ARTICLE pubs.acs.org/JPCA
Asymptotic Exchange Energy of Heteronuclear Dimers C. L. Yiu,*,†,‡ K. T. Tang,‡ and W. G. Greenwood‡ †
Department of Mathematics and ‡Department of Physics, Pacific Lutheran University, Tacoma, Washington, 98447, United States ABSTRACT: A simple expression for the asymptotic exchange energy of heteronuclear dimers is derived from the surface integral method. A five-dimensional hypersurface, consisting of all spherical surfaces centered at the nucleus of the atom with higher ionization energy, more appropriate for the case where the two atoms have different ionization energies, is used in the surface integral. All integrals are carried out analytically. Compared with the exchange energy of Smirnov and Chibisov, which is also obtained from the surface integral method with a hypersurface consisting of all infinite planes perpendicular to the internuclear axis, the present result is much simpler. The exchange energies of alkali hydrides are computed as an illustration. It is shown that the present method and the method of Smirnov and Chibisov are complementary to each other.
1. INTRODUCTION The foundation of the presently accepted explanation for the chemical bond was laid in the year 1927. In that year, the chemical bond of the singlet state of the H2 molecule was successfully described by Heitler and London using a symmetrized wave function.1 The calculation showed that the stabilization of the bond is mainly provided by the exchange integral. The same calculation demonstrated that the exchange integral is also responsible for the repulsive potential of the triplet state. In the asymptotic region, where the overlap integral is vanishing, the exchange integral is equal to half of the exchange energy defined as Ex = Es Et, where Es and Et are, respectively, the energy of the singlet state and the triplet state. Exchange energy is fundamentally important for understanding not only covalent chemical bonds but also polyatomic potential surfaces,2 charge exchange in atomic collisions,3 and magnetism in many-body systems.4 In 1964, Herring and Flicker5 derived a surface integral expression for the exchange energy6 and, with a perturbed wave function, obtained the exact asymptotic exchange energy of H2. The surface integral method corrects a physically unacceptable conclusion of the HeitlerLondon treatment, according to which for H2, the singlet state lies above the triplet state at distances larger than 50 atomic units.7,8 The basic idea of the surface integral method was already developed in 1951 by Firsov9 and independently by Holstein,10 Landau and Lifshitz,11 and Herring12 to treat the analogous problem in H2þ. This approach takes account of the actual physical process of the two electrons in the molecule trading places.2 Regardless of whether one accepts its physical interpretation, the results of the surface integral method should still be valid because they follow from the rigorous mathematical formulation of the multidimensional continuity equation of quantum mechanics. In 1993, we used the surface integral method with the zerothorder product wave function to calculate the exchange energy of the H2 molecule.13 Without any further approximation either in r 2011 American Chemical Society
the formulation or in the evaluation of the integral, we obtained the exchange energy in a closed analytic form. The results closely follow the exact numerical values of Kolos and Wolniewicz14 in the entire range over eight orders of magnitude, and in the asymptotic region, it does not exhibit the incorrect behavior of the symmetrized HeitlerLondon treatment. For systems other than H2, only asymptotic atomic wave functions can be expressed in a closed analytic form.15 Using these asymptotic wave functions, Smirnov and Chibisov16 in 1965 developed a surface integral for the asymptotic exchange energy between any two s-shell electrons. In 1970, Duman and Smirnov17 generalized this surface integral method to multielectron systems. In 1994, we corrected an error of a factor of 4.5 in the result reported by Duman and Smirnov and obtained the exchange energy of He2.18 Adding this exchange energy to the damped dispersion series according to the TangToennies potential model19 resulted in a remarkably accurate He2 potential. To apply the theory to systems with p and higher angular momentum states, the angular momentum coupling factor of Duman and Smirnov17 was reexamined and was found to reduce to a simple counting procedure, consistent with the idea that exchange occurs only between two electrons with the same spin.20 The asymptotic wave function of the valence electron of any atom can be expressed in terms of the Whittaker function.21 If it is approximated by its leading term, which is independent of l, it will behave like an s electron. Thus, in principle, it is possible to predict the asymptotic exchange energy for any two-atom system, as long as we know the exchange energy of two s electrons. This was demonstrated in the case of homonuclear rare gas dimers.22
Special Issue: J. Peter Toennies Festschrift Received: February 1, 2011 Revised: May 18, 2011 Published: May 22, 2011 7346
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In ref 16, there are exchange energy expressions of two s electrons for both homonuclear and heteronuclear dimers. However, the expression for heteronuclear dimers is very complicated and is valid in the case where the ionization energies of the two atoms are close. In this paper, we will derive an exchange energy expression with a different hypersurface that is appropriate for the case where the ionization energies of the two atoms are significantly different. From the surface integral over this new five-dimensional hypersurface, a simple analytic expression is obtained for the asymptotic exchange energy of heteronuclear dimers. We will use an example to explicitly demonstrate that good numerical values of the exchange energy can be obtained with this simple expression. All numerical values in this paper are in atomic units unless explicitly stated otherwise.
