Semiclassical theory for the quantum defect function of diatomic molecules

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2003 J. Phys. B: At. Mol. Opt. Phys. 36 3697

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INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS B: ATOMIC, MOLECULAR AND OPTICAL PHYSICS

J. Phys. B: At. Mol. Opt. Phys. 36 (2003) 3697–3705 PII: S0953-4075(03)64705-X

Semiclassical theory for the quantum defect functionof diatomic molecules

H Nakamura1,2 and E A Solov’ev2,3

1 Department of Functional Molecular Science, The Graduate University for Advanced Studies,Myodaiji, Okazaki 444-8585, Japan2 Department of Theoretical Studies, Institute for Molecular Science, Myodaiji,Okazaki 444-8585, Japan3 Macedonian Academy of Sciences and Arts, PO Box 428, 1000 Skopje, Macedonia

Received 12 June 2003Published 22 August 2003Online at stacks.iop.org/JPhysB/36/3697

AbstractThe simple analytical expression for the quantum defect function, �lm(R),of a diatomic molecule is derived using the uniform semiclassical approachincluding the effects of hidden crossing. �lm(R) is a function of the internucleardistance R and the effective charge at each nucleus is also a function of R.Thus, in the case of homonuclear molecules this expression gives universaldependence on R. The derived expression is applied to H2 molecules. Thequantum defects are found to be in good agreement with the ab initio numericaldata by Wolniewicz and Dressler (1994 J. Chem. Phys. 100 444).

1. Introduction

Recently, molecular superexcited states (SESs) have attracted much attention bothexperimentally and theoretically because of their peculiar properties and the remarkableprogress in synchrotron and laser technologies [1–4]. The SESs are classified into two types:the first kind is doubly or inner-shell excited states which are embedded in the ionizationcontinuum and the second kind is rovibrationally excited Rydberg states which can autoionizeby the transfer of rovibrational excitation energy into the ionization channel. The three kindsof states, i.e. the two kinds of SESs and the ionization continuum, play decisive roles in thedifferent dynamics of highly excited states of molecules. For instance, the autoionization of thefirst kind of SES is determined by the electronic coupling between this state and the ionizationcontinuum, and that of the second kind is controlled by the quantum defect. The MQDT(multi-channel quantum defect theory) presents a very powerful methodology for treating thevarious SES dynamics [2, 5–8]. In the case of the dynamics of the second kind of SES, theR-dependence of quantum defects is especially important. In order to fully comprehendSES dynamics, the interplay between spectroscopic experimentation, quantum chemicalcalculations of various basic physical quantities and MQDT-type analyses is inevitable [9]. Itis not easy to accurately determine the absolute values for quantum defects theoretically, but it

0953-4075/03/173697+09$30.00 © 2003 IOP Publishing Ltd Printed in the UK 3697

3698 H Nakamura and E A Solov’ev

is rather easy to do so by fitting the experimental term values. The R-dependence should,however, be estimated quantum chemically. Thus, its analytical expression, if any, shouldbe quite useful in investigating SES dynamics. Such an analytical expression for a diatomicmolecule is presented in this paper. It consists of two parts. The first part is the quantum defectof the united atom at R = 0, which implicitly takes into account multi-electron correlations.The second part is the quantum defect of the one electron two-centre Coulombic problemwhich provides R-dependence. By semiclassically analysing the Schrodinger equations of thetwo-centre Coulombic problem expressed in spheroidal coordinates, the R-dependence of thequantum defect is derived analytically.

This paper is organized as follows: in section 2 the basic model treated here is presented.That is to say, the Schrodinger equations of the two-centre Coulomb problem in the spheroidalcoordinates and the definition of the R-dependent quantum defect are provided. Here theclassical origin of the appearance of the hidden crossings is also discussed. In section 3,the uniform semiclassical analysis is presented and the analytical expression of the quantumdefect is derived by following [10]. Numerical applications of the formula obtained are carriedout in section 4. Direct comparison is made with the ab initio quantum chemical data of theH2 molecule. The formula is found to work pretty well and thus is expected to be usable inpractical applications. Concluding remarks will follow in section 5.

