Dispersion interaction between the hydrogen atom and hydrogen like ions

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Dispersion interaction between the hydrogen atom and hydrogen like ions

F.N. Yousif, G.C. Shukla

To cite this version:

F.N. Yousif, G.C. Shukla. Dispersion interaction between the hydrogen atom and hydrogen like ions.

Journal de Physique, 1980, 41 (2), pp.101-107. �10.1051/jphys:01980004102010100�. �jpa-00209221�

Dispersion interaction between the hydrogen ^{atom and} hydrogen ^{like ions}

F. N. Yousif and G. C. Shukla

Department ^of Physics, ^Basrah University, Basrah, Iraq

(Reçu le 30 octobre 1978, ^{révisé le} 7 ’mai 1979, accepte ^{le 23} juillet 1979)

Resume.

Nous présentons ^{le calcul des} interactions de dispersion à deux corps ^entre les multipôles ^{de l’atome}

et l’ion hydrogène d’une part, ^{et entre ceux des ions} hydrogène ^d’autre part. ^Les polarisabilités dynamiques ^des multipôles ^sont déduites à l’aide de l’analogue variationnel de l’équation différentielle permettant de passer d’un modèle hydrodynamique ^{à un modèle} quantique. ^Les approximants ^{de Padé sont} également utilises pour déterminer cette polarisabilité dynamique ^à partir des coefficients de Cauchy ^du multipôle. ^Les coefficients de l’énergie ^de dispersion qui apparaissent dans le calcul des énergies d’interactions des systèmes H-He+, H-Li++, He+-He+, ^{Li++-Li++ et He+-Li++ sont obtenus} jusqu’au ^{terme R-10.}

Abstract.

The two-body dispersion interactions between the multipole ^{of the} hydrogen ^{atom and the} multipole

of the hydrogen like ions and between multipoles ^{of two} hydrogen ^{like ions have been worked out. The} frequency dependent multipole polarizabilities ^have been worked out by ^the variational analog ^{to the} differential equation

obtained from the hydrodynamic ^model analogy ^to quantum mechanics. Also, ^the [2, 1] ^Pade’ approximants

are used to construct the frequency dependent multipole polarizabilities ^{from the} multipole Cauchy’s coefficients.

The dispersion energy coefficients which appear in the expressions ^{for the} interaction _energies of the H-He +, H-Li++, He+-He+, ^{Li++-Li++ and He+-Li++} systems ^have been obtained up to the term R-10.

Classification Physics ^Abstracts

31.20

1. Introduction.

Recently [1] ^we formulated a

differential equation ^and its variational analog using

the hydrodynamic analogy ^to quantum mechanics in order to study ^the multipoles ^{of the} hydrogen ^atom.

The differential equation ^{for the} multipoles ^{of the} hydrogen ^{atom was solved} by using ^{a Frobenius} type solution [2] enabling ^{us to} compute Cauchy’s ^coeffi-

cients for the multipoles ^{of atomic} hydrogen. ^Further-

more we cosstructed Pade’ approximants ^{for fre-}

quency dependent multipole polarizabilities ^{of the} hydrogen ^{atom to work out the} dispersion energy between interacting hydrogen pair [3]. The extension of this differential equation ^{to the} hydrogen ^{like ions} (Z

2, 3,

^{etc. for} He+, ^Li++... etc.) ^reads

where

For Z

1 it reduces to the hydrogen ^{atom as dis-}

cussed [1]. ^In present ^{work we} computed ^the Cauchy

moments for the multipoles ^{of He + and Li + +} by solving the above differential equation ^{for Z}

²

and 3 respectively in the identical manner used in our

earlier work [2]. ^{We list the} resulting Cauchy ^moments

for multipole polarizabilities aK and multipole ^shield- ing ^factors yK in ^{table I and table II for He+ and} Li+ + respectively. The variational equation ^{of the} hydrogen ^{atom has} already been solved by ^the

authors. The variational analog ^to the differential

equation ^{for the} hydrogen like ions is discussed in section 2. The dispersion energy between interacting pairs involving H, He+ ^{and Li+ +} is discussed in section 3. The work is concluded with a discussion

containing ^a comparison ^of present work with that of earlier works.

