High-precision nonadiabatic calculations of dynamic polarizabilities and hyperpolarizabilities for the lowlying vibrational-rotational states of hydrogen molecular ions
aa r X i v : . [ phy s i c s . a t o m - ph ] J u l High-precision nonadiabatic calculations of dynamicpolarizabilities and hyperpolarizabilities for the lowlyingvibrational-rotational states of hydrogen molecular ions
Li-Yan Tang , , Zong-Chao Yan , , , Ting-Yun Shi , and James F. Babb State Key Laboratory of Magnetic Resonance and Atomicand Molecular Physics and Center for Cold Atom Physics,Wuhan Institute of Physics and Mathematics,Chinese Academy of Sciences, Wuhan 430071, P. R. China Department of Physics, University of New Brunswick,Fredericton, New Brunswick, Canada E3B 5A3 and ITAMP, Harvard-Smithsonian Center for Astrophysics,60 Garden Street, Cambridge, Massachusetts 02138, USA (Dated: October 9, 2018) bstract The static and dynamic electric multipolar polarizabilities and second hyperpolarizabilities of theH +2 , D +2 , and HD + molecular ions in the ground and first excited states are calculated nonrelativis-tically using explicitly correlated Hylleraas basis sets. The calculations are fully nonadiabatic; theBorn-Oppenheimer approximation is not used. Comparisons are made with published theoreticaland experimental results, where available. In our approach, no derivatives of energy functions norderivatives of response functions are needed. In particular, we make contact with earlier calcula-tions in the Born-Oppenheimer calculation where polarizabilities were decomposed into electronic,vibrational, and rotational contributions and where hyperpolarizabilities were determined fromderivatives of energy functions. We find that the static hyperpolarizability for the ground stateof HD + is seven orders of magnitude larger than the corresponding dipole polarizability. For thedipole polarizability of HD + in the first excited-state the high precision of the present method facil-itates treatment of a near cancellation between two terms. For applications to laser spectroscopy oftrapped ions we find tune-out and magic wavelengths for the HD + ion in a laser field. In addition,we also calculate the first few leading terms for long-range interactions of a hydrogen molecularion and a ground-state H, He, or Li atom. PACS numbers: 31.15.ac, 31.15.ap, 34.20.Cf . INTRODUCTION Polarizabilities and hyperpolarizabilities of molecules can describe linear and nonlinearoptical phenomena, such as light scattering from gases and solids and the Kerr effect, and dy-namic (or frequency-dependent) values are helpful in designing optical materials and in gaug-ing electric field responses for experiments. While calculations are challenging, there are nu-merous calculated results for many molecules—static and dynamic polarizabilities and hyper-polarizabilities are available properties in many mature quantum chemistry programs—yetactual fully nonadiabatic ab initio results (obtained without use of the Born-Oppenheimerpicture) are rare. In previous studies, it was demonstrated [1–4] that a theory based onthe explicitly correlated Hylleraas basis set expansion yielded high accuracy nonadiabaticproperties of three-body systems. In this paper, we extend the formalism contiguously tomultipolar dynamic electric polarizabilities and dynamic second hyperpolarizabilities of thehydrogen molecular ion and its deuterium containing isotopologues in the ground and firstexcited states. While the formalism presented here is purely nonrelativisitic, the nonadia-batic theory on which it is based is well-tested beyond order α Ry as progress in calculationsof energies of HD + , for example, are now at the level where the uncertainties in transitionfrequencies are of the order of 70 kHz, with unknown effects contributing at order α Ry [5],while refinement of nonadiabatic calculations on simple molecules continues using differentapproaches [6–9]. A comparison of nonrelativistic results for energies is given in Sec. III.The present calculations, we believe, are of great value for several potential applications.While our approach intrinsically includes rotational and vibrational degrees of freedom itdispenses with the Born-Oppenheimer approximation. Use of the Born-Oppenheimer ap-proximation facilitates the breakdown of polarizabilities and hyperpolarizabilities into “elec-tronic”, “vibrational”, and “rotational” components and the theoretical underpinnings ofthis picture are well-established, but there are different formulations and subtleties in exe-cuting such calculations [10–16]. We show how our results provide insight into these descrip-tions, allowing direct comparisons with earlier Born-Oppenheimer results, and in Sec. IIIwe use these insights, for example, to resolve a discrepancy found by Olivares Pil´on andBaye [17] in comparing nonadiabatic and Born-Oppenheimer calculations of the dynamic3lectric quadrupole polarizability. Our method avoids the cumbersome Born-Oppenheimerseparation, our tabulated nonadiabatic data can be valuable for estimations or extrapola-tions of “electronic”, “vibrational”, and “rotational” contributions, when combined withavailable Born-Oppenheimer calculations [15, 18]. In addition, our nonadiabatic approachdoes not require derivatives of an energy function [11, 19] nor derivatives of response func-tions [20], which can introduce additional numerical loss of precision, but it does providedefinitive convergence-based error bars thereby allowing us to gauge the accuracy of previousresults for hyperpolarizabilities calculated using gradients of fields.There is much recent interest in trapping molecular ions for precision measurements (oftime [21] and of mass [22], for example) and for realizing quantum computing [23]—in thesecases the responses of ions to applied fields are important considerations [24] and our cal-culations can serve as useful models or references for future studies. We find, for example,that the hyperpolarizabilities of H +2 and D +2 are much larger than the dipole