Oscillator Strengths for Rydberg States in the Polar Molecule NeH

Journal of Molecular Spectroscopy 211, 71–81 (2002) doi:10.1006/jmsp.2001.8465, available online at http://www.idealibrary.com on

Oscillator Strengths for Rydberg States in the Polar Molecule NeH P. G. Alcheev, V. E. Chernov, and B. A. Zon Voronezh State University, University Sq., 1, Voronezh 394693, Russia E-mail: [email protected] Received June 8, 2001; in revised form September 2, 2001

We calculate oscillator strengths for Rydberg electron transitions in polar molecules with a simple analytical procedure of taking into account the effects of nonspherical symmetry of molecular core field due to its dipole moment. Such effects result in nonzero oscillator strengths values for some transitions which are forbidden in widely used atom-like models of molecular Rydberg states. For the allowed transitions we also report the difference between the atom-like calculations and the calculations C 2002 Elsevier Science that account for the dipole moment of the molecular core.  Key Words: NeH; oscilaltor strengths; quantum defect; polar molecules; Rydberg states.

This semiphenomenological method is closely related to the well-known Simons model potential method (13, 14) used in Ref. (15) where oscillator strengths were calculated for electronic transitions to highly excited Rydberg states in NeH. These calculations (15) generalize the atomic calculations based on Simons model potential method. In their earlier works (16, 17 ) the authors of Ref. (15) noted the importance of correct account for angular dependence of the molecular wavefunctions. However, in the calculations presented in (15), only atomic orbitals nlλ are used, l and λ being the orbital momentum of the Rydberg electron and its projection onto the molecular axis. But such atom-like description of molecular Rydberg states is only approximate. Due to the nonspherical symmetry of the molecular core potential, the orbital momentum is not a good quantum number resulting in l-mixing. To provide an account for molecular symmetry a simple formalism of dipole-spheric functions was proposed in Refs. (18, 19). With the help of this formalism we re-calculate the oscillator strengths reported in (15), as well as the oscillator strengths for the transitions forbidden in atom-like approximation (e.g., s → d, etc.); the exclusion is removed due to the above-mentioned l-mixing. Instead of the “atomic” orbital momentum l its noninteger counterpart, the socalled quasimomentum, is used. In this work the oscillator strengths are calculated for the equilibrium internuclear distance values R = R0 . The results for other R values can be obtained in the same way if the correspondent quantum defects µ(R) is known from other theoretical works (9) or from the experiment. Let us to give a brief outline of the present paper. Section 2 summarize the results of the earlier works (18, 19). The Section 2.1 presents contains formalism of quasimomentum and dipole-spherical functions. The radial electron wavefunctions are given in Section 2.2 where we briefly discuss some

I. INTRODUCTION

For the past two decades great attention has been paid to socalled excimer molecules, whose properties are widely used for laser generation in the UV range. As an example of excimer molecules, rare gas hydrides can be referred to, whose ground terms are repulsive so that they can exist only in excited states. These molecules are also known as Rydberg molecules (1). Spectroscopic features of the rare gas hydrides were investigated experimentally and theoretically in a number of works (see, e.g., (2–11)). In particular, Refs. (7, 8) dealt with calculations of transition probabilities from low-excited electronic terms to the ground dissociative X 2  + state as well as transitions between low-excited Rydberg terms. Due to the difficulty of ab initio calculations, the transitions between low- and highly excited Rydberg levels (with principal quantum number n > 5) had not been calculated up to recent time. A poweful formalism for studing the highly excited Rydberg states is the quantum defect theory (QDT) (12). In QDT formalism, the Rydberg electron is considered to move far from the atomic or molecular core, so that its interaction with the core is almost Coulomb. The difference of this interaction from the Coulomb one is accounted by the quantum defect µ, and the energy spectrum is given by the Rydberg formula E na = −

Z2 , 2(n − µa )2

where Z is the core charge, n is the principal quantum number of the Rydberg electron, the subscript a stands for the other quantum numbers. Once the µa values are given (theoretically or experimentally), the QDT makes it possible to claculate oneelectron wavefunctions and therefore matrix elements in simple and straightforward way. 71

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linkages of the proposed formalism with the quantum defect theory (QDT). Different methods for calculation of the radial integrals are considered in Section 3. The basic calculation formulae together with the tables of the calculated oscillator strengths and discussion are presented in Section 4. All formulae use atomic units. The spectroscopic notations nlλ for molecular orbitals are given in the untied atom limit.

(Tˆ + Vˆ ) = E .

[2]

Here       1 ∂ ∂ ∂2 1 ∂ ∂ 1 Tˆ = − 2 r2 + sin ϑ + 2 2r ∂r ∂r sin ϑ ∂ϑ ∂ϑ sin ϑ ∂ 2 ϕ