2. PHYSICS AND MATHEMATICS OF EXCHANGE ENERGY For a system of two atoms, each with one valence electron, the asymptotic exchange energy ΔE is defined as the energy difference between the singlet and the triplet states when the internuclear distance R is much greater than the individual atomic radii. Let r6 be the Laplacian operator in the six-dimensional (r1,r2) space, where r1 and r2 are the position vectors of the two valence electrons. With r1 and r2 given by the regular threedimensional Laplacian for electron 1 and electron 2, the sixdimensional gradient operator can be defined as r = (r1,r2). With these notations, the asymptotic exchange energy can be computed by the following surface integral5,6 Z ΔE ¼ 2 ðPΦÞrΦ 3 n ^ ds5 ð1Þ Σ
where Σ is a five-dimensional hypersurface in the six-dimensional space (r1,r2); P is the exchange operator such that P(r1,r2) = (r2,r1); Φ is what Herring called the “localized wave function” defined as 1 Φ ¼ pffiffiffi ðΦs þ Φt Þ 2
ð2Þ
where Φs and Φt are singlet and triplet wave functions, respectively. Equation 1 is based on the Pauli principle applied to the two valence electrons. Therefore, it holds for either homonuclear or heteronuclear systems. This equation was explicitly derived from the six-dimensional Gauss theorem for the H2 system in ref 13. It is easily seen that all steps are equally valid in the present case. The first step in the application of eq 1 is necessarily the choice of the hypersurface Σ. In principle, the choice of this surface can be arbitrary. In practice, its choice is closely bound up with the symmetry of the localized wave function Φ or the lack of it. In the case of homonuclear systems, the choice is based on the physical picture that as the interatomic distance R becomes much greater than the atomic radius (i.e., in the asymptotic region), the surface of equal charge density of a pair of valence electrons becomes approximately an ellipsoid, and at the same time, the charge density decreases exponentially as the size of the ellipsoid increases. Consequently, the ellipsoid with significant charge density virtually collapses into the line joining the two atomic nuclei, confining the valence electrons to face each other. This is where the exchange interaction occurs. This is also why the
Figure 1. Coordinate system for two electrons. Nuclei A and B, with ionization energies (1/2)R2 and (1/2)β2, respectively, are separated by internuclear distance R. The polar angles are shown as θ1 and θ2. The electrons also have independent azimuthal angles that are suppressed in the two-dimensional diagram. The electron coordinates are centered on the nucleus of larger ionization energy, so that β is presumed to be larger than R.