polarizabilitiesby four orders of magnitude, which confirms [25] that the Stark shift of H +2 immersed athigh field strength would be influenced by the hyperpolarizability. For the ground stateof HD + , the sign of static dipole polarizability and hyperpolarizability are opposite, sug-gesting that the hyperpolarizability should be considered in experimental analyses, sincethe Stark shifts for this system would tend to cancel each other. In Sec. III we presenthighly accurate calculations of Stark shifts, tune out and magic wavelengths, and nonlineardynamic hyperpolarizabilities for HD + in the ground and excited states. Finally, the multi-polar polarizabilities that we compute enter as parameters in the long-range “polarizationpotential” [26–28], which are effective potential expansions, for the interactions of an elec-tron with the the molecular ion isotopologues. We also calculate the long-range dispersioninteractions between H, He, or Li and each of the H +2 isotopologues in their ground or firstexcited states.In this work, the 2006 CODATA masses [29] of the proton and the deuteron are adopted [ ? ], where m p = 1836 . m e , (1) m d = 3670 . m e , (2)4nd m e is the electron mass, and atomic units are used throughout unless specifically men-tioned. The polarizabilities and hyperpolarizabilities are presented in atomic units [15];conversion factors to SI units are given in, for example, the reviews by Bishop [13] andby Shelton and Rice [15]. In this nonrelativistic study we neglect finite temperature ef-fects [15, 31], hyperfine structure [21, 32], and we do not consider the first hyperpolarizability(which is only non zero for HD + ). II. THEORY AND METHODA. Hamiltonian and Hylleraas basis
In the present work, we treat the hydrogen molecular ion as a three-body Coulombicsystem; the calculations are fully nonadiabatic (the Born-Oppenheimer approximation isnot used). Taking one of the nuclei as particle 0, the electron is chosen as particle 1 andthe other nucleus is seen as particle 2. In the center of mass frame, the Hamiltonian can bewritten as H = − X i =1 µ i ∇ i − m X i>j ≥ ∇ i · ∇ j + q X i =1 q i r i + X i>j ≥ q i q j r ij , (3)where µ i = m i m / ( m i + m ) is the reduced mass between particle i and particle 0, q i is thecharge of the i th particle, r i is the position vector between particle i and particle 0, and r ij = | r i − r j | is the inter-particle separation.The wave functions are constructed in terms of the explicitly correlated Hylleraas coor-dinates as φ ijk ( r , r ) = r i r j r k e − αr − βr Y LMℓ ℓ (ˆ r , ˆ r ) , (4)where r j e − βr sufficiently represents the vibrational modes between the nuclei if j and β arechosen big enough [1], Y LMℓ ℓ (ˆ r , ˆ r ) is a vector-coupled product of spherical harmonics, Y LMℓ ℓ (ˆ r , ˆ r ) = X m ,m h ℓ m ; ℓ m | LM i Y ℓ m (ˆ r ) Y ℓ m (ˆ r ) , (5)and the nonlinear parameters α and β are optimized using Newton’s method. All terms inEq.(4) are included such that i + j + k ≤ Ω , (6)5here Ω is an integer, and the convergence for the energy eigenvalue is studied as Ω isincreased progressively. The computational details used in evaluating the necessary matrixelements of the Hamiltonian are given in Ref. [33]. B. Polarizability and Hyperpolarizability
When the hydrogen molecular ion is exposed to a weak external electric field E , thesecond-order Stark shift for the rovibronic state is∆ E = − E α ( ω ) + α ( T )1 ( ω ) g ( L, M )] , (7)where L is the angular momentum with magnetic quantum number M , g ( L, M ) is the only M -dependent part, g ( L, M ) = 3 M − L ( L + 1) L (2 L − , L ≥ , (8)and ω is the frequency of the external electric field in the z -direction. The dynamic scalarand tensor dipole polarizabilities, respectively, are α ( ω ) and α ( T )1 ( ω ); when ω = 0, they arecalled, respectively, the static scalar and tensor dipole polarizabilities. The derivation of theexpressions for the dynamic polarizabilities α ( ω ) and α ( T )1 ( ω ) are similar to those describedin Ref. [34]. In particular, for the case of rovibronic ground-state with L = 0, α ( ω ) = α ( P, ω ) , α ( T )1 ( ω ) = 0 , (9)with α ( L a , ω ) following the general expression of 2 ℓ -pole partial dynamic polarizabilities, α ℓ ( L a , ω ) = 8 π (2 ℓ + 1) (2 L + 1) X n ∆ E n |h n L k T ℓ k nL a i| ∆ E n − ω , (10)where n and n , respectively, label the initial state and the intermediate state and ∆ E n = E n − E n is the difference between the initial and intermediate state energies. The detailedformula for the 2 ℓ -pole transition operator T ℓ in the center of mass frame is given in Ref. [35].For the rovibronic excited-state with L = 1, α ( ω ) and α ( T )1 ( ω ) can be written α ( ω ) = α ( S, ω ) + α ( P, ω ) + α ( D, ω ) , (11) α ( T )1 ( ω ) = − α ( S, ω ) + 12 α ( P, ω ) − α ( D, ω ) , (12)6here α ( P, ω ) denotes the contribution of nucleus 2 and electron 1 both being in p con-figuration to form a total angular momentum of P . The expressions for other multipoledynamic polarizabilities are derived similarly to those for the dipole polarizabilities [34–36].The fourth-order Stark shift for the rovibronic state can be written in the form,∆ E = − E (cid:20) γ ( − ω σ ; ω , ω , ω ) + γ ( − ω σ ; ω , ω , ω ) g ( L, M ) + γ ( − ω σ ; ω , ω , ω ) g ( L, M ) (cid:21) , (13)where g ( L, M ) is only dependent on the angular momentum quantum number L and mag-netic quantum number M , g ( L, M ) = 3(5 M − L − L )(5 M + 1 − L ) − M (4 M − L (2 L − L − L − , L ≥ , (14)and ω i are the frequencies of the external electric field in the three directions with ω σ = ω + ω + ω . The dynamic scalar second hyperpolarizability is γ ( − ω σ ; ω , ω , ω ), and thedynamic tensor second hyperpolarizabilities are γ ( − ω σ ; ω , ω , ω ) and γ ( − ω σ ; ω , ω , ω ).(From this point on, we will omit “second” when referring to the hyperpolarizabilities.)When all ω i = 0, the functions are called static hyperpolarizabilities. In particular, for therovibronic excited-state with L = 0 only the dynamic scalar hyperpolarizability remains andit is γ ( − ω σ ; ω , ω , ω ) = 16 π (cid:2) T (1 , , ω , ω , ω ) + 245 T (1 , , ω , ω , ω ) (cid:3) , (15)where T ( L a , L b , L c ; ω , ω , ω ) = X P (cid:20) X kmn h n L k T µ k mL a ih mL a k T µ k nL b ih nL b k T µ k kL c ih kL c k T µ k n L i (∆ E mn − ω σ )(∆ E nn − ω − ω )(∆ E kn − ω ) − δ ( L b , L ) X m h n L k T µ k mL a ih mL a k T µ k n L i (∆ E mn − ω σ ) × X k h n L k T µ k kL c ih kL c k T µ k n L i (∆ E kn + ω )(∆ E kn − ω ) (cid:21) , (16)the P P implies a summation over the 24 terms generated by permuting the pairs ( − ω σ /T µ ),( ω /T µ ), ( ω /T µ ), and ( ω /T µ ), where the superscripts µ i are introduced for the purposeof labeling the permutations [37]. 7 II. RESULTS AND DISCUSSIONA. Energies
