 were successfully interpreted in previous theoretical work [4, 5, 6, 7], where another method based on MQDT was applied. Thus, the present treatment can also be tested against the previous theoretical studies. Our treatment of photoionization is similar to the one developed by Stephens and Greene [6, 7], and employed in Refs. [5, 6, 7] for interpretation of two photoionization experiments 2 by Bordas et al. [4] and by Mistr´ık et al. [5]. Both experiments were interpreted using a full rovibronic frame transformation [6, 7]. The present treatment has several differences from the one proposed by Stephens and Greene. The first difference is the use of the adiabatic hyperspherical approximation [20, 21, 22] for the representation of vibrational wave functions. Stephens and Greene used the exact three-dimensional vibrational wave functions. The second difference is the correction of the incompatibility between the form for the reaction matrices used in Refs. [5, 6, 7, 23] and the quantum defect parameters of Jahn-Teller coupling used in the mentioned studies. In fact, the values of Jahn-Teller quantum defect parameters used in Refs. [5, 6, 7, 23] are compatible with an alternative form of the reaction matrix, which was adopted in Refs. [24, 25, 26]. In the present work we use the same form of K-matrix as in Refs. [5, 6, 7, 23] and quantum defect parameters from Ref. [5]. Thus, Jahn-Teller parameters δ and λ from [5] should be multiplied by π − to be used in the present study. The third difference is in the symmetrization of the total rovibrational wave functions of the H+ ion. In Refs. [5, 6, 7] the symmetrization is made 3 according to the procedure proposed by Spirko and Jensen [27]: Rotational and vibrational parts of the total wave function are symmetrized separately and in two-step procedure. In the present treatment we symmetrize the total wave function only once at the very final step. This greatly simplifies the construction of wave functions of a required symmetry. The fourth difference is in calculation of dipole transition moments. Calculating the dipole moment into a final state, Stephens and Greene accounted only for the diagonal component of the final state wave function. Our treatment accounts for all non-diagonal wave function components contributing to the dipole transition element. The article is organized as follows. Section II describes construction of the total wave function of H+ and compares our method of the construction with the method proposed in 3 Ref. [27]. In Sec. III, we build up the scattering matrix that represents the collision between an electron and an ion. Section IV presents a derivation of dipole transition moments and oscillator strengths for H . We discuss results of our calculation and compare those results 3 with experimental data in Sec. V. Section VI states our conclusions. Atomic units are used in the article unless otherwise stated. 3 II. SYMMETRY OF THE TOTAL WAVE FUNCTION OF H+ 3 In this study we consider only p-wave scattering (or half-scattering) of the electron from the molecule. As demonstrated in Refs. [5, 7], higher electronic partial waves make much smaller contribution than the p-wave to the photoionization spectrum. Similar to our study of H+ dissociative recombination, we chose the molecular axis Z along the main symmetry 3 axis of the molecule. Directions of two other axes, X and Y, are shown in Fig. 3 of Ref. [9]. A. Total wave function The total wave function Φn.sym of the ion can be represented as a sum of terms, each of t which is product of three factors [9]: Φnt.sym = ΦIgIRN+K+m+(α,β,γ)Φv(Q). (1) In the above equation, α,β, and γ are three Euler angles defining the orientation of the molecular fixed axis with respect to the space fixed coordinates system. Below, we describe briefly the construction of all three factors in the product of Eq. (1). A more detailed description is given in Ref. [9]. The rotational part (α,β,γ) of the total wave function in Eq. (1) is the symmetric R top wave function for H+, which is proportional to the Wigner function [28]. The quantum 3 numbers N+, K+, and m+ refer to the total angular momentum N+ and its projections on the molecular Z-axis, K+, and the laboratory z-axis, m+. The transformation properties of the symmetric top wave function under the D group, are given in Table II of Ref. [9]. 