exchange interaction is essentially reduced to a one-dimensional problem, at least so long as the differential equation for the correction to the localized wave function is concerned. The key role of this internuclear line, as well as the symmetry of the diatomic system, suggests that the most convenient choice of Σ is a family of planes perpendicular to the internuclear line. Clearly, this choice of Σ is not appropriate for a heteronuclear system because there is no longer symmetry between the atoms. As will be shown, in the asymptotic region, the equal-chargedensity surfaces with significant charge density are approximately spheres centered around the atom with larger ionization energy, in contrast to the internuclear line for the homonuclear system. An appropriate choice of Σ should therefore coincide with this family of spheres. Yet, there is another physical aspect that distinguishes the homonuclear system from the heteronuclear system. In the asymptotic region, the zeroth-order approximation of the localized wave function Φ can be represented by the London approximation, namely, the product of two unperturbed wave functions of the valence electrons. In the homonuclear system, the constricted equal-charge-density surfaces force the electrons together, causing the necessity to take into account the modification of the London approximation.23 However, in a heteronuclear system, the electrons have ample space to avoid each other. Therefore, we expect the modification due to the electron electron interaction to be not as significant. This was demonstrated in our previous calculations of the H2 system.13 The exchange energy calculated from the surface integral with the London approximation is about 20% too large compared with the exact value at the equilibrium distance of the triplet state (8a0) because of the lack of correlation. However, at the equilibrium distance of the chemical bond (1.4a0), the exchange energy calculated from the surface integral with the London approximation is only 0.7% smaller than the exact value. This is because at such small distances, the two electrons are not confined into a line to face each other. Therefore, we shall use only the London approximation as our localized wave function in our calculation.
3. COMPUTATION OF THE ASYMPTOTIC EXCHANGE ENERGY We shall compute the asymptotic exchange energy of a system consisting of two atoms A and B. The ionization energy 7347
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of atom A is (1/2)R2, and that of B is (1/2)β2. We assume β > R. We place the nucleus of atom A at the point A and the nucleus of atom B at the point B, both on the z-axis, as shown in Figure 1. The vector pointing from A to B is denoted as R. The vectors ri and the angle θi (i = 1,2 and 0 e θi e π) are the position vectors and the polar angles of the two electrons relative to the origin O, which is at the nucleus of atom B. The localized wave function of this system is Φðr1 , r2 Þ ¼ jR ðR þ r1 Þ 3 jβ ðr2 Þ
ð3Þ
Therefore the five-dimensional surface integral of eq 1 can be written as Z Z Z ΔE ¼ 2
ð4Þ
Aβ jβ ðr2 Þ ¼ pffiffiffiffiffiffijr2 jð1=βÞ 1 eβjr2 j 4π
ð5Þ
ðPΦÞ
∂Φ 1 ð2πÞ2 u4 du 3 sin θ1 dθ1 3 sin θ2 dθ2 ∂v 4
ð11Þ Because the dominant contribution to the asymptotic exchange energy comes from the vicinity of atom B, we expect that the nonvanishing contribution to the surface integral in eq 1 mainly comes from the region where r1 , R. This condition enables us to make the approximation
where jR (R þ r1) and jβ(r2) are the atomic wave functions of atom A and atom B, respectively. Asymptotically AR jR ðR þ r1 Þ ¼ pffiffiffiffiffiffijR þ r1 jð1=RÞ 1 eRjR þ r1 j 4π