The converged energies of the H +2 , D +2 , and HD + molecular ions from the present Hylleraascalculations for the rovibronic levels ( υ, L ) with υ ≤ L ≤ +2 and HD + , who used a different formof basis sets with pseudorandom complex exponents and the 2002 CODATA values of theproton and deuteron masses [39]. For the (0 ,
0) state of H +2 the present result contains20 significant figures, which improves by six orders of magnitude the result of Korobov.Other results in Table I are converged to at least 10 significant digits. For states ( υ ≥ , L )the energies are less accurate than the corresponding (0 , L ) states since our calculationsin this paper are for applications to “sum over states” determinations of polarizabilities.Thus, the energies in Table I for a given system and value of ( υ, L ) correspond to optimizednonlinear variational parameters for the corresponding υ = 0 state. In contrast, calculationsby Korobov [38] optimized the bases for each value ( υ, L ), and as expected, our presentvalues are systematically more positive compared to his. Recently, even more accurateenergy values for HD + were published in Ref. [40] using basis sets similar to the presentapproach, but with specific optimization and diagonalization for each separate energy level( υ, L ). (Accurate treatments of relativistic corrections to the ground and first excited stateswere presented recently for H +2 [41, 42] and for HD + [5, 41].) B. Ground-state static polarizabilities and hyperpolarizabilities
Table II presents a convergence study of the static multipole polarizabilities α (0) and α (0), and the static hyperpolarizability γ (0; 0 , ,
0) for H +2 in the rovibronic ground-state( υ = 0 , L = 0). The number of basis sets for the state of interest is indicated by N S , thenumber used for the intermediate states with P symmetry and D symmetry are indicatedby N P and N D respectively. The extrapolated values are obtained by assuming that theratio between two successive differences stays constant as the number of basis sets usedbecomes infinitely large. The static polarizabilities α (0) and α (0) converged quickly to,8espectively, twelve and eleven digits as the dimensions of the basis sets N S , N P , and N D were increased. The static hyperpolarizability, which is larger than α (0) by four orders ofmagnitude, converged to the ninth significant digit. Similar convergence tests for α (0) and α (0) of H +2 yield the extrapolated results listed in Table III.The static multiple polarizabilities and hyperpolarizabilities for the ground-state ( υ =0 , L = 0) of H +2 , HD + , and D +2 are listed in Table III. The polarizabilities and hyperpolar-izabilities for the homonuclear molecular ions H +2 and D +2 have the same magnitudes. Forthe heteronuclear ion HD + the corresponding values are much larger than those for H +2 andfor D +2 , due to the much smaller value of the first allowed transition energy. Note thatthe hyperpolarizability of HD + has opposite sign from H +2 and D +2 due to the sign of thecontribution from the two terms of Eq. (15).Table III also gives a comparison with selected previous works for the static dipole polar-izabilities in the rovibronic ground-state (0 ,
0) calculated using nonadiabatic methods (someearlier results for H +2 can be found in Ref. [47]). In order to facilitate comparison of thepresent dipole polarizabilities with those of Yan et al. [1], we repeated the calculations byusing the same nuclear masses as they used, and the resulting values are listed in the secondline. The agreement for α (0) could hardly have been better. However, the present staticdipole polarizability of H +2 is accurate to three parts in 10 , which improves by one orderof magnitude the result of Yan et al. For the static dipole polarizability of H +2 , our polar-izability of 3.168 725 805 289(1) is 0.025% different from the experimental value of 3.16796(15) [28]. For D +2 , our value is in good agreement with the less accurate result of Hilico etal. [43] and slightly larger than the result of Yan et al. [1]. The present dipole polarizability3.071 988 697 188(1) of D +2 agrees with the experimental value 3.07187(54) at the level of0.004%. For HD + , our result is much more accurate than the early result of Moss andValenzano [45]. Some other nonadiabatic calculations of the quadrupole (and higher order)polarizabilities are given in Refs. [1, 17] and we are in good agreement. There is a previousnonadiabatic calculation of the second hyperpolarizability for H +2 : Moss and Valenzano [45]find γ = 1 . × , in harmony with our result.It is interesting to examine in more detail the quadrupole polarizibility and second hy-perpolarizability calculations with previous Born-Oppenheimer treatments, where the quan-9ities are separated into “electronic”, “vibrational”, and “rotational” contributions [48]. Asexhibited in Table III, the relative magnitude of α (0) is much larger than those of α (0)and α (0), which is related to the available low-lying virtual state in the energy denom-inator (a similar argument pertains to α (0)). In the Born-Oppenheimer approach, thevirtual excitation corresponds to no change in the electronic or vibrational quantum num-ber, but a change in the rotational quantum number by 2. Bishop and Lam [48], (see theirtable 7), found α (0) = 1370 . . .
