Geometrical phase driven predissociation: Lifetimes of 2^2 A' levels of H_3
aa r X i v : . [ phy s i c s . a t o m - p h ] J a n Geometrical phase driven predissociation: Lifetimes of 2 A ′ levelsof H Juan Blandon and Viatcheslav Kokoouline
Department of Physics, University of Central Florida, Orlando, Florida 32816, USA (Dated: October 26, 2018)
Abstract
We discuss the role of the geometrical phase in predissociation dynamics of vibrational statesnear a conical intersection of two electronic potential surfaces of a D h molecule. For quantitativedescription of the predissociation driven by the coupling near a conical intersection, we developeda method for calculating lifetimes and positions of vibrational predissociated states (Feshbachresonances) for X molecule. The method takes into account the two coupled three-body potentialenergy surfaces, which are degenerate at the intersection. As an example, we apply the method toobtain lifetimes and positions of resonances of predissociated vibrational levels of the 2 A ′ electronicstate of the H molecule. The three-body recombination rate coefficient for the H+H+H → H +Hprocess is estimated. PACS numbers: ( v ′ , j ′ ) → H+H ( v, j ) scattering process (and its isotopic variants) that has been extensively studiedin theory and experiment [2, 3]. The related processes are the collisional dissociation of H and its inverse, the three-body recombination of hydrogen: H+H+H → H ( v, j )+H, which isresponsible, for example, for the formation of the first generation of stars [4]. The processesare governed by the two lowest molecular potential surfaces 1 A ′ and 2 A ′ of H (see Fig. 1).The lowest 1 A ′ potential is repulsive and leads to the H +H dissociation. The dissociationlimit of the 2 A ′ potential correlates with the H+H+H breakup and has a number of quasi-bound vibrational levels, which are predissociated towards to the H +H dissociation due tothe coupling near the CI. The predissociation of 2 A ′ vibrational states has been studied byKupperman and collaborators [5] using a time-independent scattering framework, the time-delay analysis, and a combination of Jacobi and hyperspherical coordinates. The calculationhas been done with the two-channel diabatic potential of H with non-diagonal diabaticcouplings obtained directly from first derivatives of ab initio BOA electronic wave functionsof the two interacting states. In another study by Mahapatra and K¨oppel [6], the time-dependent approach was employed by propagating a wave packet that is initially placedon the 2 A ′ PES; the lifetimes are then derived from the autocorrelation function. In Ref.[6], the authors also used a diabatic representation of the coupled H potential, but thediabatization is done differently than in Ref. [5] and does not require explicit calculationof first derivatives of the BOA electronic wave functions. The lifetimes obtained in the twostudies are significantly different.In this study, we suggest a general theoretical method to describe the nuclear dynamicsinvolving two molecular potentials coupled by a non-adiabatic interaction near the CI. As anexample, we calculate lifetimes of several 2 A ′ predissociated vibrational levels of H . Themethod can be used for other small polyatomic molecules where the CI plays an important2 φ FIG. 1: (Color online) The two lowest ab-initio potential energy surfaces of H shown as functionsof hyperangles 0 ≤ θ ≤ π/ ≤ φ ≤ π for a fixed hyper-radius ρ = 2 . a . The projection atbottom of the figure corresponds to 1 A ′ PES. role. There are two main ingredients in the proposed method: (1) The diabatization proce-dure is made in a way that accounts for the Jahn-Teller coupling between the 1 A ′ and 2 A ′ molecular states and the D h symmetry of the total vibronic wave function. (2) Nuclear dy-namics is described by Smith-Whitten hyperspherical coordinates [7], adiabatic separationsof hyperangles and hyper-radius along with the slow-variable discretization (SVD) [8, 9],and a complex absorbing potential (CAP) to obtain resonance lifetimes. Treatment of nuclear dynamics.