2. Variational equation and its solution for the

multipoles ^{of the} hydrogen ^{like ions.}

The variational

equation ^{for the} multipoles ^{of the} hydrogen ^{like ions}

is :

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01980004102010100

102

Table I.

Numerical values for Cauchy ^moments of multipole polarizabilities aK, multipole shielding factors 7K and calculated resonance frequencies for ^{the He+ atom} (in a.u.) for L ^{= 1 to 6. The notation a} + b represents

a x 10±b.

-

Table II.

Numerical values for Cauchy ^moments of multipole polari=abilities ^aK, multipole shielding ’/Úc!or.B "K, and calculated _resonance frequencies for ^{Li+ + atom} (in a.u.) , for ^L

^{1 to 6. The notation a} + b represents

a x 10:tb.

104

Note that the above equation follows from our earlier formulation for atomic hydrogen [1] with the diffe-

rence that the ground ^{state wave} function of the

hydrogen like ions is :

which for Z

2 and 3 refer to He+ and Li+ + res-

pectively.

For solving the variational _equ. (1), the choices for the arbitrary ^functions FL(R ) ^and GL(R ) ^{are as}

follows :

and

The above choices are determined from our previous

work for the multipoles ^{of atomic} hydrogen [1]. ^For

the static case (co

0) ^the GL(R) ^{function vanishes} [1] ;

so equ. (1) simplifies ^{to :}

This equation for static multipoles ^{of the} hydrogen

like ions results in convergence with the first two terms in FL(R) ^function. Retaining ^{the first two terms of}

the function FL(R), i.e.,

and substituting equ. (6) ^into equ. (5), ^we get

The same function is obtained from the differential

equation approach [2]. ^The expression for static mul-

tipoles ^for hydrogen like ions is :

For the dynamic multipole polarizabilities (w =1= 0),

the role of the GL(R ) function is significant. ^In solving

equ. (1) ^with FL(R ) ^as given by equ. (6) ^and taking ^the

first three terms in the function GL(R), i.e.,

we have achieved convergence in (J.,2L(W). ^{The final} expression ^for (J.,2L(W) ^{is as follows :}

Table III.

The values of rx2L(0), A 1, Bl, B2 ^and cor

(in ^atomic units) of the helium ion (He+) for ^various

values of ^L from ^{case II. The notation a} + b repre- sents a x 10::tb.

where a2L(o) is the static multipole polarizabilities, A 1, B, and B2 ^are constants.

The values of the static multipoles a2L(o), the ^the constants A 1, BI ^and B2 ^{and the resonance} frequen-

cies _cor for L ⁼ 1 to 6 for He + and Li+ + are as in tables III and IV respectively.

Table IV.

The values of a2L(o), A 1, Bl, B2 ^and co,

(in a.u.) of ^{lithium ion} (Li+ +) for ^{various values} of L from ^{case II. The notation a} + b represents ^{a x} 10:f:b.

3. The dispersion energy between interacting pair

a and b is given by [4].

where the multipole-multipole dispersion interaction

C(La, Lb) ^between interacting pair ^{a and b is :}

with

and aa(iy) ^and ab(iy) ^{refer to the} respective frequency dependent multipole polarizabilities ^of systems ^{a and} b at an imaginary frequency ^{co =} iy.

Using ^the expression ^{for the} frequency dependent multipole polarizabilities ^{for the} hydrogen ^{atom as}

obtained earlier [5] ^{and the} frequency dependent multipole polarizabilities ^{for He+ and Li+ + as listed}

Table V.

The values of multipole-multipole ^dis- persion force coefficients C(La, Lb) (in a.u.) for ^He- He+, H-Li+ +, He+-He+, Li+ +-Li+ + and He+-Li+ +

for ^{various values} of La ^and Lb from the variational

approach. ^{The notation a} + ^b represents ^{a x} 10:tb.