69 a.u., and 1362 .
24 a.u., where the relatively largerrotational contribution reflects the low-lying virtual excitation. In a recent paper, OlivaresPil´on and Baye [17] compared their total nonadiabatic calculation of α (0) for the groundstate to a second order perturbation theoretic sum over the first four vibrational states (theirEq. (25)) using matrix elements from their nonadiabatic calculation. They found that thenonadiabatic result was greater by an additive factor of 4 . α (0). The missing quantity is supplied by Bishop and Lam’s “electronic”component of 4 .
8. Evidently, the partial sum of Olivares Pil´on and Baye does not convergeto the correct value simply because of the neglect of higher electronic excitations.The magnitude of the static hyperpolarizability can also be understood along similarlines in Born-Oppenheimer picture. Earlier work using finite field methods by Bishop andSolunac [25] and by Adamowicz and Bartlett [12] established that nonadiabatic effects werenot the source of the large hyperpolarizability. Subsequently, Bishop and Lam [48] calculated γ (0) = 11537 .
16, with electronic, vibrational, and rotational contributions of, respectively,29 .
76, 568 .
7, and 10945 .
13, where again the larger rotational contribution is mainly due tothe virtual transition where the rotational quantum number changes by 2.Dynamic hyperpolarizabilities pertain to the four nonlinear optical processes (cf.Refs. [15, 37, 49]): Thus, the quantity γ ( − ω ; ω, ,
0) is the dc Kerr effect, γ ( − ω ; ω, ω, − ω )represents degenerate four-wave mixing (DFWM), γ ( − ω ; 0 , ω, ω ) is electric-field-inducedsecond-harmonic generation (ESHG) and γ ( − ω ; ω, ω, ω ) is third-harmonic generation(THG). 10n the Born-Oppenheimer approach, the rotational contributions to the dynamic hyper-polarizabilities for the dc Kerr and DFWM processes at optical wavelengths are expected tobe comparable to γ (0) while the rotational contributions to the ESHG and THG processesare expected to be much reduced in comparison to γ (0) [15]. For H +2 , we calculated thedc Kerr, DFWM, and ESHG hyperpolarizabilities at a wavelength of 632.8 nm. Using theavailable Born-Oppenheimer calculations of the electronic contributions from Bishop andLam [50] (their tables 2–4) (at the H +2 equilibrium internuclear distance 2 a.u.) and thevibrational contributions (their table 7), we estimated the rotational contributions by sub-traction from our nonadiabatic values. The results are given in Table IV. The nonadiabaticcalculations were carried out using the methods described herein with the largest basis set( N s , N p , N d ) = (2840 , , − C. Dynamic dipole polarizabilities and hyperpolarizabilities for the rovibronicground-state of HD + Since the transition (0 , → (0 ,
1) is a forbidden transition for the H +2 and D +2 ions, thefirst allowed transitions are at about ω = 0 . +2 and ω = 0 . +2 ,corresponding to “electronic transitions” (in the Born-Oppenheimer picture) and which arenot in the visible spectrum. Thus, in this subsection we concentrate only on the dynamicdipole polarizability and hyperpolarizability of the HD + system, for which optical transitionscan occur. Table V presents selectively some values of dynamic dipole polarizabilities andhyperpolarizabilities for ground-state HD + . All of the values are accurate to at least ninesignificant figures. The effect of the (0 ,
0) to (0 ,
1) resonance near the energy 2 . × − , seeTable I, on the quantities tabulated is apparent.Figs. 1–3 show the dynamic dipole polarizability α ( ω ) of HD + in the ground state as afunction of wavelength λ = c/ω in µ m. The perpendicular lines represent the positions of11esonant transitions. That there are many resonance transitions as λ → µ m is evident inFig 1. However, for the wavelengths λ = 4 − µ m, shown in Fig. 2, and the wavelengths λ = 10 − µ m, shown in Fig. 3, there is only one transition in each range. In theinserts for Figs. 2 and 3 the plots are magnified to show the positions where α ( ω ) = 0.In Fig. 2, the transition (0 , → (1 ,
1) occurs at λ = 5 . µ m (or photon energyof 0.008907 a.u.) and α ( ω ) = 0 at λ = 5 . µ m (0.009022 a.u.). In Fig. 3, thetransition (0 , → (0 ,
1) occurs at λ = 227 . µ m and α ( ω ) = 0 occurs at λ =20 . µ m. Our results for the (0 ,
0) state are in good agreement with the less accurateresults of Koelemeij [18], who combined the nonadiabatic polarizability calculations of Mossand Valenzano [45] with vibrational-rotational energies and electric dipole matrix elementscalculated in the Born-Oppenheimer picture to obtain values of α ( ω ) in the infrared. InFig. 4 the various hyperpolarizabilities (dc Kerr, DFWM, ESHG, and THG) are plottedover the energy range 0 < ω < × − a.u. The first resonant transition is prominentnear 2 . × − a.u. Note that sign changes for ESHF and THG occur at lower energiesand sign changes for DFWM, ESHG, and THG occur at higher energies as well, due to thecomplicated perturbation theoretic expressions. D. First excited-state static polarizabilities and hyperpolarizabilities
Table VI shows a convergence study of the static scalar and tensor dipole polarizabilitiesfor H +2 in the rovibronic excited-state ( υ = 0 , L = 1). The integer N ( pp ′ ) P represents thenumber of intermediate states used when the electron and one nucleus are both in excitedstates of p symmetry to form the total angular momentum L = 1. The contribution ofthe configuration α (( pp ′ ) P ) to α (0) is about 20%, as shown in Table VI. The final staticscalar and tensor dipole polarizabilities are both converged to the ninth figures. Calcu-lations of α (0) for D +2 were also carried out with similar results. Results for the staticscalar and tensor dipole polarizabilities for HD + are presented in Table VII and there isa partial cancellation between two intermediate symmetries, which can be seen by com-paring columns 2 and 4. For the largest basis set, α ( S ) = − .