3h The vibrational symmetry of H+ and H is described by the group C . C is a subgroup 3 3 3v 3v of D : D = σ C , where σ is the operation of reflection with respect to the plane 3h 3h h 3v h ⊗ of three nuclei. For our discussion of the vibrational symmetry of H+, it is convenient to 3 use normal coordinates Q , Q , and Q (for definitions, see, for example, Ref. [5]). Q 1 x y 1 describes the symmetric stretch mode. The motion along this coordinate is characterized by the (approximate) quantum number v and by the corresponding frequency ω . Normal 1 1 coordinates Q and Q correspond to two vibrational modes having the same frequency of x y oscillations ω . Vibrations along Q and Q are characterized by the approximate numbers 2 x y v and v , correspondingly. The total vibrational energy can be approximated E = ω (v + x y 1 1 1/2)+ω (v +v +1). (Thevibrationalquantumnumbers v ,v andv andthecorresponding 2 x y 1 x y 4 energy E arenotexact aslongastheionicmolecular potentialis notexactly harmonic.) Due to the degeneracy of the Q and Q modes, the two-dimensional vibrational motion along x y the Q and Q coordinates can be equivalently represented in polar vibrational coordinates x y ρ and φ. Then, instead of quantum numbers v and v , it is convenient to define v = v +v x y 2 x y andl , wherel isassociatedwiththemotionalongφcoordinate: vibrationalangularmotion 2 2 around the symmetry axis. Thus, the vibrational energy is determined only by the quantum numbers v and v : E = ω (v + 1/2) + ω (v + 1). The number v shows how many 1 2 1 1 2 2 2 vibrational quanta are in the asymmetric mode. The number l determines how many of the 2 asymmetric quanta v contribute to the vibrational angular momentum, v l v . In 2 2 2 2 − ≤ ≤ reality, due to the anharmonicity of potentials, the vibrational energy of states with same v 1 ˜ and v but different l are slightly different. However, pairs of states with l , where l = 3k 2 2 2 2 ± 6 ˜ (here and below, k is any integer number), are strictly degenerate. This is a consequence of the fact that the D symmetry group has doubly degenerate representations. Thus, 3h vibrational wave functions, Φv(Q) = |v1,v2l2i, (2) of the ion are specified by the triad of quantum numbers v ,vl2. The quantum number l can 1 2 2 have values v , v +2...v 2,v and it controls the symmetry of the vibrational wave 2 2 2 2 − − − ˜ ˜ functions. States with l = 3k, with an integer k = 0, can be of A or A symmetry. In order 2 1 2 6 to distinguish the two symmetries using the number l , we will label states A with positive 2 1 ˜ ˜ ˜ l , l = 3k, k 0 and A states with negative l . A pair of states with l = 3k having both 2 2 2 2 2 ≥ 6 signs of l constitutes the degenerate pair of functions, that transform according to the E 2 representation. In contrast to the numbers v ,vl2, the classifications with symmetry labels 1 2 A , A , or E are exact. 