II. BASIC MODEL: RYDBERG ELECTRON IN COULOMB–DIPOLE FIELD

A Rydberg electron with angular momentum1 high enough (l > 1) spends a significant part of time at a long distance from the molecular core. Thus only long-range terms of the core potential influence the motion of such electron. This concept is also known as long-range interaction model (see, e.g., (20, 21)). Apart from the Coulomb field of the residual molecular ion, the role of such long-range potentials can be played by the terms appearing in the multipole expansion of the core potential. In nonpolar molecules the most long-range term is due to the core quadrupole moment. In polar molecules the core possesses a moment of lower multipolarity, i.e., a dipole moment. So the theory of Rydberg states in polar molecules should be based on an account for the core dipole moment. Such an account was proposed in (18, 19) for adiabatic approximation and in (22) for diabatic approximation. Adiabatic (or Born–Oppenheimer, BOA) approximation is characterized by rapid motion of electron around slowly rotating core so that the projection λ of the electron orbital momentum onto the molecular ζ axis. According to the general principles of quantum mechanics, the quantitative condition of BOA validity is reduced to smallness of the rotational spectrum intervals compared with the difference between the Rydberg levels, Z 2 µ 2B(J + + 1)  , n3

system with the polar ζ axis along the dipole moment d, the Schr¨odinger equation for the Rydberg electron in a Coulomb + “point dipole field” can be written as

[1]

where B is the rotation constant of the core, J + is its angular moment, and µ is the difference between quantum defects of two electron states with principal quantum numbers closest to n. This condition is different somewhat from traditional concept of BOA applicability which was investigated for ground and low-excited electron terms. Namely, the appearance of quantum defect in [1] indicates that BOA does not require slow core rotations compared with electron motion along the Kepler orbit; however, the core rotation should be slow compared with precession of the Kepler ellipse itself. This result can be obtained in both classical and quantum mechanics frameworks and is due to the well-known degeneracy of Coulomb spectrum (23). In conditions of BOA the electronic states are not significantly perturbed by molecular rotations. Choosing spherical coordinate 1 It will be shown below that, although this quantum number is not conserved in molecules, some noninteger counterparts can be introduced.

is the electron kinetic energy operator and Z (d · r ) Vˆ = − + r r3

[3]

is the interaction of the electron with the core, i.e., with residual molecular ion having dipole moment d and charge Z . II.1. Accounting for Molecular Symmetry: l-Mixing and Noninteger Quasimomentum in Angular Wavefunctions Unlike the spherically symmetric atomic potential, the molecular core potential possesses only axial symmetry. Thus the angular part of the Rydberg electron wavefunctions should differ from familiar spherical harmonics which describe an electron in central field of atoms. Indeed, we introduce dipole-spherical functions Zη (d; ϑ, ϕ) satisfying the equation  −

1 ∂ sin ϑ ∂ϑ

 sin ϑ

∂Z ∂ϑ

 +

 1 ∂ 2Z + 2d cos ϑZ = ηZ sin2 ϑ ∂ 2 ϕ [4]

and standard boundary conditions: 2π -periodicity in azimuthal angle ϕ and regularity at the polar angle ϕ = 0, π values. These Z functions allows one to separate the variables in Eq. [2]: (r ) = Rη (r )Zη (d; ϑ, ϕ). The equation for the radial functions can be written as     1 d 2 dR η Z r − R = 0. R + 2 E + r 2 dr dr r2 r

[5]

So the eigenvalues η determine a modified centrifugal energy which now includes interaction with point dipole. A similar procedure is performed in model potential (MP) theory widely used in atomic and molecular calculations (13, 14, 24). However, we emphasize that in the modified centrifugal energy has clear physical origin in the present model whereas it is introduced only empirically in the MP theory. It is not surprising that such simple and clear problems connected with electron movement in a point dipole field could not

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remain without attention during the development of nonrelativistic quantum mechanics. So we cite some relevant publications here. It seems that the above-mentioned dipole-spherical functions Z were first used in analysis of the Stark effect for a symmetric rotator (25). A similar problem was also considered in (26) for weak electric fields and in (27) for strong fields. The Bremsstrahlung effect in electron scattering on a polar molecule was considered in (28). In that work a representation was given for the η eigenvalues in terms of continuous fractions. The references (29, 30) dealt with two-electron excitations of atoms which display dipole interaction due to the linear Stark effect in the hydrogen atom. The work (28) contains references to other works in whose the functions Z appear. For our purposes, it is sufficient to obtain the expansion of the dipole-spherical functions over the familiar spherical harmonics, Zη (d; ϑ, ϕ) ≡ Zλ (d; ϑ, ϕ) =



alλ (d)Ylλ (ϑ, ϕ),

[6]

l≥|λ|

where the λ subscript means the electron angular momentum projection onto the molecular axis. This quantum number is conserved due to axial symmetry of the molecular potential. Substitution [6] into [4], one can obtain the recurrence relation for the coefficients alλ :  2d

l 2 − λ2 4l 2 − 1

1/2

λ al−1 + 2d



(l + 1)2 − λ2 (2l + 1)(2l + 3)

= [ηλ − l(l + 1)]alλ

1/2

λ al+1

[7]

The coefficients alλ determining the l-state admixture to the -state are plotted in Figs. 1 and 2 as functions of l for different dipole moments and different λ molecular states. Passage to the spherically symmetric limit of zero dipole moment in the eigenvalues ηλ (d) −→ ( + 1); d→0

alλ (d) −→ δl d→0

[8]

determines uniquely the solution of the homogeneous difference eigenvalue problem [7], [8]. Since the diagonal terms in [7] are much less than the off-diagonal ones at l → ∞, we can readily see that ηλ −→ ( + 1); →∞

alλ −→ δl . l→∞

[9]