024 382 526 724 a.u and α ( D ) = 133 .
405 246 966 154 a.u.; thus, when the two terms are added a loss of two sig-12 .0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.02.93.03.13.23.33.4 (0,0) (2,1)(0,0) (3,1)(0,0) (5,1)(0,0) (4,1) d y na m i c po l a r i z ab ili t y ( a . u . ) photon wavelength ( m ) FIG. 1: (Color online) Dynamic dipole polarizability α ( ω ) (in a.u.) for the rovibronic ground-state ( υ = 0 , L = 0) of the HD + ion for photon wavelengths from 0 to 4 µ m. The resonances(0 , → ( υ,
1) in the dynamic polarizability are marked. nificant figures results. Similar calculations were performed to obtain the static multipolepolarizabilities α (0) and α (0) of the H +2 , D +2 , and HD + ions in their first excited-states( υ = 0 , L = 1). Our results for α (0) and α ( T )1 (0) for all three molecular ions are in agree-ment with the recent results of Schiller et al [32], which are accurate to 8 significant digits.Table VIII summarizes the final values of the static multipole polarizabilities and hyper-polarizabilities for the H +2 , D +2 and HD + ions in their first excited-states ( υ = 0 , L = 1).From this table, we can see that dipolar and octupolar quantities for HD + are much largerthan those for H +2 and D +2 , especially for the hyperpolarizability, due to the allowed low-lying virtual state entering in the HD + case. For HD + (0 , α (0) = 3 .
990 667 in a nonadiabatic calculation. For H +2 (0 , γ = 4 634 .
39 in the Born-Oppenheimer approximation.13 d y na m i c po l a r i z ab ili t y ( a . u . ) (0,0) (1,1) O photon wavelength ( m ) FIG. 2: (Color online) Dynamic dipole polarizability α ( ω ) (in a.u.) for the rovibronic ground-state ( υ = 0 , L = 0) of the HD + ion for photon wavelengths from 4 to 10 µ m. The resonance(0 , → (1 ,
1) in the dynamic polarizability is marked. In the inset the region where α ( ω ) = 0around 5 . µ m is shown in greater detail. E. Static Stark shift
The static Stark shift ∆ E for the rovibronic ground-state (0 ,
0) of a hydrogen molecularion in an electric field of strength E is∆ E = − E α (0) − E γ (0; 0 , ,
0) (17)and the relative ratio between the second term and the first term is written as X = γ (0; 0 , , E α (0) . (18)This ratio determines the extent to which the Stark shift is influenced by the hyperpolar-izability at high field strengths. Using the values of Table III, at E = 6 . × − a.u. ∼ (334 kV/cm), we find X = 1 . × − for H +2 , X = 2 . × − for D +2 , and X = − . + . When E = 2 . × − a.u. ∼ (1087 kV/cm), we find X = 1 . × − for H +2 ,14
50 100 150 200 250 300-4x10 -3x10 -2x10 -1x10 d y na m i c po l a r i z ab ili t y ( a . u . ) photon wavelength ( m ) (0,0) (0,1) O FIG. 3: (Color online) Dynamic dipole polarizability α ( ω ) (in a.u.) for the rovibronic ground-state( υ = 0 , L = 0) of the HD + ion for photon wavelengths from 10 to 300 µ m. The (0 , → (0 , µ m where α ( ω ) = 0. X = 2 . × − for D +2 , and X = − .
032 for HD + . So the hyperpolarizability effect is moresignificant for the HD + system compared to either the H +2 or D +2 system. In particular, itcan cancel the Stark shift from the dipole polarizabilities.The leading term of static Stark shift ∆ E for the transition (0 , → (0 ,
1) of hydrogenmolecular ions in the electric field strength E is∆ E = − E (cid:2) α (0 , (0) − α (0 , (0) (cid:3) , (19)where α (0 , (0) and α (0 , (0) represent the static dipole polarizabilities for the ground-state(0 ,
0) and excited-state (0,1) respectively. Using the present values from Tables III andVIII, we obtain ∆ α (0) = α (0 , (0) − α (0 , (0) = − .
009 577 675 711 a.u. for H +2 , ∆ α (0) = − .
004 601 675 812 a.u. for D +2 , and ∆ α (0) = 391 .