v¼0
and to the lowest order of 1/R jR ðR þ r1 Þ = jR ðRÞeRr1
and
R|Rþr1|β|r2|
The factor e dominates the behavior of the function Φ, as given by eq 3. The surface RjR þ r1 j þ βjr2 j ¼ constant
ð12Þ
jR þ r1 j ¼ ðR 2 þ 2R 3 r1 þ r12 Þ1=2 R þ r1 cos θ cos θ1
Thus ¼ jR ðRÞeRr1
Φðr1 , r2 Þ
Aβ ð1=βÞ 1 βr2 cos θ1 p ffiffiffiffiffiffi r2 e 4π
pffiffi pffiffi Aβ ¼ jR ðRÞpffiffiffiffiffiffi eR cos θ1 ðu þ vÞ= 2 βðu vÞ= 2 ðu vÞð1=βÞ 1 2ð1=2Þðð1=βÞ 1Þ 4π
ð13Þ is the equal-charge-density surface. We define two new variables u and v as 1 u ¼ pffiffiffi ðr1 þ r2 Þ 2
1 v ¼ pffiffiffi ðr1 r2 Þ 2
ð6Þ
∂Φ ∂v
v¼0
1 pffiffiffi 2 dr1 dr2 ¼ 1 pffiffiffi 2
)
)
With the Jacobian determinant, we have 1 pffiffiffi 2 du dv ¼ du dv 1 pffiffiffi 2
ð7Þ
Thus, the volume element in the six-dimensional (r1,r2) space is dV
¼ ¼
ð2πÞ2 r12 dr1 sin θ1 dθ1 3 r22 dr2 sin θ2 dθ2 1 ð2πÞ2 ðu2 v2 Þ2 du dv 3 sin θ1 dθ1 3 sin θ2 dθ2 4 ð8Þ
Now let the hypersurface Σ be v = 0, or r1 = r2 (denoted as r). Following Herring and Flicker,5 we call the part of the 6-D space that is separated by the surface Σ and that contains the point (r1, r2) = (R,0) the near-side. It is considered as the “inner” part of the surface Σ. Because at this point v = (1/21/2)(R 0) > 0, the normal vector n ^ of Σ points in the direction of decreasing v. Hence, in eq 1 ∂ n ^3r ¼ ∂v 1 ds ¼ ð2πÞ ðu2 v2 Þ2 du 3 sin θ1 dθ1 3 sin θ2 dθ2 4 2
PΦðr1 , r2 Þ
ð10Þ
¼ jR ðRÞeRr2
Aβ ð1=βÞ 1 βr1 cos θ2 p ffiffiffiffiffiffi r1 e
4π
v¼0
Aβ ¼ pffiffiffiffiffiffijR ðRÞeRr cos θ2 βr r ð1=βÞ 1 4π
v¼0
ð15Þ
In the last two equations, we have used the fact u = 21/2r. With these expressions, eq 11 becomes ΔE ¼ 2π½jR ðRÞ2 A2β K
ð16Þ
with Z K ¼
¥
r
ð1=βÞ þ 3 2βr
(Z
e
1
1
0
ð9Þ
Also, from dV, we obtain
Aβ 1 ¼ jR ðRÞpffiffiffiffiffiffi2ð1=2Þðð1=βÞ 1Þ pffiffiffiðR cos θ1 βÞuð1=βÞ 1 4π 2 pffiffi 1 1 uð1=βÞ 2 eðR cos θ1 þ βÞu= 2 β Aβ n ¼ jR ðRÞpffiffiffiffiffiffi ðR cos θ1 βÞr ð1=βÞ 1 8π 1 ð1=βÞ 2 1 r ð14Þ eðR cos θ1 þ βÞr β
#
eRry dy
ð1=βÞ 2 1 1 eRrx dx β
Z
1
1
h
ðβ RxÞr ð1=βÞ 1
) dr
ð17Þ
where x = cos θ1 and y = cos θ2. These integrals can be carried out analytically; the details are shown in the Appendix. The final result is a relatively simple 7348
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expression # " 4π 2 1 1 A2 ½jR ðRÞ2 ΔE ¼ ð1=βÞ Γ 2 Rβ β ðβ RÞð2=βÞ þ 1 ðβ þ RÞð2=βÞ þ 1 β
ð18Þ
4. EXCHANGE ENERGY OF ALKALI HYDRIDES As illustrations, we will compute the asymptotic exchange energy of CsH and LiH. For CsH, the ionization energy of cesium Ei(Cs) is 0.143 au, and that of hydrogen Ei(H) is 0.5 au. Therefore pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi β ¼ 2Ei ðHÞ ¼ 1 ð19Þ R ¼ 2Ei ðCsÞ ¼ 0:535 For the amplitude of the asymptotic wave function, we use the BatesDamgaard normalization24 AR ¼
ð2RÞ1=R ðRÞ1=2 ¼ 0:467 1 þ1 Γ R
Aβ ¼
ð2βÞ1=β ðβÞ1=2 ¼2 1 Γ þ1 β
ð20Þ
ð21Þ
Therefore, eq 18 becomes " # 8π 1 1 AR ð1=RÞ 1 RR 2 p ffiffiffiffiffi ffi R ΔEðCsHÞ ¼ e R ð1 RÞ3 ð1 þ RÞ3 8π ¼ 7:884R 1:738 e1:07R
ð22Þ This exchange energy is shown as the solid line in Figure 2a. For LiH, the ionization energy of lithium Ei(Li) is 0.198 au. Therefore pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð23Þ R ¼ 2Ei ðLiÞ ¼ 0:630 AR ¼
ð2RÞ1=R ðRÞ1=2 ¼ 0:808 1 þ1 Γ R
ð24Þ
Figure 2. (a) Exchange energy of CsH. Present results from eq 18 (solid line) are presented together with the ab initio results of Geum et al.25 (crosses) and the results of Smirnov and Chibisov16 obtained by a direct integration of eq 26 (triangles). The dashed line represents the power expansion of Smirnov and Chibisov shown in eq 27. Corresponding results are displayed for LiH in (b).