FIG. 1. Distribution of the squared coefficients alλ in Eq. [6] over l for different λ-states and dipole moments (: d = 0.1 a.u., : d = 1.0 a.u., : d = 2.0 a.u.,
: d = 3.0 a.u.).

noninteger quantum number as ˜ ˜ + 1). η ≡ (

In actual practice, the property [9] allows pass from the infinitedimensional eigenvalue problem [7], [8] to the corresponding reduced finite-dimensional eigenvalue problem. The latter can be solved by standard methods of computational linear algebra; the accuracy increases with increase of the maximal l value in the sum [6]. It seems useful to introduce the concept of “quasimomentum” for an electron in dipole-Coulomb field. We define this

As seen from [8], the  value has a simple meaning: it is the convenient (integer) angular moment value to which the quasimomentum tends in the limit of small dipole moment. The noninte˜ ger value of the quasimomentum ˜ = (λ), as well as admixture of spherical harmonics with l =  to the angular function [6] is the essential difference between the Rydberg states BOA theory for polar molecules and the corresponding theory for atoms.

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In the first nonvanishing perturbation approximation we obtain from [7] the Zon–Watson formulae, ˜ =  + 2d 2

( + 1) − 3λ2 , ( + 1)(42 − 1)(2 + 3)

2 ˜ = − d 2 , 3 alλ = δl + d

 = 0; 

δ−1,l 



 alλ = δl −

 ≥ 1;

dδ+1,l +1

2 − λ 2 δ+1,l − 2 4 − 1  + 1



[10]  ( + 1)2 − λ2 , 4( + 1)2 − 1  ≥ 1;

( + 1)2 − λ2 , 4( + 1)2 − 1

 = 0.

The obtained formulae demonstrate that the conditions [9] are fulfilled. II.2. Radial Functions and Bound State Energies As solutions of Eq. [5], the radial functions R(r ) ≡ Rkλ (r ) formally do not differ from the solutions of the corresponding “pure-Coulomb” problem (without the dipole potential). The only difference is that Rkλ (r ) includes noninteger quasimomentum ˜ (instead of the integer angular momentum l as it takes place in the pure-Coulomb case) as well as noninteger principal quantum number. ˜ d) + 1, ηkλ = k + (λ,

[11]

where k = 0, 1, 2, . . . is the radial quantum number. We present explicit expressions for the normalized radial functions of discrete spectrum states in terms of Whittaker functions, Gaussian hypergeometric functions, and Laguerre polynomials (31) (x = 2Zr/ηkλ ): 

˜ (k + 2+2) k!  ˜ 2Z 3/2 x  e−x/2 1 F1 (−k, 2˜ + 2; x) (k + 2˜ + 2) = 2 k! ηkλ (2˜ + 2)  2Z 3/2 k! ˜ ˜ = 2 [12] x  e−x/2 L 2k +1 (x). ηkλ (k + 2˜ + 2)

˜ 1 (x) 2Z 3/2 Mηkλ ,+ 2 Rkλ (r ) = 2 ˜ ηkλ x(2 + 2)

As distinct from H-like levels, the energy of bound Rydberg states in polar molecules depends, as expected, on core dipole ˜ and on its projection λ, moment, on quasimomentum , E kλ = −

Z2 Z2 = − ,  2 dip 2 2ηkλ 2 k +  + 1 − µλ

[13]

FIG. 2. Distribution of the squared coefficients alλ in Eq. [6] over l for different λ-states and dipole moments (: d = 0.1 a.u., : d = 1.0 a.u., : d = 2.0 a.u.,
: d = 3.0 a.u.). dip

where we introduce the dipole quantum defect µλ , determined by the interaction with the core dipole moment. This quantum defect causes the difference of the noninteger molecular quasimomentum ˜ from the integer orbital momentum  appearing in atoms, (i.e. at d = 0): dip ˜ =  − µλ → 

at

d → 0.

[14]

Figure 3 shows the dependence of quasimomenta on dipole moment.

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or the Bates–Damgaard QDT method. A brief sketch of these methods is given in the next section. III. RADIAL MATRIX ELEMENTS OF ELECTRON TRANSITIONS

III.1. Simons Model Potential (MP) To find the one-electron wavefunction in this method, the model potential Z 1  ˆ Vˆ = − + 2 Al P l r r l

FIG. 3. Quasimomenta of some λ-states as functions of dipole moment d. Dashed lines correspond to perturbation formulae [9].

The value η becomes complex at ˜ ≤ −1/4. As a result, some bound state energies E kλ becomes also complex. This should be interpreted as the Rydberg electron “falling to the center” (32). In our case, absence of stable bound states means that the point dipole approximation is inapplicable. Some calculations (see also Fig. 3) show that the falling to the center takes place for the states with low  (i.e., “penetrating states”). The values ˜ < −1/4 arise at d > 0.64 a.u. This value is low enough to consider that the point dipole approximation seems quite restrictive. The cause is that the point dipole potential has a singularity at the origin while the real molecular potential should not contain any singularities. A simple way to overcome this difficulty is the following: we introduce an additional potential Vc (r ) which has a singularity at the origin so that the resulting potential V (r ) + Vc (r ) is regular at the origin. Without loss of generality we can assume that Vc (r ) = 0 outside the effective core radius, i.e. for r ≥ rc . Then we can apply the QDT methods; the quantum defect will effectively involve Vc (r ) and account for real (nonsingular) core potential. In fact, it results in some addition to the quasimomentum value (33) in the radial functions without any changes in the angular functions. The wavefunction constructed in such a way will adequately describe the electron at r > rc . In QDT frameworks, the Rydberg electron energy levels εkλ = −