1 2 In the present treatment the relative phase of degenerate states v ,vl2 with l = 3k˜ is | 1 2 i 2 6 slightly different from the one used in Ref. [29]. Transformations of v ,vl2 with the present | 1 2 i choice of phases are summarized in Table III of Ref. [9]. The third factor of the total ionic wave function is the nuclear spin wave function. The nuclear spin states are classified according to the total spin I = 3/2 or I = 1/2. These states are constructed as described in Ref. [9]. The result is ΦI(I = 3/2), ΦI (I = 1/2), 0 1 − and ΦI (I = 1/2) wave functions, transformed according to the A E representation +1 1 ⊕ of the respective symmetry group S of three identical particle permutations. The state 3 ΦI(M = 3/2) transforms according to the A representation; the states ΦI (I = 1/2) and 0 1 1 − 5 ΦI (I = 1/2) transform according to the E representation. +1 The transformation of the total wave function Φn.sym is determined by the quantum t numbers K+,l , and g . The final step in the construction of the total wave function is an 2 I appropriatesymmetrizationofΦn.sym. Sincethetotalwavefunctionshouldbeantisymmetric t with respect to (12) we determine Φ : t 1 Φ = Φn.sym(K+,l ,g ) ( 1)N+s Φn.sym( K+,l , g ) , (3) t √2 t 2 I − − 2 t − 2′ − I (cid:16) (cid:17) In the above equation s = 1 for all vibrational states excluding A , for which s = 1; 2 2 2 − ˜ ˜ l = l if l = 3k 1, and l = l if l = 3k. The condition for antisymmetry with respect 2 − 2′ 2 ± 2 2′ 2 to (12) is specified explicitly. If the symmetrization is trivial, i.e. both terms in Eq. (3) are identical, then the wave function is Φ = Φn.sym(K+,l ,g ). (4) t t 2 I It is only possible if g = 0, K+ = 0 and l = l . I 2 2′ The fermionic nature of nuclei also requires that the wave function Φ should be anti- t symmetric with respect to operations of (13) and (23). It is only possible if Φ transforms t according to the A or A representations of the D (M) group. This condition can be ′2 ′2′ 3h written as G˜ = 3k˜, where G˜ = K+ + l + g . The determination of the total symme- 2 I try has one exception from the above rule. Namely, when the symmetrization is trivial: l = l , K+ = 0. For this case, the rovibrational part of the product (4) has A or A 2 − 2′ 1 2 symmetry, thus, g can only be 0. Finally, the overall parity of the total state, which is I determined as transformational under the operation of total inversion E , is determined by ∗ the number K+: The parity is even if K+ is even and the parity is odd if K+ is odd. In this study we are primarily interested in the ortho-modification of the H molecule of 3 A symmetry. Thus, Eq. (3) is reduced to ′2 Φ = 1 (α,β,γ) v ,vl2 ( 1)N+s (α,β,γ) v ,vl2′ (5) t √2 RN+K+m+ | 1 2 i− − 2|RN+−K+m+ | 1 2 i (cid:16) (cid:17) or, when K+ = 0 and l = l , to 2 2′ Φ = (α,β,γ) v ,vl2 . (6) t RN+K+m+ | 1 2 i Only states with G˜ = K+ + l = 3k˜ and with even K+ are allowed. Again there is an 2 exception, when the symmetrization is trivial: When K+ = 0 and N+ is even (rotational symmetry is A ), l = 3k˜ can only be negative (A vibrational symmetry); when K+ = 0 ′1 2 2 and N+ is odd, l = 3k˜ must be positive or zero. 2 6 B. H+ vibrational dynamics in an adiabatic hyperspherical approach 3 As in our study of H+ dissociative recombination [8, 9, 23], we employ the adiabatic 3 hyperspherical approach to describe the vibrational dynamics of H+ and H in three di- 3 3 mensions. In this approach, three hyperspherical coordinates, the hyper-radius R and two hyperangles θ, ϕ, represent three vibrational degrees of freedom [9, 20, 21, 22]. In our calculations we use accurate potential surfaces of H+ from Refs. [31, 32]. 