316 177 674 2 a.u. for HD + . Thus the15 .0 1.0x10 -4 -4 -4 -4 -15-10-5051015 d y na m i c h y pe r po l a r i z ab ili t y ( a . u . ) photon energy (a.u.) -9 Kerr -9 DFWM -9 ESHG -9 THG
FIG. 4: (Color online) Dynamic hyperpolarizabilities (in a.u.), see text, for the rovibronic ground-state ( υ = 0 , L = 0) of the HD + ion for photon energies ω ≤ . second-order Stark shift will be larger for HD + than for either the H +2 or D +2 ion. F. Tune-out and magic wavelengths of HD + At certain laser frequencies where the dynamic polarizability vanishes it may be possibleto eliminate the shift induced by an applied laser field [51]—these frequencies are knownas tune-out frequencies or wavelengths. In addition, there might exist laser frequencies foran ion in two different states where the radiation induced shifts are equal (because thedynamic polarizabilities are equal at those frequencies): These frequencies are known as magic frequencies or wavelengths.For the first excited-state (0 ,
1) of HD + , the dynamic dipole polarizability is α ,M ( ω ) = α ( ω ) + α T ( ω ) 3 M − L ( L + 1) L (2 L − , (20)where M is the magnetic quantum number. In Table IX we list some of low-lying (in16 .0005 0.0006 0.0007 0.0008 0.0009 0.0010-100-80-60-40-20020 photon energy (a.u.) d y na m i c po l a r i z ab ili t y ( a . u . ) (0,0) (0,1), M=0 (0,1),|M|=1 FIG. 5: (Color online) Dynamic dipole polarizabilities α ( ω ) (in a.u.) of HD + for photon energiesbetween 0.0005 and 0.001 a.u. The solid black line denotes the dynamic polarizabilities of groundstate ( υ = 0 , L = 0). The dashed red and dotted blue lines represent the dynamic polarizabilities ofthe first excited state( υ = 0 , L = 1) with M = 0 and | M | = 1 respectively. The magic-wavelengthfor the transition (0 , → (0 ,
1) is marked by the arrow. energy) tune-out wavelengths for the ground state and the first excited state of HD + . Thepositions of magic-wavelengths between the ground-state and the first excited-state of HD + are marked by the arrows in Figs. 5–7, there are no magic-wavelengths in the visible lightrange. In Table X we list the values of the magic wavelengths indicated in Figs. 5–7. G. Long-range interactions
Spectroscopic measurements of the Rydberg states of the hydrogen molecules H and D have been performed by several groups [26, 52–54]. The data can be explained in termsof the long-range polarization potential model, in which, among other terms, the multipolepolarizabilities of the parent molecular ions H +2 or D +2 enter as parameters in the effective po-tentials of the multipole expansion of the ion interaction with the distant charge [52, 55, 56].An elaborate polarization potential model was developed for analysis of experiments on the17 .0070 0.0075 0.0080 0.0085 0.0090 0.0095 0.0100-10-50510 d y na m i c po l a r i z ab ili t y ( a . u . ) photon energy (a.u.) (0,0) (0,1), M=0 (0,1),|M|=1 FIG. 6: (Color online) As for Fig. 5, but for photon energies between 0.007 and 0.01 a.u. highly-excited Rydberg states of the hydrogen and deuterium molecules [26–28, 57–59]. Itsapplication yielded the experimental values for the static polarizabilities [28] given in Ta-ble III. Our nonadiabatic results for α (0) and higher multipoles do not appear to be readilyapplicable to this particular model, which utilizes a separation of higher order polarizabilitiesinto electronic, vibrational, and rotational contributions. For example, fits of the measuredspectra utilize the electronic and vibrational components of α (0); the rotational componentis treated as a higher order perturbation [27, 57] and handled separately.We used the dynamic multipole polarizabilities to calculate the long-range dispersioncoefficients C , C , and C for the interaction between a ground state H, He, or Li atomand a ground state H +2 , D +2 , or HD + ion. The results are given in Table XI. The detailedexpressions for the coefficients were given in Refs. [35] and [36]. For the atoms we usedmethods described previously. For H, the energies and matrix elements are obtained usingthe Sturmian basis set to diagonalize the hydrogen Hamiltonian [60], while for He and Li,the wave functions are expanded as a linear combination of Hylleraas functions [35, 60].When the atom is in the ground state but the molecular ion (denoted by “b”) is in anexcited L b state with magnetic quantum number M b , the leading terms of the second-order18 .0160 0.0165 0.0170 0.0175 0.0180012345 (0,0) (0,1), M=0 (0,1),|M|=1 photon energy (a.u.) d y na m i c po l a r i z ab ili t y ( a . u . ) FIG. 7: (Color online) As for Fig. 5, but for photon energies between 0.016 and 0.018 a.u. interaction energy are V ab = − C M b R − C M b R − · · · , . (21)The detailed expressions for C M b and C M b are given in Refs. [35] and [36].Table XII lists the dispersion coefficients of H +2 , D +2 , and HD + ions in the first excitedstate ( L b = 1) interacting with the ground-state H, He, and Li atoms. As above, the atomicproperties were taken from Ref. [60]. Note that the precision of the calculated C M b and C M b for the excited-state HD + interacting with H and He atoms is less than that for the H +2 andD +2 ions. In the case of interactions with Li, the accuracy of the coefficients is limited bythe accuracy of the Li calculations. IV. CONCLUSION
We calculated the static and dynamic multipole polarizabilities and hyerpolarizabilitiesfor the ground and first excited states of H +2 , D +2 , and HD + in the non-relativistic limit byusing correlated Hylleraas basis sets without using the Born-Oppenheimer approximation.For the static dipole polarizability of H +2 , the present value is the most accurate to date.19he hyperpolarizabilities were calculated without derivatives (not using finite field methods)for H +2 and its isotopomers. The present high precision values can not only be taken as abenchmark for testing other theoretical methods, but may also lay a foundation for investi-gating the relativistic and QED effects on polarizabilites and hyperpolarizabilities and assistin planning experimental research on hydrogen molecular ions. Acknowledgments
We are grateful to Prof. J. Mitroy for comments and to Prof. W. G. Sturrus for helpfulcorrespondence. This work was supported by the National Basic Research Program of Chinaunder Grant Nos. 2010CB832803 and 2012CB821305 and by NNSF of China under GrantNos. 11104323, 11274348. Z.-C.Y. was supported by NSERC of Canada and by the com-puting facilities of ACEnet and SHARCnet, and in part by the CAS/SAFEA InternationalPartnership Program for Creative Research Teams. ITAMP is supported in part by a grantfrom the NSF to the Smithsonian Astrophysical Observatory and Harvard University. [1] Z.-C. Yan, J.-Y. Zhang, and Y. Li, Phys. Rev. A , 062504 (2003).[2] J.-Y. Zhang and Z.-C. Yan, J. Phys. B , 723 (2004).[3] A. K. Bhatia and R. J. Drachman, Phys. Rev. A , 205 (1999).[4] A. K. Bhatia and R. J. Drachman, Phys. Rev. A , 032503 (2000).[5] Z.-X. Zhong, P.-P. Zhang, Z.-C. Yan, and T.-Y. Shi, Phys. Rev. A , 064502 (2012).[6] H. Olivares Pil´on and D. Baye, Phys. Rev. A , 032502 (2013).[7] M. Stanke and L. Adamowicz, J. Phys. Chem. A , 10129 (2013).[8] V. I. Korobov and Z.-X. Zhong, Phys. Rev. A , 044501 (2012).[9] D. Kedziera, M. Stanke, S. Bubin, M. Barysz, and L. Adamowicz, J. Chem. Phys. , 084303(2006).[10] D. Bishop, L. Cheung, and A. Buckingham, Molec. Phys. , 1225 (1980).[11] P. K. K. Pandey and D. P. Santry, J. Chem. Phys. , 2899 (1980).[12] L. Adamowicz and R. J. Bartlett, J. Chem. Phys. , 4988 (1986), ; , 7250E (1987).