Z2 , 2 2νkλ

νkλ = k +  + 1 − µλ

[15]

is introduced, where Pˆ l is the projector onto the subspace of spherical harmonics with a given angular moment l. The phenomenological constants Al are to be borrowed from an experimental spectrum. The radial wavefunctions are defined for all 0 ≤ r ≤ ∞ and they satisfy the equation     1 d 2 dR eff (eff + 1) Z r − R = 0, [18] R + 2 E + r 2 dr dr r2 r where the effective orbital quantum number eff = eff (l, λ) is expressed in terms of the MP constant as eff (eff + 1) = l(l + 1) + 2Al . The values of eff can be determined based on an experimental Rydberg spectrum [15]:

eff = Z / −2εkλ − k − 1. The effective orbital momentum eff = eff (l) defined in such a way coincides with the quasimomentum [14] in the pure-dipole case, i.e. at µshort λ = 0. Note again that in atomic calculations the potential [17] is introduced phenomenologically, while in the above-described Rydberg molecule model this potential has a clear physical meaning: in Eq. [5] it describes the point dipole potential, which is absent in atoms. The solution of [18] can be expressed in terms of the Gaussian hypergeometric function:

[16]

are expressed in terms of the total quantum defect µλ ≡ µtot λ = dip short µλ + µshort , where the “short-range” quantum defect µ λ λ = ηkλ − νkλ is due to the difference between the real molecular core potential and the point dipole potential. So, for simple description of real Rydberg states in polar molecules, the angular functions introduced in Section 2.1 can be used, the radial functions being constructed using an experimental spectrum [15] based on, e.g., the Simons model potential

[17]

2Z 3/2 [(k + 2eff + 2)]1/2 2 (k!)1/2 (2eff + 2) νkλ eff    2Zr 2Zr −Zr/νkλ . e1 F1 −k, 2eff + 2; × νkλ νkλ

RMP keff λ (r ) = keff λ | r  =

[19] Radial matrix elements for dipole transitions are expressed in terms of the Appell hypergeometric function of two

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variables (24), k1 eff1 λ1 |r |k2 eff2 λ2    eff2  eff1 1 ν 1 ν2 2ν1 2ν2 = 4Z 2 ν1 + ν2 ν 1 + ν2 ν1 + ν2 (eff1 + eff2 + 4) (2eff1 + 2)(2eff2 + 2)   (ν1 + eff1 + 1)(ν2 + eff2+1 ) 1/2 × k1 !k2 !  × F2 eff1 + eff2 + 4; −nr1 , −nr2 ; 2eff1 + 2, 2eff2 ×

+ 2;

 ν1 ν2 , , ν1 + ν2 ν1 + ν2

where ν1,2 is calculated using [15]. The node number k of the wavefunction is calculated according k = n − n cl , where n is borrowed from spectroscopic notation (in the united atom limit) of the nlλ orbital, n cl being the number of closed electron shells n cl = 2 for NeH). III.2. Bates–Damgaard (BD) Method The radial wavefunction of the Rydberg electron in QDT is considered to be defined outside the core (i.e., for r > rc ), where the effective core radius rc is an undefined parameter which can be assumed to be equal to the left classical turning point rc = ( + 12 )/2Z . Outside the core (r > rc ) the radial wavefunction is expressed in terms of the Whittaker function (12, 34), √   ∂µλ (νkλ ) (−1)k Z BD ˜ Rk λ 1 + λ | r  = (r ) = ν kλ ˜ r νkλ ∂νkλ   − 12 2Zr ˜ ˜ × (νkλ − )(1 +  + νkλ ) , Wνkλ ,+ ˜ 1 2 νkλ [20]

FIG. 4. Radial wavefunctions of 3sσ state in BD method (upper curve) and in MP method (lower curve): the difference is decreased due to the r 3 factor in [21].

sidered in (34). In Fig. 5 we show the difference between the molecular radial wavefunction of the 3 pπ state (d = 1.147) and the corresponding atomic function of the 3 p state (d = 0) with the same quantum defect (i.e., with the same effective principal quantum number νkλ ). III.3. Davydkin–Zon (DZ) WKB QDT Method As the WKB approximation to the BD QDT method, a simple procedure for calculation of matrix elements for transition

where k is to be defined according to Eq. [16]. The function [20] has nonphysical divergence at r → 0. Nevertheless, since the integrand in the matrix element  ∞ ν1 ˜1 λ1 |r |ν2 ˜2 λ2  = Rk1 ˜1 λ1 (r )Rk1 ˜ 1 λ1 (r )r 3 dr [21] rc

3

contains an r factor, the value of the integral depends insignificantly on rc . This value is close to the matrix element [19], calculated in the MP framework (although the corresponding MP wavefunctions RBD ˜ (r ) and Rkλ (r ) are significantly different at k λ MP small r ). The difference between RBD ˜ (r ) and Rkλ (r ) decreases k λ with νkλ (see Fig. 4). The new approach proposed in the present work consists in using the non-integer quasimomentum values in [20] and [21] instead of the integer values which appear in “atomic” case con-

FIG. 5. Molecular radial wavefunction of 3pπ state (d = 1.147 a.u., solid line) and atomic wavefunction of 3p state (d = 0, dashed line) with the same quantum defect.