3 The hyperspherical coordinates used are symmetry-adapted coordinates: Each operation of the C group–a permutation of instantaneous positions of the three nuclei–is described 3v with an appropriate change in the hyperangle ϕ only. With this respect, the hyperspherical coordinates are similar to the normal coordinates of H+ [5, 7, 9], where all operations from 3 C involve the polar angle φ uniquely. For example, the effect of the (123) operation is a 3v cyclicpermutationofthethreeinternuclear distances. Inthehyperspherical coordinates,itis realized by adding the angle 2π/3 to ϕ as determined in Eqs. (23) of Ref. [9]. The operation (12) exchanges the internuclear distances r and r . Equations (23) of Ref. [9] show that 2 3 this operation corresponds to a mirror reflection about the axis ϕ = [ π/2,π/2]. Only (12) − the angle ϕ is changed, into ϕ = (12)ϕ = π ϕ. This operation is exhibited in Fig. 1: The ′ − operation (12) exchanges nuclei 1 and 2, transforming the triangle T into T . The figure a b also shows all three symmetry axes, corresponding to three binary permutations (12), (23), and (13). These symmetry properties of the hyperspherical vibrational coordinates simplify our treatment appreciably. In the adiabatic hyperspherical method we first solve the vibrational Schr¨odinger equa- tion at a fixed hyper-radius R [9], obtaining a set of energies U and corresponding eigen- i functions Φ (θ,ϕ). Changing R, we obtain a set of adiabatic potential curves U (R) and i i adiabatic hyperspherical eigenstates Φ (θ,ϕ;R). As mentioned above, each element from i the C symmetry group is represented by a corresponding transformation involving only 3v the hyperangle ϕ. The hyper-radius is not involved in the C operations. Thus, the vi- 3v brational hyperspherical states Φ (θ,ϕ;R) and curves U (R) can be classified according to i i irreducible representations of the group C . Namely, each state Φ (θ,ϕ;R) and correspond- 3v i ing curve U (R) can be labeled by either the A , A , or E irreducible representation. The i 1 2 representation E is two-dimensional. Thus, two degenerate E-components Φ (θ,ϕ) will be i labeled by E and E . Their linear combinations are also good vibrational eigenstates. a b 7 Several low ionic hyperspherical curves U+(R) are shown in Fig. 2. We specify the i pair vl2 of quantum numbers for first several states. As in the familiar Born-Oppenheimer 2 approximation for diatomic molecules, curves of the same symmetry do not cross, whereas curves of different symmetries may cross. One can see from Fig. 2 that low-lying potential curves with the same v are almost degenerate. This is because the anharmonicity is quite 2 small for low states. However, atv = 4 the potential curves with quantum numbers vl2 = 44 2 2 and vl2 = 40,2 are already significantly separated. 2 Similar to Ref. [9], from real-valued E-states obtained after a diagonalization at fixed R, in the space of the two hyperangles θ and ϕ, we construct “helicity” [38] E states: 1 vl2 = (E +iE ); | 2 i √2 a b vl2′ = v l2 = 1 (E iE ). (7) | 2 i | 2− i √2 a − b The sign of l in the above equation is chosen in such a way that (123) transforms the state 2 v l2 as | 2± i (123)|v2±l2i = e±2π3il2|v2±l2i. (8) For example, if l = 2, then l = 2 and l = 2 (see discussion in Sec. IIC). Finally, | 2| 2 − 2′ we multiply all real-valued vibrational functions A by i in order to obtain a real reaction 2 matrix K. Once the adiabatic hyperspherical potential curves U+(R) are determined, we calculate i vibrational energies, E , by solving the adiabatic hyper-radial equation, i,v 1 ∂2 +U+(R) ψ+ (R) = E ψ+ (R), (9) −2µ∂R2 i i,v i,v i,v (cid:20) (cid:21) where v,i v ,vl2 . In solution of Eq. (9), we seek solutions (Siegert states) that { } ≡ { 1 2 } obey outgoing wave boundary conditions [14, 15] at a finite hyper-radius R and which are 0 normalized as in Ref. [9]. Inclusion of Siegert states into the treatment allows us to represent the dissociation of the neutral H formed during the collision between H+ and the incident e . 