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3. For eachvalue of ( υ, L ) in the first column, the first row gives the present result resulting from a singlediagonalization of the lowest υ state for a given L . Where a value of ( υ, L ) has a second row (H +2 and HD + ) the entry on the second row lists the result of Korobov [38], for which each value isthe result of a separate minimization (see text for further discussion). The number in parenthesesrepresents the computational uncertainty in the last digit. ( υ, L ) H +2 D +2 HD + (0,0) − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − ABLE II: Convergence of static multipole polarizabilities α (0), α (0), and hyperpolarizability γ (0; 0 , ,
0) (in a.u.) for the rovibronic ground-state ( υ = 0 , L = 0) of the H +2 ion. N S , N P , and N D , respectively, represent the number of basis sets for the initial-state of S symmetry, interme-diate states of P symmetry, and intermediate states of D symmetry. The extrapolated values foreach quantity are listed on the last line with the computational uncertainties of the last digits inparentheses. α (0) α (0) γ (0; 0 , , N S , N P ) value ( N S , N D ) value ( N S , N P , N D ) value(420,532) 3.168 723 735 424 03 (420,561) 1371.890 552 022 99 (420,532,561) 11479.750 406 991(680,695) 3.168 725 614 348 09 (680,727) 1371.894 443 542 72 (680,695,727) 11479.793 416 663(1036,1120) 3.168 725 797 655 76 (1036,954) 1371.894 963 825 42 (1036,1120,954) 11479.795 141 858(1255,1388) 3.168 725 804 884 54 (1255,1225) 1371.895 138 590 14 (1255,1388,1225) 11479.804 857 235(1504,1697) 3.168 725 805 220 47 (1504,1544) 1371.895 140 761 38 (1504,1697,1544) 11479.805 065 320(1785,2050) 3.168 725 805 275 76 (1785,1915) 1371.895 141 217 43 (1785,2050,1915) 11479.805 067 728(2100,2450) 3.168 725 805 286 34 (2100,2342) 1371.895 141 236 83 (2100,2450,2342) 11479.805 069 686(2451,2900) 3.168 725 805 288 58 (2451,2829) 1371.895 141 237 55 (2451,2900,2829) 11479.805 069 814Extrapolated 3.168 725 805 289(1) Extrapolated 1371.895 141 24(1) Extrapolated 11479.805 07(1) ABLE III: Static polarizabilities and hyperpolarizabilities (in a.u.) of H +2 , HD + , and D +2 ionsin the ground-state ( υ = 0 , L = 0). The numbers in parentheses represent the computationaluncertainties obtained by extrapolation. The first line gives the present values calculated using theCODATA 2006 masses. The second line gives the present values calculated using m p = 1836 . m d = 3670 . γ denote powers of ten. α (0)Author and Reference H +2 D +2 HD + Present a b et al. [1] 3.168 725 802 67(1) 3.071 988 695 7(1) 395.306 328 7972(1)Moss and Valenzano [45] 395.306Bhatia and Drachman [4] c et al. [43] 3.168 725 803(1) d e et al. [28] f α (0) α (0) α (0) γ (0; 0 , , +2 +2 + − a Using CODATA 2006 masses. b Using m p = 1836 . m d = 3670 . c Using the excitation energy of the first transition from Ref. [46]. d This value, without error bar, was also obtained by Olivares Pil´on and Baye [17] e Including relativistic corrections of O ( α ) f Experiment ABLE IV: For H +2 , estimation of rotational contributions to the dc Kerr, DFWM, and ESHGprocesses, in the Born-Oppenheimer picture, using tabulated electronic and vibrational valuesand the present nonadiabatic values, at wavelength of 632.8 nm. The values for the “Electronic”component” correspond to the internuclear distance of 2 a.u.component dc Kerr DFWM ESHGNonadiabatic (total) 4028.6 8445.1 14.631Electronic (Ref. [50]) 54.3 56.2 58.3Vibrational (Ref. [50]) 187.21 388.87 -8.65Rotational (row 1-(row 2+row3)) 3787 8000 -35.0TABLE V: Dynamic dipole polarizabilities and hyperpolarizabilities (in a.u.) for HD + in theground-state ( υ = 0 , L = 0) for photon energies ω ≤ . ω × α ( ω ) dc Kerr DFWM ESHG THG0.2 399.273 277 88(1) − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − ABLE VI: Convergence of static scalar and tensor dipole polarizabilities (in a.u.) for the firstexcited-state ( υ = 0 , L = 1) of the H +2 ion. ( N S , N