where JðR, β, RÞ
Therefore
¼ A2R A2β 22ð2=ðR þ βÞÞ Γ
ΔEðLiHÞ ¼ 40:44R 1:175 e1:26R
ð25Þ
This exchange energy is shown as the solid line in Figure 2b. For these systems, ab initio calculations using a large-scale multireference configuration interaction method were carried out by Geum et al.25 for the potential energies of the lowest singlet and triplet states. The exchange energies as the difference between these ab initio numerical values are shown in Figure 2 as crosses for both CsH (Figure 2a) and LiH (Figure 2b). It is seen in the figure that the present asymptotic results agree with these ab initio calculations very well for internuclear distances greater than 10 au, especially for CsH. According to Smirnov and Chibisov,16 the exchange energy of the heteronuclear dimer is given by ΔE ¼ R ð2=RÞ þ ð2=βÞ ð1=ðR þ βÞÞ 1 eðR þ βÞR JðR, β, RÞ ð26Þ
1 2 2 þ ð1=ðR þ βÞÞ ½HðR, β, RÞ þ Hðβ, R, RÞ Rþβ Rþβ
R þ β ð2=RÞ ð2=ðR þ βÞÞ HðR, β, RÞ ¼ 2β Z 1 eððy 1Þ=βÞ þ Rðβ RÞy ð1 yÞð2=βÞ ð1=ðR þ βÞÞ 0
β R 2 ð1=ðR þ βÞÞ y ð1 þ yÞð2=RÞ ð2=βÞ þ ð1=ðR þ βÞÞ 1 þ dy βþR Because this is very complicated, they suggested to expand the integral J in powers of R(R β) 1 JðR, β, RÞ ¼ J0 ðR, βÞ þ Rðβ RÞJ1 ðR, βÞ þ R 2 ðβ RÞ2 J2 ðR, βÞ 2
ð27Þ 7349
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The Journal of Physical Chemistry A The values of the integrals J0, J1, and J2 are given to be J0 ¼ 4:96 J2 ¼ 0:92 J0 ¼ 20:1 J2 ¼ 3:52
J1 ¼ 1:05 for the CsH interaction J1 ¼ 3:00 for the LiH interaction
By a direct numerical integration of eq 26, we have calculated the exchange energies of CsH and LiH. They are shown in Figure 2 as triangles. The dashed lines in Figure 2 represent eq 27, which are the approximate power expansion of Smirnov and Chibisov. In the homonuclear case, we expect that the dominant contribution to the exchange energy will come from the ellipsoidal region between the nuclei because the electrons’ probability distribution is mainly concentrated along the internuclear axis, as noted previously by Gor’kov and Pitaevski.6 In the heteronuclear case, however, we expect the electrons to be attracted preferentially toward the side with larger ionization energy, and the imbalance occurs quickly as the ionization energies become disparate. For systems involving hydrogen and an alkali, such as CsH, where R and β differ by nearly a factor of 2, the result is that the electrons become concentrated near to the hydrogen nucleus, approaching a spherical distribution that is treated most naturally by our coordinate system that is centered on the hydrogen nucleus. On that basis, we can expect our results to agree more closely with ab initio calculations for CsH than those for LiH, and that is indeed the case in Figure 2. The theory of Smirnov and Chibisov,16 however, is based on a cylindrical geometry centered along the line between the nuclei and patterned after the model of Herring and Flicker.5 That geometry could be expected to apply more naturally to the case of nuclei with more similar ionization energies, where the electrons are concentrated mainly in a narrow cylinder around the internuclear axis. That the ab initio results agree with the calculations of Smirnov and Chibisov more closely for LiH than those for CsH represents the complementary character of that theory and the present work.