2. The two-centre Coulombic problem: classical analysis of hidden crossings

The binding energy of the highly excited electron can be approximately expressed as

Enlm(R) = − Z 2

2[n − �lm(R)]2, (1)

where R is the internuclear distance, nlm are the spherical quantum numbers in the unitedatom limit, Z is the charge of the molecular ionic core, �lm(R) is the quantum defect functionwhich does not depend on n. The quantum defect �lm(R) can be split into two terms

�lm(R) = δl + �TCClm (R), (2)

where δl is the quantum defect of the united atom at R = 0 and �TCClm (R) is a shift due to the

separation of two-Coulomb centres (TCCs).The TCC problem is described by[

− h2

2me� − Z1

r1− Z2

r2

]�nlm(r) = εnlm(R)�nlm(r), (3)

where r1 and r2, respectively, are the distances from the electron to the first and second nucleihaving charges Z1 and Z2 which are equal to 1/2 in the case of the homonuclear molecularion and have nontrivial R-dependence in the heteronuclear case. The eigenenergy εnlm(R) isrelated to the quantum defect �TCC

lm (R) by

�TCClm (R) = n − Z√−2εnlm(R)

. (4)

The Schrodinger equation (3) is separated in the prolate spheroidal coordinates: ξ =(r1 + r2)/R, η = (r1 − r2)/R, ϕ = arctan(x/y), where x and y are the ordinary Cartesiancoordinates in the frame of reference with the z-axis directed along the internuclear axis.Representation of the electron wavefunction in the form

�nlm(r) =√

(ξ2 − 1)(1 − η2)Fnlm(ξ)�nlm(η)eimϕ

Semiclassical theory for the quantum defect function of diatomic molecules 3699

leads to the following system of equations:

d2 Fnlm(ξ)

dξ2+

{1

h2 P2(ξ) − m2 − 1

(ξ2 − 1)2

}Fnlm(ξ) = 0, (5)

d2�nlm(η)

dη2+

{1

h2 Q2(η) − m2 − 1

(1 − η2)2

}�nlm(η) = 0, (6)

where

P(ξ) =√

aξ − λ

ξ2 − 1− p2, Q(η) =

√bη + λ

1 − η2− p2,

are the effective momenta, λ is a separation constant,

p = √−2meεnlm(R)R/2, a = me(Z1 + Z2)R, b = me(Z2 − Z1)R.

Using the uniform semiclassical approach, we take into account the effects of hiddencrossings to obtain a simple analytical expression for the quantum defects. Formally thehidden crossings are the branch points of the multi-sheet energy Riemann surface in thecomplex R-plane connecting pairwise the sheets related to different quantum energy levelson the real R-axis. Equations (5) and (6) are not time dependent, which is very importantfor the identification of the hidden crossings. So we first discuss the classical mechanicscorresponding to equations (5) and (6), which is helpful in understanding the origin of hiddencrossings.

The classical equations of motion for lz = mh = 0 (m ∼ 1 as h → 0) [11]

dξ

dt= 4(ξ2 − 1)P(ξ)

R2(ξ2 − η2),

dη

dt= 4(1 − η2)Q(η)

R2(ξ2 − η2), (7)

can be recast into the separable forms

dξ

dτ= (ξ2 − 1)P(ξ),

dη

dτ= (1 − η2)Q(η), (8)

by introducing the new time variable

τ =∫ t 4 dt ′

R2[ξ2(t ′) − η2(t ′)],

which is a generalization of Kepler’s anomaly for the TCC. According to equations (8) at smallvalues of R the electron oscillates along ξ between the two turning points

ξ1,2 = a ∓ √a2 − 4λp2 + 4 p4

2 p2(9)

and rotates along η. With an increase in the internuclear separation R at some moment abecomes equal to λ and ξ1 = 1, i.e. the internal caustic disappears here. At a = λ theCoulombic singularity at ξ = 1 in P2(ξ), which generally reduces the zero of the first orderin the factor (ξ2 − 1) to the standard turning point singularity

√ξ − 1, vanishes and the ‘time’

τ logarithmically diverges there. ξ(t) = 1 is a periodic orbit coinciding with the internuclearaxis. This type of periodic orbit also follows from the well known property that the Coulombicsingularity repels the head-on colliding electron irrespective of the energy. As a result theelectron oscillates between two nuclei along the internuclear axis. However, such an orbit isunstable. A small deviation from the axis leads to the spiral trajectory shifting the electronexponentially from the internuclear axis. At positive energy the motion along such a trajectory(in the inverse direction) leads to the capture of the incoming electron and to the appearance ofbroad resonances [12]. As will be shown later, at negative energy the unstable periodic orbitsgive rise to the hidden crossings of the adiabatic states.