106

in tables III and IV, ^the dispersion ^force coefficients for

H-He+, H-Li++, He+-He+, ^{Li++-Li++ and} He+- Li+ + interactions for L ⁼ 1 to 3 are obtained as

reported ^{in table V. The} following expressions ^{for the} dispersion energy between interacting pairs ^of H, He+ and Li++ result :

and

Note that a ± b in equs. (14) ^to (18) implies ^{a x 1 O:t b.}

The lower and the _upper bounds of the multipole polarizabilities ^{for the He+ and Li+ +} using [2, 1] ^Pade’

approximants ^are calculated as in our earlier work [3]. ^The dispersion ^force coefficients C(L., Lb) ^for H-He+,

H-Li+ +, He+-He+, ^{Li+ +-Li+ + and He+-Li+ +} interactions, ^for L ⁼ 1 to 3 are tabulated along with the lower limit, ^the upper limit and the mean values of the dispersion force coefficients C(La, Lb) ^{in table VI.} The result-

ing dispersion energy expressions ^{read :}

Note that a + b in equs. (19) ^to (23) implies ^{a x 1 O:t b.}

Table VI.

The values of multipole-multipole dispersion force coefficients C(La, Lb) (in a.u.) for H-He+, H-Li+ +, He+-He+, Li + +-Li+ ⁺ and He+-Li+ ⁺ for ^{various values} of La ^and Lb ⁱⁿ [2, 1 Pade’approximant. ^The superscript I, ^{u and m stand} respectively for ^{the lower} limit, upper limit and the mean value of C(La, Lb). ^{The nota-}

tion a + b represents ^{a x} l0 ±b.

Discussion. The comparison ^{of values of} C(La, Lb) ^{for H-He+ with} those of Bartolotti and

Tyrrel’s [8] ^{are as} in table VII. It should be stressed that Bartolotti and Tyrrel have also utilized the method of hydrodynamic ^{model to quantum mecha-}

nics to obtain C(La, Lb) ^{for such an} interacting pair.

However, ^we have achieved a different convergence value for the multipoles ^{of the} hydrogen ^{and the} helium-plus ion than obtained by them. This _pre- sents two different estimates for C(La, Lb) ^{of the}

H-He+ pair ^{from the same model. We} have listed all the 26 Cauchy ^{moments for the} multipoles ^{of the} hydrogen like ions as these are needed to construct up ^to [13, 12] ^Pade’ approximants ^{to work out the}

oscillator strength ^of hydrogen like ions [9]. Also,

Table VII.

Comparison of the values for C(La, Lb) of ^{the H-He+} pair ; La ^and Lb ^stand for multipoles of ^{H and} multipoles of ^He+ respectively. ^{The nota-}

tion a + b represents ^{a x} 10±b.

the work is in _progress to work out induction effect for the present ^model [10].

References [1] SHUKLA, ^{G. C. and} BARBARO, M., Phys. ^{Rev. 15A} (1977) ^23.

[2] YOUSIF, F. N., TONDON, V. K. and SHUKLA, ^G. C., Phys. ^Rev

17A (1978) ^1269.

[3] YOUSIF, F. N. and SHUKLA, G. C., J. Chem. Phys. ⁷⁰ (1979)

314.

[4] DALGARNO, A., ^Adv. Phys. ² (1962) ^281.

[5] YOUSIF, ^F. N., M. Sci. Thesis, ^Basrah University (1978), unpublished.

[6] LANGHOFF, ^P. W. and KARPLUS, M., Application of ^Pade’

approximants ^to dispersion force ^and optical polarizability computations in the Pade’ Approximant in Theoretical

Physics, ^edited by Baker G. A. Jr. and Gammel J. L.

(Academic, ^New York) ^1970.

[7] BETHE, ^{H. A. and} SALPETER, ^E. E., Quantum ^Mechanics of

one-two-electron atoms, page 9, equ. (2.13) (Springer- Verlag, Berlin) 1957; ^the multipole resonances occur for n = L + 1, n’ = ^1.

[8] BARTOLOTTI, ^{L. J. and} TYRRELL, J., ^Chem. Phys. ^{Lett. 39} (1976) ^19.

[9] SULEMAN, F. I. and SHUKLA, ^G. C., ^to be submitted to J.

Physique.

[10] SHUKLA, ^{G. C.} (under preparation).