5. SUMMARY AND DISCUSSION In this paper, we have derived a simple analytic expression for the asymptotic exchange energy of a heteronuclear diatomic moelecule. The expression is derived from the surface integral formulation of the exchange energy. On the basis of the understanding that in a heteronuclear dimer, most charge density in the asymptotic region is preferentially distributed around the nucleus of the atom with higher ionization potential, a new five-dimensional hypersurface consisting of all spherical surfaces centered around this nucleus is introduced in the surface integral. This hypersurface is different from the conventional one used in the surface integral method. With this new hypersurface and the asymptotic approximation of the wave functions, the integral can be carried out analytically. A comparison with the multireference CI calculations for the alkali hydrides suggests that the present analytic expression is highly accurate, especially in cases where the constituent atoms have very different ionization potentials. The functional dependence of the present results of eq 18 can be written as CFR (R), where FR(R) is the asymptotic charge density of the atom with a smaller ionization potential
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and C is the amplitude. This functional form is in agreement with several previous studies. In the so-called Fermi formula, C is expressed in terms of scattering length. 26,27 In 1991, Patil used symmetrized perturbation theory to study the interactions of alkali rare gas systems and expressed C in terms of a combination of various expectation values of rare gases.28 More recently, Geum et al. approximated C with an empirical constant independent of R and β for the exchange energy of alkali hydrides.25 In all of these expressions, the R dependence is given by FR(R). It is gratifying to see that the present surface integral results provide not only an explanation of the functional form but also an explicit expression for the amplitude C. All of these expressions do not reduce to the results for homonuclear dimers in the case of equal ionization potentials. In the present case, the problem is not due to the surface integral formulation but to the approximation that we introduced in eq 12. This approximation is based on the fact that for β > R, the charge density is mostly around B, that is, r1 , R. However, for R = β, this is not the case, and this approximation makes the interal diverge. (See the first integral of eq 32.) Without this approximation, the five-dimensional integration has to be carried out numerically. Such a calculation would be even more complicated than the formula of eq 26. In the future, we plan to test eq 18 further by examining the exchange energy of the alkali rare gas interactions. For such systems, the difference between R and β is even larger than that of alkali hydrides. It would be interesting to see if the angular momentum coupling factor discussed in the Introduction is still valid for these heteronuclear systems.