3700 H Nakamura and E A Solov’ev

3. The uniform semiclassical approximation

The quantum defect for the two-centre Coulombic problem �TCClm (R) in the leading

semiclassical order was obtained in [10]. Following [10], we will derive an analyticalexpression for the quantum defect which is uniform near a ∼ λ. In the limit p → 0 thequasi-angular equation (6) does not depend on the energy and the semiclassical approach givesthe expression for the separation constant [13]

λ = [(l + 1/2)h]2 +b2

8[(l + 1/2)h]2, (10)

which is valid in the interval 0 � R � [(l + 1/2)h]2/|Z1 − Z2| including the region wherethe hidden crossing occurs. A much more complicated case is the quasi-radial problem wherethe Coulombic singularity is changed from repulsion to attraction at a = λ. To construct theuniform asymptote near a ∼ λ we will divide the ξ -axis into two overlapping parts. At ξ < ξ2

we can use the Whittaker equation{d2

dx2−

[1

h2

(2γ

x− 1

)+

4µ2 − 1

4x2

]}M−iγ /h ,µ

(2i

hx

)= 0 (11)

as a comparison equation to reproduce the first- and second-order poles in the quasi-radialequation (5). Assuming Fnlm(ξ) in the form

Fnlm(ξ) = [x ′(ξ)]−1/2 M−iγ /h ,µ

(2i

hx(ξ)

),

we obtain a nonlinear equation for the scaling function x(ξ)

[x ′(ξ)]2

[1 − 2γ

x(ξ)− h2(4µ2 − 1)

4[x(ξ)]2

]= P2(ξ) − h2(m2 − 1)

(ξ2 − 1)2− h2

2{x, ξ}, (12)

where

{x, ξ} ≡ x ′′′(ξ)

x ′(ξ)− 3

2

[x ′′(ξ)

x ′(ξ)

]2

is Schwarz’s derivative. At h → 0 the solution of equation (12) can be expanded in series overthe small parameter h2. Then in the first approximation the scaling function x(ξ) is determinedby the transcendental equation

2γ arcsin(√

x(ξ)/2γ)

+√

2γ x(ξ) − x(ξ)2 =∫ ξ

1|P(ξ)| dξ. (13)

The solution x(ξ) must be smooth. This leads to the condition that the first turning pointof equation (5) must coincide with the turning point of the comparison equation (xt = 2γ ),i.e. x(ξ1) = xt . It determines the first index γ of the Whittaker function as

γ = 1

π

∫ ξ1

1|P(ξ)| dξ. (14)

The second index µ is determined by the condition that the residues of the second-order polesmust coincide in equations (5) and (11). This leads to the result µ = m/2.

Far right of the turning point xt , the Whittaker function reaches its asymptotic form andthe solution can be presented as

Fnlm(ξ) ∼ cos

(1

h

[x − γ − γ ln

(2x

γ

)]− χ(γ ) − π

2

), (15)

Semiclassical theory for the quantum defect function of diatomic molecules 3701

where

χ(γ ) = π(m − 1)

4− γ

h

(1 − ln

γ

h

)+

1

2iln

[�([m + 1]/2 − iγ /h)

�([m + 1]/2 + iγ /h)

](16)

is the Coulombic phase shift. From equation (12) the scale function x(ξ) is determined in thisregion by

√x2(ξ) − 2γ x(ξ) − 2γ ln

(√x(ξ)

2γ+

√x(ξ)

2γ− 1

)=

∫ ξ

ξ1

P(ξ∗) dξ∗ (17)

or at large values of x by

x(ξ) − γ − γ ln

(2x(ξ)

γ

)=

∫ ξ

ξ1

P(ξ∗) dξ∗. (18)