P , N ( pp ′ ) P , N D ) α ( S ) α (( pp ′ ) P ) α ( D ) α (0) α ( T )1 (0)(124,140,185,131) 0.650 846 694 354 224 0.599 484 436 191 1.903 824 885 459 3.154 156 016 01 − − − − − − − − − − − − TABLE VII: Convergence of static scalar and tensor dipole polarizabilities (in a.u.) for HD + inthe rovibronic excite-state ( υ = 0 , L = 1). ( N S , N P , N ( pp ′ ) P , N D ) α ( S ) α (( pp ′ ) P ) α ( D ) α (0) α ( T )1 (0)(124,140,104,150) -130.074 770 552 639 0.547 263 971 717 133.368 575 021 506 3.841 068 441 117.011 545 036(240,290,221,325) -130.024 339 444 310 0.600 501 791 517 133.405 220 142 988 3.981 382 490 116.984 068 326(420,532,406,616) -130.027 278 013 066 0.600 795 791 278 133.404 624 757 699 3.978 142 536 116.987 213 433(680,890,675,815) -130.024 394 969 388 0.608 486 409 588 133.405 156 819 278 3.989 248 259 116.988 122 492(1036,1388,1044,1055) -130.024 382 831 359 0.609 081 936 428 133.405 235 157 018 3.989 934 262 116.988 400 284(1504,1697,1271,1340) -130.024 382 564 424 0.609 175 971 976 133.405 245 134 960 3.990 038 543 116.988 446 037(1785,2050,1529,1674) -130.024 382 532 136 0.609 261 536 706 133.405 246 685 012 3.990 125 689 116.988 488 632(2100,2450,1820,2061) -130.024 382 527 321 0.609 275 874 989 133.405 246 931 565 3.990 140 279 116.988 495 772(2451,2900,2299,2505) -130.024 382 526 724 0.609 281 615 158 133.405 246 966 154 3.990 146 055 116.988 498 638Extrapolated 3.990 148(2) 116.988 499(1) ABLE VIII: Static polarizabilities and hyperpolarizabilities (in a.u.) of H +2 , HD + , and D +2 ionsin the first excited-state ( υ = 0 , L = 1). The numbers in parentheses represent the computationaluncertainties. The numbers in the square brackets denote powers of ten.System α (0) α ( T )1 (0) α (0) α (0)H +2 − a D +2 − + b System γ (0) γ (0)H +2 − +2 − + − a Ref. [17] b Ref. [45]
TABLE IX: Tune-out wavelengths (in a.u.) for HD + . The numbers in parentheses represent thecomputational uncertainties. State M Tune-out wavelengths(0 ,
0) 0 0.002 215 386 568(1) 0.009 036 752 923(1) 0.017 178 225 41(1)(0 ,
1) 0 0.001 578 607 28(1) 0.008 614 811 326(1) 0.009 169 305 51(1) 0.017 340 149(5)(0 , ± TABLE X: Magic wavelengths expressed as photon energies (in a.u.) between the ground-stateand the first excied-state of HD + molecular ions. The values correspond to the marked arrows inFigs. 5–7. The number in parentheses represents the computational uncertainty. M Magic-wavelengths0 0.000 768 659 980(1) 0.009 260 494 92(1) 0.016 800 815 862(5) 0.017 170 143 339(1) 0.017 343 53(1) ± ABLE XI: Long-range dispersion coefficients C , C and C (in a.u.) for a ground state H +2 ,D +2 , or HD + ion interacting with a ground-state H, He, or Li atom. The numbers in parenthesesrepresent the computational uncertainties.System C C C H +2 -H 4.891 143 017 14(1) 90.316 962 31(1) 1807.210 076(2)D +2 -H 4.797 060 197 49(1) 87.850 021 22(1) 1756.323 945(2)HD + -H 5.381 569 069 96(1) 99.592 513 40(2) 2023.687 265(3)H +2 -He 2.195 917 825 1(1) 28.404 530 92(1) 368.784 69(1)D +2 -He 2.161 390 926 5(1) 27.641 661 03(1) 357.632 88(1)HD + -He 2.344 144 702 7(3) 31.043 628 96(3) 416.428 89(1)H +2 -Li 47.684(2) 2838.66(3) 168607(1)D +2 -Li 46.411(2) 2754.84(2) 163881(1)HD + -Li 66.498(2) 3354.24(2) 196257(1) ABLE XII: Long-range dispersion coefficients C M b and C M b (in a.u.) for an H +2 , D +2 , or HD + ion in the first excited-state with magnetic quantum number M b interacting with a ground-stateH, He, or Li atom. The numbers in parentheses represent the computational uncertainties.System M b C M b C M b H +2 -H 0 5.542 473 599 4(1) 114.730 417 8(1)D +2 -H 0 5.417 791 267 5(1) 110.820 998 3(1)HD + -H 0 6.233 633(2) 136.097 48(2)H +2 -H ± +2 -H ± + -H ± +2 -He 0 2.449 741 778 8(1) 36.967 531 92(1)D +2 -He 0 2.404 518 138 9(2) 35.706 083 48(2)HD + -He 0 2.659 058(1) 43.666 737(2)H +2 -He ± +2 -He ± + -He ± +2 -Li 0 55.361(1) 3500.16(2)D +2 -Li 0 53.635(1) 3377.72(2)HD + -Li 0 81.810(1) 4431.18(2)H +2 -Li ± +2 -Li ± + -Li ±1 59.047(1) 2772.99(1)