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TABLE 1 Angular Q-Factors for Different Transitions Depending on the Core Dipole Moment Core dipole moment d, a.u. Transition

0 (15)

0.2

0.4

0.6

0.8

1.0

1.5

2.0

2.5

(s) ↔ (s) (s) ↔ ( p) (s) → ( p) (s) ↔ (d) (s) → (d) ( p) ↔ ( p) ( p) ↔ (d) ( p) → ( p) ( p) → (d) ( p) → (s) ( p) → ( p) ( p) ↔ ( p) ( p) → (d) ( p) ↔ (d) ( p) ↔ (d) (d) ↔ ( p) (d) → ( p)

0 1/3 2/3 0 0 0 4/15 0 2/5 1/3 0 0 1/15 2/5 4/5 2/15 1/5

0.017 0.321 0.660 0.001 0.001 0.006 0.266 0.007 0.400 0.330 0.003 0.003 0.066 0.398 0.799 0.133 0.200

0.062 0.290 0.641 0.003 0.004 0.020 0.265 0.025 0.399 0.320 0.013 0.013 0.065 0.394 0.796 0.130 0.199

0.119 0.251 0.616 0.005 0.008 0.034 0.264 0.050 0.399 0.308 0.025 0.028 0.063 0.386 0.792 0.126 0.199

0.178 0.214 0.589 0.007 0.011 0.042 0.264 0.077 0.400 0.295 0.038 0.048 0.061 0.376 0.786 0.121 0.200

0.232 0.182 0.563 0.009 0.014 0.043 0.264 0.102 0.403 0.282 0.051 0.072 0.057 0.366 0.778 0.115 0.202

0.337 0.128 0.510 0.009 0.016 0.027 0.263 0.154 0.416 0.255 0.077 0.146 0.048 0.329 0.756 0.096 0.208

0.409 0.099 0.470 0.007 0.015 0.007 0.255 0.191 0.428 0.235 0.095 0.227 0.037 0.292 0.731 0.075 0.214

0.462 0.081 0.440 0.005 0.013 0.000 0.237 0.216 0.436 0.220 0.108 0.307 0.028 0.260 0.705 0.057 0.218

between Rydberg states with |ν| ≡ |ν2 − ν1 |  ν¯ ≡ was proposed in (35, 36): 1 λ1 |r |2 λ2  =

ν¯ 2 πZ



π

√

ν1 ν2

dξ (1 − ε cos ξ )2 × cos[φ(ξ )

where the Kronecker delta accounts for double degeneration of the final state due to λ-doubling. The dependence of the factor Q on the dipole moment appears due to the angular matrix elements with the account of [6]:  ˜1 λ1 Y1,λ1 −λ2 ˜2 λ2

0

− ν(ξ − ε sin ξ ) + π(ν − )], [22]

2 2 ε = 1 − (1 + 2 + 1) /2ν ,  = 2 − 1 ; √ cos ξ − ε 1 − ε2 cos φ = , sin φ = sin ξ. 1 − ε cos ξ 1 − ε cos ξ This formula does not include the undefined rc parameter which is inherent in QDT: in the quasiclassical approximation this parameter coincides with the classical turning point. Due to its simplicity, the DZ method is successfully used for the calculation of transitions between highly excited Rydberg states (37, 38). IV. OSCILLATOR STRENGTHS OF RYDBERG TRANSITIONS IN THE NeH MOLECULE

We use the standard definition (coinciding with that used in (15)) of oscillator strengths (OS) for electron transitions in a molecule, f (n 1 ˜1 λ1 → n 2 ˜2 λ2 ) =

Q=

2 ε k 2  2 λ2 − ε k 1  1 λ1 3 ×Q(˜1 λ1 →˜2 λ2 )|k1 1 λ1 |r |k2 2 λ2 |2 ,

 2 4π  2 − δ0,λ2 ˜1 λ1 Y1,λ1 −λ2 ˜2 λ2 , 3

[23]

λ1

= (−1)





(2l1 +

l1 l2

 ×

l1 −λ1

1 λ1 − λ2

1)(2l2 l2 λ2



l1 1 l2 + 1) 0 0 0



aλ11l1 (d)aλ22l  (d) 2

[24]

In Table 1 we present values of Q-factors for different transitions in an abstract molecule depending on its core dipole TABLE 2 Quantum Defects of Low-Excited States of NeH State

µ from (39)

12  + (3s) 32  + (4s) 62  + (5s) 22  + (3 p) 12 + (3 p) 52  + (4 p) 32 + (4 p) 42  + (3d) 22 + (3d) 12 + (3d) 72  + (4d) 42 + (4d) 22 + (4d)

1.5647 1.5750 1.5513 0.8418 0.8127 0.7662 0.7388 −0.0208 −0.0345 −0.0485 −0.0568 −0.0708 −0.0936

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µ extracted from (9) 1.7175 1.6740 1.6011 0.9272 0.8418 0.8805 0.8076 0.0215 −0.0105 −0.0785