To treat the vibrational dynamics in hyper-spherical coordinates ρ, θ, φ , we represent the vibrational wave function ψ ( ρ, θ, φ ) as the3xpansion ψ ( ρ, θ, φ ) = P k y k ( ρ, θ, φ ) c k in the basis of non-orthogonal basis functions y k = π j ( ρ ) ϕ a,j ( θ, φ ). In this expression, ϕ a,j ( θ, φ ) is the hyperspherical adiabatic (HSA)state calculated at a fixed hyper-radius ρ j , H ρ j ϕ a,j ( θ, φ ) = U a ( ρ j ) ϕ a,j ( θ, φ ) , (1)with the corresponding eigenvalue U a ( ρ j ). H ρ i is the two-channel vibrational Hamiltonianwith the hyper-radius fixed at ρ j . If ρ j changes continuously, energies U a ( ρ j ) and the wavefunctions ϕ a,j ( θ, φ ) form the HSA curves U a ( ρ ) and channel functions ϕ a,ρ ( θ, φ ), respectively,(see Figs. 2, 3, 4). In the basis y k , the Schr¨odinger equation for the total wave function ψ ( ρ, θ, φ ) takes the form of a generalized eigenvalue problem for coefficients c k ≡ c ja X j ′ ,a ′ h h π j | − ~ µ d dρ | π j ′ iO ja,j ′ a ′ + h π j | ˆ U a ( ρ ) | π j ′ i δ aa ′ i c j ′ a ′ = E vibn X j ′ ,a ′ h π j | π j ′ iO ja,j ′ a ′ c j ′ a ′ . (2)with overlap matrix elements O ja,j ′ a ′ = h ϕ a,j ( θ, φ ) | ϕ a ′ j ′ ( θ, φ ) i , (3)that replace the familiar non-adiabatic couplings between the ϕ a,ρ ( θ, φ ) channels. This wayof representing the non-adiabatic hyperspherical couplings provides an important advantageover the familiar method of dealing with the couplings using the first and second derivativesof ϕ a,ρ ( θ, φ ) with respect to hyper-radius. In order to obtain lifetimes of predissociatedvibrational levels, we place a CAP at a large value of the hyper-radius (for details, see Ref.[9]). The total three-body rotational angular J momentum is 0 in the present calculation. Diabatic basis for the coupled H potential. We use the adiabatic ab initio A ′ and 2 A ′ PES from Ref. [10], which will be referred as V and V . The dependence of thepotentials on the two hyperangles is shown in Fig. 1. In our approach, we also use a diabaticrepresentation of the interaction potential. However, we derive the non-diagonal diabaticcoupling elements from the ab initio PESs without calculating them explicitly as derivativesof Born-Oppenheimer electronic states. This way of representing the non-adiabatic couplingshas been very successful in diatomic molecules: If the two ab initio
PESs have an isolatedavoided crossing, the vibrational dynamics is well represented by a 2 × Hyper-radius ( a o ) -0.2-0.100.10.20.3 E n e r gy ( E h ) A ’2 A ’ 3H(1s)H ( v,j )+ H(1s) FIG. 2: (Color online) Hyperspherical adiabatic potential curves, obtained from uncoupled 1 A ′ (black curves) and 2 A ′ (green curves) PESs of H . Different dissociation limits for the 1 A ′ familycorrespond to different v and j . Here, we only show the curves of the A irreducible representation. between the ab initio PESs at the avoided crossing. The diabatic non-diagonal coupling inour model is derived in the following way.The two 1 A ′ and 2 A ′ electronic states become degenerate at the equilateral configurationand should be referred to as two components of the E ′ irreducible representation of the D h symmetry group. It is convenient to use the basis functions | E + i and | E − i in the two-dimensional E ′ space [11, 12]. Although the electronic states 1 A ′ and 2 A ′ for clumpednuclei are classified according to the C s symmetry group, the vibronic states of H shouldbe classified according to the D h group. The most general form of the diabatic potential inthe basis of | E ± i for an arbitrary geometry isˆ V = A Ce if Ce − if A , (4)where A , C , and f are real-valued functions of the three hyperspherical coordinates [13].The diagonal elements A are the same because of the degeneracy of E ± states. The functions A and C transform in the D h symmetry group according to the A representation, and f has the following property under the C symmetry operator: C f = f + 2 π/
3. The A and5 functions are uniquely determined from the 1 A ′ and 2 A ′ PESs: A = ( V + V ) / C = ( V − V ) /
2. The actual form of function f can be derived near the CI: It is equal tothe phase of the asymmetric normal mode distortion [12, 13]. Although this form of f isderived near the CI, we will use it everywhere. This is justified because (1) the transitionbetween adiabatic states occur only near the CI, (2) far from the CI the phase factor e ± if does not play a role as long as the symmetry property mentionned above is satisfied.The vibrational wave functions ψ obtained from Eq. (2) have two components ψ ± corre-sponding to the two diabatic E ′± basis functions. In the adiabatic basis, corresponding tothe 1 A ′ and 2 A ′ electronic states, the two components ψ , of the ψ function have the form ψ = ( ψ − e if/ + ψ + e − if/ ) / √ ,ψ = i ( ψ − e if/ − ψ + e − if/ ) / √ . (5)After applying the C symmetry operator three times, the molecule returns back to itsoriginal position, however the components ψ , change sign ψ , → − ψ , because f → f +2 π .It is a well-known property of adiabatic states in the presence of CI, which is often referredas geometrical or Berry phase effect. Because the adiabatic electronic wave functions | i and | i also change sign, the total vibronic wave functionΨ = ψ | i + ψ | i = ψ + | E + i + ψ − | E − i (6)is unchanged after the identity operator C is applied. Results.