’ APPENDIX To evaluate the K integral of eq 17, let us first define Z 1 1 eRry dy ¼ ðeRr eRr Þ Y ðrÞ ¼ Rr 1
ð1=βÞ 2 # 1 1 eRrx dx ðβ RxÞr β 1 ð1=βÞ 2 Z 1 1 1 ðβ RxÞeRrx dx Y ðrÞ r 1=β 1 β 1
Z XðrÞ
¼ ¼
1
"
ð28Þ
ð1=βÞ 1
ð29Þ By change of variable t = β Rx Z 1 Z eβr β þ R rt Rrx ðβ RxÞe dx ¼ te dt R βR 1 This integral can be evaluated with integration by parts. Putting the result into X(r), we have 1 Rr fe ½ðβ þ RÞr 1 Rr 2 ð1=βÞ 2 1 Rr 1 e ½ðβ RÞr 1g Y ðrÞ ð30Þ β
XðrÞ ¼ r ð1=βÞ 1
Now, the integral K of eq 17 can be written as Z ¥ K ¼ r ð1=βÞ þ 3 e2βr Y ðrÞXðrÞ dr
ð31Þ
0
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The integrand can be simplified somewhat, and the integral becomes
Z ¥ Z 1 ¥ ð2=βÞ 1 2ðβ RÞr R2 K ¼ ðR þ βÞ r 2=β e2ðβ RÞr dr r e dr β 0 0 Z ¥ Z ¥ 2 2β r 2=β e2βr dr þ r ð2=βÞ 1 e2βr dr β 0 0 Z Z ¥ 1 ¥ ð2=βÞ 1 2ðβ þ RÞr r2=β e2ðβ þ RÞr dr r e dr þ ðβ RÞ β 0 0
ð32Þ 29
Using the identity Z ¥
xν 1 eμx dx ¼
0
1 ΓðνÞ μν
ð33Þ
the six integrals of the last equation can be written as 2 2 Γ þ1 Γ 1 β β R2 K ¼ ðR þ βÞ ½2ðβ RÞð2=βÞ þ 1 β ½2ðβ RÞ2=β 2 2 Γ þ1 Γ 2 β β 2β þ ð2=βÞ þ 1 β ½2β ½2β2=β 2 2 Γ þ1 Γ 1 β β þ ðβ RÞ ð34Þ ½2ðβ þ RÞð2=βÞ þ 1 β ½2ðβ þ RÞ2=β
(13) Tang, K. T.; Toennies, J. P.; Yiu, C. L. J. Chem. Phys. 1993, 99, 377. (14) (a) Kolos, W.; Wolniewicz, L. J. Chem. Phys. 1965, 43, 2429. (b) Kolos, W.; Wolniewicz, L. Chem. Phys. Lett. 1974, 24, 437. (15) Patil, S.H.; Tang, K.T. Asymptotic Methods in Quantum Mechanics; Springer-Verlag: Berlin, Germany, 2000; Chapter 2. (16) Smirnov, B. M.; Chibisov, M. I. Sov. Phys. JETP 1965, 21, 624. (17) Duman, E. L.; Smirnov, B. M. Opt. Spectrosc. USSR 1970, 29, 229. (18) Tang, K. T.; Toennies, J. P.; Yiu, C. L. Phys. Rev. Lett. 1995, 74, 1546. (19) Tang, K. T.; Toennies, J. P. J. Chem. Phys. 1984, 80, 3726. (20) Kleinekathofer, U.; Tang, K. T.; Toennies, J. P.; Yiu, C. L. J. Chem. Phys. 1995, 103, 6617. (21) Bardsley, I. N.; Holstein, T.; Junker, B. R.; Sinka, S. Phys. Rev. A 1975, 11, 1911. (22) Kleinekathofer, U.; Tang, K. T.; Toennies, J. P.; Yiu, C. L. J. Chem. Phys. 1997, 107, 9502. (23) Carr, W. J., Jr.; Ashkin, M. J. Chem. Phys. 1965, 42, 2796. (24) Bates, D. R.; Damgaard, A. Philos. Trans. R. Soc. London, Ser. A 1949, 242, 101. (25) Geum, N; Jeung, G. H.; Derevianko, A.; C^ote, R.; Dalgarno, A. J. Chem. Phys. 2000, 115, 5984. (26) Fermi, E. Nuovo Cimento 1934, 11, 157. (27) Smirnov, B.M. Physics of Atoms and Ions; Springer-Verlag: New York, 2003; p 270. (28) Patil, S. H. J. Chem. Phys. 1991, 94, 8089. (29) Gradshteyn, I.S.; Ryzhik, I.M. Table of Integrals, Series, and Products, 4th ed.; Academic Press: New York, 1965; p 317.
By the definition of the gamma function Γ(x þ 1) = xΓ(x), this expression can be simplified as 2 2 Γ Γ 2R 2R β β R2 K ¼ βðβ RÞ ½2ðβ RÞ2=β βðβ þ RÞ ½2ðβ þ RÞ2=β ð35Þ It follows
# " 2 2 1 1 K ¼ 1=β Γ 2 Rβ β ðβ RÞð2=βÞ þ 1 ðβ þ RÞð2=βÞ þ 1 ð36Þ
This is the result shown in eq 18.
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dx.doi.org/10.1021/jp2010925 |J. Phys. Chem. A 2011, 115, 7346–7351