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TABLE 3 Oscillator Strengths for the Transitions from the 12 Σ+ (3s) State (Multiplied by 103 ) d = 1.147 a.u.; µ from (39)

d = 1.147 a.u.; µ from (9)

Final state

d=0 Ref. (15)

ω21 , cm−1

MP

BD

DZ

ω21 , cm−1

MP

BD

DZ

32  + (4s) 62  + (5s) 2  + (6s) 2  + (7s) 2  + (8s) 2  + (9s) 2  + (10s)

0 0 0 0 0 0 0

34607 44042 47723 49572 50629 51290 51731

31.6 6.59 2.66 1.35 0.785 0.498 0.336

31.6 5.97 2.32 1.15 0.649 0.403 0.267

39.0 10.0 5.07 3.17 2.22 1.67 1.32

46436 57221 61049 62955 64040 64716 65165

19.9 3.08 1.31 0.683 0.402 0.256 0.174

17.8 1.97 0.806 0.405 0.231 0.144 0.096

23.5 4.18 2.31 1.51 1.09 0.835 0.668

22  + (3p) 52  + (4p) 2  + (5p) 2  + (6p) 2  + (7p) 2  + (8p) 2  + (9p) 2  + (10p)

255 29.5 9.45 4.3 2.35 1.43 0.94 0.65

29709 42775 47146 49262 50444 51171 51650 51981

125 14.4 4.62 2.11 1.15 0.701 0.46 0.319

150 16.3 5.24 2.39 1.31 0.796 0.523 0.363

159 19.6 7.34 3.94 2.53 1.8 1.37 1.09

41180 55444 60254 62533 63790 64555 65056 65401

110 14.3 4.77 2.2 1.21 0.739 0.486 0.337

134 15.9 5.2 2.39 1.3 0.794 0.521 0.361

146 20 7.78 4.28 2.79 2. 1.53 1.23

12 (3p) 32 (4p) 2 (5p) 2 (6p) 2 (7p) 2 (8p) 2 (9p) 2 (10p)

510 63.2 20.7 9.54 5.31 3.2 2.1 1.46

30331 42950 47225 49304 50469 51187 51660 51989

418 51.8 17 7.82 4.29 2.62 1.73 1.2

414 52.3 18 8.49 4.72 2.91 1.93 1.35

447 66.9 27.4 15.3 10 7.22 5.51 4.39

43160 55953 60477 62650 63859 64599 65085 65422

354 54.5 19 9.01 5.01 3.09 2.04 1.42

301 48. 17.4 8.38 4.71 2.92 1.94 1.35

346 66.9 29.5 17.1 11.4 8.32 6.41 5.14

42  + (3d) 72  + (4d) 2  + (5d) 2  + (6d) 2  + (7d) 2  + (8d) 2  + (9 d) 2  + (10 d)

0 0 0 0 0 0 0 0

41243 46600 48977 50277 51065 51578 51930 52183

2.84 1.05 0.518 0.293 0.182 0.121 0.084 0.061

3.89 1.16 0.576 0.326 0.203 0.135 0.094 0.068

4.29 1.57 0.864 0.54 0.368 0.266 0.201 0.156

54351 59787 62293 63650 64467 64996 65359 65618

1.76 0.733 0.367 0.209 0.13 0.087 0.061 0.044

3.23 1.24 0.603 0.339 0.21 0.139 0.097 0.07

3.82 1.63 0.87 0.533 0.356 0.254 0.189 0.145

22 (3 d) 42 (4 d) 2 (5 d) 2 (6 d) 2 (7 d) 2 (8 d) 2 (9 d) 2 (10 d)

0 0 0 0 0 0 0 0

41351 46646 49001 50291 51073 51584 51935 52186

4.49 1.68 0.833 0.472 0.294 0.195 0.137 0.099

5.13 1.56 0.789 0.451 0.282 0.188 0.132 0.096

5.9 2.2 1.23 0.771 0.525 0.379 0.285 0.222

54612 59897 62349 63683 64487 65010 65369 65625

2.66 1.14 0.575 0.33 0.206 0.138 0.096 0.07

3.24 1.37 0.691 0.395 0.247 0.165 0.115 0.084

4.38 2.02 1.09 0.668 0.445 0.314 0.231 0.176

moment. As is seen, nonspherical symmetry of the molecular core plays a significant role. Quantum defects for the NeH molecule was reported in (39); they can also be extracted from the results published in (9, Table 1). The present work uses both versions; they are presented in Table 2. This dependence on the internuclear distance R can easily be incorporated into our results since there exist some ab initio calculations of µ(R) (9). One can simply change µ → µ(R) in all the formulae. However, as seen from Table 2, the µ(R) values extracted from different sources

vary significantly even for the equilibrium internuclear distance R = R0 . The results of OS calculation for different transitions (reported in (15)) are given below in Tables 3, 4 and 5. We used three methods (MP, BD, and DZ) to calculate the radial matrix elements. The transition frequences ω21 given in Tables 3–5, ω21 = εk2 2 λ2 − εk1 1 λ1 ,

[25]

were calculated using the Rydberg formula [15]. Therefore the ω21 are very sensitive to the µ values, so a reasonable accuracy

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TABLE 4 Oscillator Strengths for the Transitions from the 22 Σ+ (3 p) State (Multiplied by 103 ) d = 1.147 a.u.; µ from (39)

d = 1.147 a.u.; µ from (9)

d=0 Ref. (15)