Equation (18) gives the phase in the asymptote (15). On the other hand, for the semiclassicalsolution which exponentially decreases at ξ � ξ2, the asymptote in the classically allowedregion is

Fnlm(ξ) ∼ cos

(1

h

∫ ξ2

ξ

P(ξ∗) dξ∗ − π

4

). (19)

Matching the two forms of asymptotes (15) and (19) gives the modified quantization condition

1

h

∫ ξ2

ξ1

P(ξ) dξ − χ(γ ) = π(nr + 3/4), (20)

where nr = 0, 1, 2, . . . is the quasi-radial quantum number. This quantization condition is atranscendental equation for the energy eigenvalue. In the limit p → 0, however, this can besolved explicitly. In this limit the second turning point ξ2 (together with nr) goes to infinityand the integral in the left-hand side of equation (20) diverges due to the Coulombic tail in theeffective potential U(ξ). To extract this diverging term explicitly let us represent the integralin the form ∫ ξ2

ξ1

[P(ξ) −

√a

ξ− a

ξ2

]dξ +

π

2

√aξ2 + o(1/ξ2). (21)

After this regularization we can put p = 0 in the integral which is reduced now to the completeelliptic integral. Using the definition of the quantum defect function (4) and the relationshipn = nr + l, one can obtain the following result in the limit n → ∞ (see also [10]):

�TCClm (R) = l +

1

4− 1

πχ(γ ) − 2

√a + λ

π hE

(2a

a + λ

), (22)

where E(x) is the complete elliptic integral of the second kind as defined in [14]. At p = 0,equation (14) is reduced to

γ = 2√

a + λ

π

[K

(λ − a

λ + a

)− E

(λ − a

λ + a

)], (23)

where K (x) is the complete elliptic integral of the first kind.Quantization condition (20) is valid for a � λ when ξ1 � 1. However, we can analytically

continue it towards larger values of R where γ becomes negative. Here the integral in (20)must be split into two terms—the first from ξ1 < 1 to 1 and the second from 1 to ξ2. It is easyto show that the first integral is purely imaginary and is compensated by the imaginary part

3702 H Nakamura and E A Solov’ev

Figure 1. The real part of �TCClm (R) in the complex R-plane; Z1 = Z2 = 1/2, l = 2, m = 0.

coming from ln(γ /h) in χ(γ ). So, we obtain by analytic continuation the correct quantizationcondition for a > λ

1

h

∫ ξ2

1P(ξ) dξ = π(nr + 3/4) +

π(m − 1)

4

− γ

h

(1 − ln

|γ |h

)− 1

2iln

[�([m + 1]/2 + iγ /h)

�([m + 1]/2 − iγ /h)

]. (24)

4. Comparison with the exact calculations

In figure 1 the real part of the quantum defect (22) is plotted as a function of the real andimaginary parts of R for the case Z1 = Z2 = 1/2, l = 2, m = 0. One can see the logarithmicsingularity which is the limiting point of the infinite series of the branch points (the so-calledS-series of hidden crossings [15]) connecting the states n, l, m and n + 1, l, m in a pairwiseway. This singularity comes from the logarithm of the �-function (see equation (16)) whichhas a pole when its argument is equal to zero: γ ± ih(m + 1)/4 = 0. In the classical limith → 0 this condition transforms into the condition for the periodic orbits: γ = 0. When themulti-dimensional electron trajectory collapses into the periodic orbit at a = λ (γ = 0), thecorresponding semiclassical/quantum eigenfunction dramatically increases near this orbit. Inturn, the neighbouring states must be subject to violent changes to keep the orthogonalizationconditions. As a result a strong interaction arises between the adiabatic states in this region,i.e. hidden crossings.

We compare our results with the exact data by Wolniewicz and Dressler [16] for the excitedpotential energies of the H∗

2 molecule. Since the ionic core in the united atom limit (1sσhydrogen-like ground state of He+) spherically deviates from the Coulombic super-symmetry,the angular momentum of the excited electron l is a good quantum number and the quantumdefects δl are well defined. Thus, H2 is the best molecule to use to check the present theory.