After solving the hyperangular part of the three-body Hamiltonian, Eq. (1), foreach BOA PES separately, we obtain two uncoupled ’families’ of adiabatic potentials U a ( ρ ),which are shown in Fig. 2. The curves belonging to the different families can cross. Whenthe coupling is turned on between the two electronic states, the crossings turn into avoidedcrossings, which is demonstrated in Fig. 3. The figure shows the HSA curves obtained bysolving Eq. (1) with the diabatic electronic potential of Eq. (4). Far from the (avoided)crossings the calculations with coupled and uncoupled electronic states produce almost thesame HSA states. The coupling changes the HSA states at (avoided) crossings only. It isalso worth to mention, that if the phase factors e ± if in Eq. (4) are neglected, the dynamicsdescribed by such diabatic potential is exactly the same as the dynamics with the uncoupledBOA PESs. It is because the transformation diagonalizing the operator ˆ V is independent ofthe nuclear coordinates if f ≡
0. Fig. 4 shows the vibrational HSA functions ϕ a,j for several6 Hyper-radius ( a o ) -0.04-0.020 E n e r gy ( E h ) (0,0 )(1,0 )(2,0 ).:(0,2 ) FIG. 3: Close up look at avoided crossings in HSA curves when the PESs are represented by Eq.(4). The figure corresponds to the frame shown in Fig. 2, where HSA curves are calculated fromthe uncoupled ab initio
PESs. The horizontal dashed lines show the positions of predissociated2 A ′ levels. adiabatic states calculated with the coupled potential ˆ V of Eq. (4).As Figs. 2 and 3 show, the ground and first excited HSA potentials of the 2 A ′ familyhave minima and can have vibrational levels that are pre-dissociated due to the couplingwith the 1 A ′ state. Although each vibrational level has components from all the HSAcurves shown in Fig. 2 (or Fig. 3), only one component is dominant. Thus, the vibrationallevels can be characterized by (1) the dominant component a and by (2) the number v of quanta along the hyper-radius. In addition, each HSA curve in the 2 A ′ family can becharacterized by the number v of quanta in the hyper-angular space and the number l ofthose v quanta along the cyclic hyperangular coordinate φ . Therefore, each predissociatedlevel can be numbered with the triad { v , v l } similar to the normal mode notations for C v molecules.Below the H+H+H dissociation limit there are only two { v } and { v } series of A predissociated levels. Their positions and lifetimes are given in Table I. Lifetimes for the7 IG. 4: (Color online) Wave functions of HSA states ϕ a,j of the A irreducible representation asfunctions of the hyperangles θ and φ for ρ j = 2 . a and several different a . Each wave function hastwo components, ϕ + and ϕ − corresponding to the two channels of the potential ˆ V . The relationshipbetween the hyperangles and the three-body configurations they represent is mapped on Fig. 6 ofRef. [13]. two series are very different. The reason for the difference is that the avoided crossingsare significantly wider for the 0 curve than for the 2 curve (see Fig. 3). The table alsocompares the obtained results with two other studies [5, 6] of the 2 A ′ predissociated levels.In Ref. [5] the non-diagonal couplings in the diabatic basis are obtained from the accurate ab initio non-Born-Oppenheimer couplings between the 1 A ′ and 2 A ′ states, but not fromPESs as in this study. The agreement with our study for the lifetime and energy of the8 v , v l } E r , τ ; this work E r , τ ; Ref. [5] E r , τ ; Ref. [6] { , } − .
85, 13 . n.a. − . ∼ { , } − .