3 3 − C. Comparison with an alternative symmetrization procedure An alternative procedure for the symmetrization of the total wave function was proposed by Spirko et al. [27]. Spirko et al. describe how rovibrational wave functions of different 8 D representations, Γ = Aζ,Aζ,Eζ, and Eζ, are obtained from the products of rotational 3h 1 2 a b and vibrational parts. We use the symbol ζ to specify the parity of a state, where ζ = or ′ ζ = . (In Ref. [27], the symbol was used for this purpose.) In the approach of Ref. [27], ′′ † the rotational states with Γ = Aζ,Aζ,Eζ, or Eζ are obtained by combining the symmetric 1 2 a b top states N+,K+,m+ (Eqs. (59)-(61) of Ref. [27]). Then products of rotational and | i vibrational states are constructed. At the next step, these products are symmetrized again to give rovibrational states of goodsymmetry. This final step is quite laborious since one has toconsider allpossible combinations ofdifferent rotationalandvibrationalstates(Eqs. (62)- (77) of Ref. [27]). To properly include all states of different nuclear spin symmetries, one would have to construct nuclear spin states andthen use a similar symmetrization procedure (Eqs. (62)-(77) of Ref. [27]) one more time. In implementing the procedure of Spirko et al. for construction of the e +H+ scattering matrix, we have found that it is difficult − 3 to obtain all states of the right symmetry. When carried out incorrectly, our scattering matrix displayed non-zero matrix elements between states of different symmetries, which of cause signals an error. For this reason, we think that our symmetrization procedure is advantageous for the present study, where we typically include and symmetrize hundreds of states. Our procedure involves only two simple steps and can be easily automated on a computer. First, the products of rotational, vibrational, and nuclear spin non-symmetrized ˜ states are constructed. For each product, the number G [or the total symmetry for the states that are trivially symmetrized, such as Eq. (6)] is determined. It describes behavior of the product under the symmetry operations (12) and (123). Then the symmetrization procedure (if it is not trivial) is accomplished by a single equation [see Eq. (3)]. To facilitate the comparison with the procedure, proposed by Spirko et al. [27], we intro- duce a “helicity” pair of degenerate E states, which transform in a uniform way, irrespective of their nature: rotational, vibrational, or nuclear spin. We determine two degenerate states E and E by their symmetry properties + | i | −i (123) E = e±i23π E± , | ±i | i (12) E = E , (10) | ±i | ∓i irrespective of coordinates: rotational, vibrational, electronic or nuclear spin. Using Ta- ble II of Ref. [9], it can be easily verified that rotational Er,ζ states are obtained from | ± i 9 N+,K+,m+ according to | i Er,ζ = N+,K+,m+ , if K+ = 3k˜+1, | + i | i Er,ζ = ( 1)N+ N+, K+,m+ , if K+ = 3k˜+1, (11) | − i − | − i Er,ζ = N+, K+,m+ , if K+ = 3k˜+2, | + i | − i Er,ζ = ( 1)N+ N+,K+,m+ , if K+ = 3k˜+2. | − i − | i In the above equation, k˜ is a non-negative integer. Vibrational Ev states are obtained | ±i using Eq. (8) as |Evi = |v1,v2±|l2|i, if |l2| = 3k˜+1, ± |Evi = |v1,v2∓|l2|i, if |l2| = 3k˜+2. (12) ± In addition to E states we introduce another pair of states E and E as a b | ±i | i | i 1 E = ( E i E ). (13) a b | ±i √2 | i± | i or 1 E = ( E + E ), a + | i √2 | i | −i 1 E = ( E E ). (14) b + | i i√2 | i−| −i The states E and E are real. Using Eqs. (10) and (14), we obtain that a b | i | i (12) E = E , a a | i | i (12) E = E . (15) b b | i −| i Using Eqs. (10), (13), and(14) andthefact thate±i23π = 1 i√3, we have forthe operation −2± 2 (123) 1 √3 (123) E = E E , a a b | i −2| i− 2 | i √3 1 (123) E = E E . (16) b a b | i 2 | i− 2| i At this stage, we can compare the present convention for E and E states with a b | i | i the one from Ref. [27]. Comparing formulas (15) and (16) with Eqs. (51)-(54) of Ref. [27], we conclude that the two conventions for E and E vibrational states coincide. a b | i | i 10