ω21 , cm−1

MP

BD

DZ

ω21 , cm−1

MP

BD

DZ

12  + (3s)

−29.5 153 1.54 0.34 0.14 0.07 0.04 0.03

−29709 4899 14333 18015 19863 20921 21582 22022

−125 74.7 0.755 0.169 0.068 0.035 0.021 0.014

−150 72.9 1.06 0.248 0.097 0.047 0.026 0.016

−159 76. 1.35 0.402 0.199 0.121 0.083 0.061

−41180 5256 16041 19869 21775 22860 23536 23985

−110 71.9 1.39 0.337 0.14 0.073 0.044 0.028

−134 65.8 1.69 0.428 0.176 0.089 0.052 0.032

−146 68.6 2.18 0.693 0.353 0.22 0.152 0.113

52  + (4p)

0 0 0 0 0 0 0

13066 17438 19554 20736 21463 21941 22273

3.98 1.08 0.464 0.246 0.147 0.096 0.066

3.4 0.932 0.401 0.212 0.127 0.083 0.057

3.68 1.17 0.595 0.37 0.258 0.193 0.152

14264 19074 21353 22610 23375 23876 24221

4.63 1.19 0.501 0.263 0.156 0.101 0.069

4.17 1.07 0.45 0.236 0.14 0.091 0.062

4.5 1.36 0.68 0.421 0.293 0.219 0.172

7.59 9.65 2.77 1.21 0.648 0.391 0.255 0.176

8.03 4.69 1.36 0.593 0.317 0.191 0.124 0.086

7.75 5.11 1.74 0.909 0.576 0.407 0.308 0.244

21.8 8.17 2.44 1.08 0.579 0.35 0.228 0.158

21.8 3.43 1.02 0.448 0.24 0.144 0.094 0.065

20.9 3.83 1.37 0.731 0.469 0.334 0.255 0.203

Final state 32  + (4s) 62  + (5s) 2  + (6s) 2  + (7s) 2  + (8s) 2  + (9s) 2  + (10s) 2  + (5p) 2  + (6p) 2  + (7p) 2  + (8p) 2  + (9p)

2  + (10p)

12 (3p) 32 (4p) 2 (5p) 2 (6p) 2 (7p) 2 (8p) 2 (9p) 2 (10p)

0 0 0 0 0 0 0 0

623 13242 17516 19595 20761 21478 21952 22280

42  + (3d) 72  + (4d) 2  + (5d) 2  + (6d) 2  + (7d) 2  + (8d) 2  + (9d) 2  + (10d)

312 42.1 14. 6.57 3.66 3.27 1.51 1.06

11534 16892 19268 20568 21356 21869 22222 22475

309 41.7 13.9 6.52 3.63 2.25 1.5 1.06

323 41.4 13.8 6.45 3.59 2.22 1.48 1.04

341 46.4 17.3 9.18 5.83 4.1 3.09 2.44

13171 18607 21113 22470 23287 23816 24179 24438

295 41.1 13.9 6.54 3.65 2.27 1.51 1.06

309 42. 14.1 6.6 3.68 2.28 1.52 1.07

327 45.2 16.9 9. 5.74 4.06 3.07 2.43

22 (3d) 42 (4d) 2 (5d) 2 (6d) 2 (7d) 2 (8d) 2 (9d) 2 (10d)

467 65.4 21.5 19.4 5.8 3.6 2.4 1.69

11642 16938 19292 20582 21365 21875 22226 22478

475 66.4 22.5 10.6 5.91 3.67 2.45 1.72

475 64.5 22. 10.4 5.82 3.63 2.42 1.71

508 73.8 28.2 15.1 9.65 6.81 5.14 4.06

13432 18717 21169 22503 23307 23830 24189 24445

451 68.1 23.6 11.2 6.32 3.94 2.64 1.86

436 67.1 23.5 11.2 6.34 3.96 2.66 1.87

471 75.1 29.3 16 10.3 7.28 5.51 4.36

in µ is required for identification of experimental spectroscopic lines before a comparison of OS calculations with the experiment. Figure 6 shows OS dependence in an abstract molecule on its core dipole moment. For NeH calculations we assumed d = 1.147 a.u. for the NeH+ ion (40). For comparison we give OS values calculated in (15) without accounting for the dipole moment (i.e., without accounting for axial molecular symmetry in the angular function). Also, we present OS for the transitions forbidden in the atom-like model (15).

1980 14773 19297 21470 22679 23419 23905 24242

There is a significant discrepancy between the “molecular” and “atomic” calculations for the ( p) ↔ (s) transitions. It is due to the fact that the angular Q-factor for these transitions is decreased by two times as the dipole moment changes from zero to the assumed value d = 1.147 a.u. (see Table 1). However, the difference of the OS for ( p) → (d), ( p) → (d) transitions from the results reported in (15) is due to the values Q(( p) → (d)) = 2/5, Q((d) → ( p)) = 1/5, Q(( p) → (d)) = 1/15, calculated in (15), which differ from our values given in Table 1.