Semiclassical theory for the quantum defect function of diatomic molecules 3703

Figure 2. The two-electron adiabatic potentials Wlm (R) (a) and effective quantum numberνlm (R) = n − �lm (R) (b) of the H2 molecule: the dashed curves are given by equation (22), thesolid curves are the adiabatic potentials Wnlσ (R), the dashed–dotted curve is the doubly exciteddiabatic potential curve and the symbols show the ab initio data by Wolniewicz and Dressler [16].In the parentheses the traditional notations of the H2 states are shown.

The dashed curves in figure 2(a) show the two-electron effective potentials of H∗2

Unlm(R) = E (+)1sσ (R) − 1

2[n − �lm(R)]2+

1

R. (25)

In equation (25) E (+)

1sσ (R) is the electron energy of the ground state of H+2 and δl were taken

from Landau and Lifshitz [17] with δs = 0.140 and δd = 0.0022. The energy Udex(R) of thedoubly excited diabatic state 1�(2pσu)

2 is shown by the dashed–dotted curve. It is obtainedby matching the data for Udex(R) at R < 2.5 au from [18] with the ab initio results for theE F-state at large values of R from [16]. The interaction of the doubly excited state 1�(2pσu)

2

with the single excited states |nlσ 〉 leads to the appearance of a sequence of avoided crossings.The solid curves are the eigenenergies Wnlσ (R) of the 6 × 6 Hamiltonian matrix with Udex(R)

andUnlσ (R) as the diagonal matrix elements and the R-independent interaction Qd,nlσ betweenthe doubly excited state 1�(2pσu)

2 and the five singly excited states |nlσ 〉. The interaction istaken to be Qd,nsσ = 0.04n−3/2 for the nsσ states and the Qd,ndσ = 0.08n−3/2 for ndσ states.At this level of interaction we obtain very good agreement between the adiabatic potentialsWnlσ (R) and the ab initio numerical data by Wolniewicz and Dressler [16]. By comparing theeffective quantum number (solid curve) with the exact data (marks) in figure 2(b) one can seethat outside a small region around the avoided crossings at R ∼ 3 au the decisive contributioncomes from the analytical expression (22). Since the R-dependence of quantum defects playsan important role as explained in the introduction, a direct comparison with the ab initio datais given in figure 3. As is seen from this figure, the analytical expression (22) gives a goodestimate of R-dependence.

3704 H Nakamura and E A Solov’ev

1 2 3 4

-0.2

0.0

0.2

0.4

0.6

0.8

quan

tum

def

ect

R / a.u.

present (n=4) diabatic (Eq.(22)) sσ, T&N (1983) dσ, T&N (1983) 4sσ, W&D (1994) 4dσ, W&D (1994)

Figure 3. The quantum defect of H2 molecules. The solid curves show the present results forn = 4, the dashed curves show the data from equation (22), the closed symbols show the datafrom [16] for n = 4 and the open symbols show the data from [18].

5. Concluding remarks

The analytical expression for providing the R-dependence of the quantum defect functionof diatomic molecules was derived by assuming that the predominant contribution to theR-dependence comes from the separation of TCC and by using the uniform semiclassicalapproximation to the corresponding Schrodinger equations using spheroidal coordinates.Although the absolute values of the quantum defects can be rather easily determined byanalysing the spectroscopic experiments using the MQDT, the R-dependence (which playsa crucial role in the dynamics) should be estimated theoretically. In this sense, the formuladerived in this paper is expected to be quite useful. In the homonuclear case, the R-dependenceturns out to be universal within the present approximation. In the case of a heteronuclearmolecule, on the other hand, the effective charges Z1(R) and Z2(R) (=1 − Z1(R)) at eachnucleus are required to estimate the quantum defects. The formula obtained was comparedwith the available ab initio data for H2, and was confirmed to be pretty accurate and usable.Once the information on the effective charges is available, this formula could be useful for theanalysis of SES dynamics.

Applications of the present formula to other homo- and heteronuclear diatomic moleculesare necessary to further clarify the applicability. This will be discussed in a future publication.

Acknowledgments

This work is partially supported by a Grant in Aid for Scientific Research, grant no 13 440 182from the Japanese Government. One of us (EAS) thanks the Institute for Molecule Sciencefor providing him with a visiting professorship during 2002–2003.

Semiclassical theory for the quantum defect function of diatomic molecules 3705

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