11, 13 . n.a. − . ∼ { , } − .
4, 14 . n.a. − .
32, n.a. { , } − .
8, 14 . n.a. − .
70, n.a. { , } − .
2, 16 . − . ∼ . − .
14, n.a. { , } − .
7, 18 . − . ∼ . − .
65, n.a. { , } − .
2, 130 . n.a. − . ∼ . E r (in units 10 − Eh) and lifetimes, τ (in fs) of pre-dissociated 2 A ′ vibrationallevels. Energies are relative to the H(1 s ) + H(1 s ) + H(1 s ) dissociation. { } resonance is very good as well as for the lifetime of the { } level. The agreement forthe energy of the { } level is not as good probably due to the somewhat special characterof the { } level: its hyper-radial wave function extends to relatively large values ∼ a of hyper-radius (and internuclear distances ∼ a ). In Ref. [6], the diabatization procedurewas based on PESs similarly as it is made in the present study, but with one importantdifference: the diabatic electronic states in [6] are non-equivalent. It means that the D h character of the vibronic wave functions in Ref. [6] is broken. The lifetimes obtained in Ref.[6] are significantly different from the present values and values of Ref. [5]. The disagreementis attributed to the choice of the diabatization procedure in [6] that does not respect thedegeneracy of the diabatic electronic wave functions.It is worthwhile to note the relatively small number N ρ = 96 of hyper-radial points ρ j required to obtain converged lifetimes and positions of resonances considering the large num-ber of sharp avoided crossings (see Fig. 3). It is an advantage of using the SVD procedurerather than the familiar h ϕ a | ∂ /∂ρ | ϕ a ′ i and h ϕ a | ∂/∂ρ | ϕ a ′ i non-adiabatic couplings betweenthe HSA states: It is not necessary to calculate the derivatives on a fine grid of hyper-radiusnear avoided crossings to describe the couplings locally. Instead, in the SVD approach,the overlap matrix elements O ia,i ′ a ′ between the adiabatic states in Eq. (3) account glob-ally for the non-adiabatic hyperspherical couplings. It allows us to reduce significantly thecomputational task.The obtained lifetimes and positions of the resonances can be used to estimate the three-9ody recombination rate coefficient k for the H+H+H → H +H reaction. Using the proba-bility P ∼ / ( τ ω ) of the transition from 2 A ′ to 1 A ′ , where ω is the frequency of oscillationin the 2 A ′ potential, and using the formula for k from Ref. [14], taking sum over totalangular momentum J up to such J max that the lowest 2 A ′ vibrational level moves above theH+H+H dissociation, and also taking the sum over all three types of vibrational levels ( A , A , and E ), we obtain value k ∼ × − cm / s at 300 K with estimated error of about50%. This value is in good agreement with other estimation [4], k ∼ . × − cm / s at300 K, derived using the detailed balance principle from the rate of the H +H → H+H+Hprocesses. Rigorous calculations for k combining the developed model for the H potential,the SVD approach, and the three-body R-matrix approach are under way.Concluding, we would like to stress that we (1) suggested a model diabatic two-channelmolecular potential that represents correctly the interaction between vibronic states neara CI. The non-diagonal as well as diagonal matrix elements of the potential are extractedfrom the ab initio adiabatic PESs of the molecule. There is no need to use ab initio non-adiabatic couplings between PESs to construct the diabatic potential. (2) We applied themodel potential for calculation of lifetimes of 2 A ′ pre-dissociated levels in H . (3) Usingthe obtained resonances, we derived the rate coefficient for the three-body recombination ofthree hydrogen atoms. The developed techniques combining the model diabatic potentialand the numerical method can also be applied to other systems where a CI is expected toplay a role.We would like to thank the Donors of the American Chemical Society Petroleum ResearchFund, the National Science Foundation under Grant No. PHY-0427460 for an allocation ofNCSA and NERSC supercomputing resources (project [1] D. R. Yarkony, Rev. Mod. Phys. , 985 (1996).[2] D. Kliner, K. Rinnen, and R. Zare, Chem. Phys. Lett. , 107 (1990).[3] D. Neuhauser, R. Judson, D. Kouri, D. Adelman, N. Shafer, D. Kliner, and R. Zare, Science , 519 (1992).[4] D. R. Flower and G. J. Harris, Mon. Not. R. Astron. Soc. , 705 (2007).
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