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TABLE 5 Oscillator Strengths for the Transitions from the 12 Π(3 p) State (Multiplied by 103 ) d = 1.147 a.u.; µ from (39)

d = 1.147 a.u.; µ from (9)

Final state

d=0 Ref. (15)

ω21 , cm−1

MP

BD

DZ

ω21 , cm−1

MP

BD

DZ

12  + (3s) 32  + (4s) 62  + (5 s) 2  + (6s) 2  + (7s) 2  + (8s) 2  + (9s) 2  + (10s)

−255 142 0.72 0.14 0.06 0.03 0.02 0.01

−30331 4276 13710 17392 19241 20298 20959 21400

−209 117 0.592 0.118 0.046 0.023 0.014 0.009

−207 139 0.657 0.112 0.036 0.015 0.007 0.004

−223 140 0.755 0.161 0.067 0.036 0.023 0.015

−43160 3277 14062 17890 19796 20881 21556 22005

−177 90.3 0.247 0.04 0.015 0.007 0.004 0.003

−150 106 0.252 0.031 0.008 0.003 0.001 0.0008

−173 106 0.279 0.042 0.014 0.006 0.003 0.002

22  + (3p) 52  + (4p) 2  + (5p) 2  + (6p) 2  + (7p) 2  + (8p) 2  + (9p) 2  + (10p)

0 0 0 0 0 0 0 0

−623 12443 16815 18931 20113 20840 21318 21650

−3.8 7.09 1.82 0.765 0.402 0.239 0.155 0.107

−4.02 9.88 2.61 1.11 0.587 0.351 0.228 0.157

−3.87 10.6 3.19 1.57 0.957 0.656 0.485 0.377

−1980 12284 17094 19374 20630 21396 21896 22241

−10.9 11. 2.4 0.959 0.49 0.288 0.184 0.126

−10.9 16.8 3.74 1.5 0.772 0.454 0.292 0.199

−10.4 17.9 4.53 2.11 1.25 0.842 0.616 0.476

32 (4p)

0 0 0 0 0 0 0

12619 16894 18973 20138 20856 21329 21658

9.26 2.51 1.07 0.569 0.341 0.222 0.153

8.37 2.34 1.02 0.544 0.327 0.213 0.147

8.99 2.88 1.46 0.903 0.627 0.468 0.367

12793 17317 19490 20699 21439 21926 22262

11.7 2.93 1.22 0.64 0.38 0.246 0.169

11.7 3.03 1.28 0.674 0.402 0.261 0.179

12.5 3.71 1.83 1.12 0.771 0.572 0.447

10911 16269 18646 19946 20733 21246 21599 21852

64.8 8.07 2.62 1.21 0.669 0.413 0.274 0.192

77.7 8.16 2.46 1.08 0.583 0.353 0.232 0.161

81.1 8.9 2.97 1.49 0.923 0.641 0.481 0.379

11191 16628 19133 20490 21307 21837 22199 22458

63.8 6.87 2.13 0.963 0.526 0.322 0.213 0.148

78.5 6.5 1.76 0.732 0.38 0.225 0.145 0.1

81. 6.62 1.96 0.933 0.562 0.387 0.289 0.229

2 (5p) 2 (6p) 2 (7p) 2 (8p) 2 (9p)

2 (10p)

42  + (3d) 72  + (4d) 2  + (5d) 2  + (6d) 2  + (7d) 2  + (8d) 2  + (9d) 2  + (10d)

78.9 9.82 3.19 1.47 0.81 0.5 0.33 0.23

22 (3d) 42 (4d) 2 (5d) 2 (6d) 2 (7d) 2 (8d) 2 (9d) 2 (10d)

237 30.6 10.1 4.68 2.59 1.6 1.07 0.75

11020 16315 18669 19959 20742 21252 21603 21855

419 54.2 17.8 8.28 4.59 2.84 1.89 1.33

488 55.3 17.2 7.74 4.2 2.56 1.69 1.18

514 61.5 21.3 10.9 6.79 4.73 3.55 2.8

11452 16738 19189 20523 21328 21850 22209 22466

413 49.1 15.7 7.21 3.97 2.44 1.62 1.14

487 49. 14.4 6.27 3.34 2.01 1.32 0.912

511 52.4 17. 8.47 5.2 3.61 2.71 2.14

12 (3d) 22 (4d) 2 (5d) 2 (6d) 2 (7d) 2 (8d) 2 (9d) 2 (10d)

474 65 21.8 10.2 5.7 3.54 2.36 1.66

11129 16389 18707 19982 20756 21262 21610 21860

916 126 42.2 19.8 11 6.85 4.57 3.21

1010 127 41.8 19.3 10.7 6.59 4.38 3.07

1080 147 53.8 28.3 17.8 12.5 9.37 7.39

11981 16963 19306 20591 21370 21879 22229 22480

900 129 43.8 20.6 11.6 7.2 4.81 3.39

925 130 43.4 20.2 11.2 6.94 4.62 3.24

1010 150 56 29.7 18.8 13.2 9.93 7.84

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REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32.

FIG. 6. Distribution of oscillator strengths (multiplied by 103 ) over n for different dipole moments (: d = 0.1 a.u., : d = 1.0 a.u., : d = 5.0 a.u.).

33. 34. 35. 36. 37.

ACKNOWLEDGMENTS We are grateful to K. Kawaguchi, A. V. Stolyarov, and J. K. G. Watson for their interest in this work.

38. 39. 40.

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