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Modeling Molecular Interactions by Analytic Potentials:Analytic Perturbation Treatment∗Eugene S. Kryachko
One of the chief concepts of molecular interactions is the con-cept of the molecular potential. This work focuses on the classof so-called spiked potentials. It is analytically demonstrated thatdespite the common belief that the traditional perturbation the-ory is not applicable to study the class of potentials combiningthe harmonic oscillator with a singular potential of the type λr−α ,
this work aims to propose and develop such treatment on thebasis of the appropriate choice of the zero-order exactly solvableHamiltonianHo = −(�2/(2m))∇2
r +U2 whereU2(r) = αr2+β/r−2.© 2012 Wiley Periodicals, Inc.
DOI: 10.1002/qua.24136
All of what is scientific in chemistry is physics— the rest is cooking.
L.D. Landau[1]
Preamble: Instead of Routine Introduction
From time to time, each scientist feels him(her)self to enable andeven, to some extent, duty to have a look at the very roots† in thatarea of science where this scientist suggests that he has a right tosay something nontrivially worth. From time to time, everybodyinternally feels such need to highlight a root from his own angleof view, at least to highlight its tiny part. Generally speaking, thisis actually a prerogative of articles only, rather than reviews andmonographs where the author attempts and intends to presentthe material in a unique manner. However, the view points ofthe author and the readers on this matter may diverge, evenquite essentially. It is primarily concerned this Festschrift articlethat author intends to say something quite nontrivial regardingthe jubileer, nontrivial from his point of view that by no meansdoes not imply that this is nontrivial for the readers too. IlyaKaplan grew in the 50s and 60s of the previous century‡ when,at least in the USSR, the air was full of physics, of theoreticalphysics in particular, with the icons, such as Lev D. Landau andVladimir A. Fock, and many others. Kaplan’s supervisor of his PhDwas Alexander S. Kompaneets (1914–1974) who was the “firstpupil of L. D. Landau” [1, p. 323] because he was the first whocompletely passed the famous Landau’s theorerical-minimumexam. Kompaneets was well known for his generalization of theThomas–Fermi equation on inhomogeneous systems that wasthe basis of the modern density functional theory.
The Heydays: Molecular Interactions
Some people think that there is an air betweenmolecules.
A. S.Kompaneets[2]
Molecular interactions hold stable the molecular world that sur-rounds us. As a pupil of Kompaneets, I. Kaplan always attempted
†Postulate:There always exists a root to any nontrivial problem.‡See the related memoirs by I. Kaplan in this issue.
to demonstrate that the space between molecules where molec-ular interactions in fact operate is not empty.[3] This is clearlyseen in his books[4, 5] (see also the review[6]). As follows fromthe title of Kaplan’s recent book, one of the chief concepts ofmolecular interactions is the concept of the molecular potential.The problem of its modeling, that is, its description by someanalytical formula, is definitely the key problem in this area,which started by Boscovich [5, p. 5] who modeled the interac-tion between molecules at large intermolecular distances r bya Newton’s law, r−2. Another familiar model potentials are theharmonic one obeying the law r2 and the Coulomb, r−1. The lat-ter describes the interaction of two charged particles. Togetherwith the former, it composes a so-called Hookean molecularmodel.[7] In fact, the potential of the Hookean model is referredto so-called spiked-type potential,[8, 9] which belongs to the classof model potentials treated below.
Spiked Potential
Trivialize the problem.
L.D. Landau
Consider a 3D Schrödinger equation Hp� = E� , where Hamil-tonian Hp = −(�2/(2m))∇2
r + Up and Up is a spiked-typepotential[8, 9]
Up(r) = αr2 + β
rp(1)
where α = mω2/2 β > 0, p > 0. Its minimum,
Up
(r(p)o
) = (2α)p
p+2 (β)2
p+2
[1
2p
2p+2 + p
− pp+2
](2)
E. S. Kryachko
Bogoliubov Institute for Theoretical Physics, Kiev 03680,Ukraine
E-mail: [email protected]
∗Faithfully dedicated to Ilya Kaplan on the occasion of his 80th birthday.
© 2012 Wiley Periodicals, Inc.
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this potential reaches at
r(p)o =[pβ
2α
] 1p+2
. (3)
Obviously,
limp→∞ r(p)o = 1, lim
p→∞Up
(r(p)o
) = α, limp→∞
d2Up
dr2
∣∣∣∣r(p)o
= ∞. (4)
that implies that if the spiked component of Up(r) or simply, aspike is singular enough, or, equivalently, if p is sufficiently large,the harmonic force constant of Up(r) at its minimum becomeslarge as well. Spike also contributes to shifting the minimum ofthe potential, by location, from r = 0 to r = 1, and by magnitude,from zero upward to α determined in Eq. (1). It is also noteworthythat
limp→∞
dr(p)o
dp= 0 and lim
p→∞(−1)nUnp
(r(p)o
) = ∞ (5)
where Unp(r) designates the nth derivative of Up(r) with respect
to r.On the other hand, approaching β to zero results in r
(p)o goes to
zero and Up(r(p)o ) tends to zero, too. However,Up(r
(p)o ) approaches
then 2α(p + 2), and
lim(−1)nU(n)p
(r(p)o
) = ∞. (6)
Equation (6) fully explains the Klauder’s phenomenon[10–14] con-sisting in that a spiked-like perturbation does not converge tothe original harmonic potential, as β goes to zero with finite p
(see Ref. [13]). In particular, for p = 4, one easily derives that
r(4)o = (2β/α)1/6, U2
(r(4)o
) = 3(α/2)2/3β1/3. (7)
A standard representation of the total eigenwavefunction �(r) =[ψ(r)/r]Ylm(θ , φ) transforms the 3D Scrödinger equation into the1D one,
d2ψ
dr2+[e − Ar2 − B
rp− l(l + 1)
r2
]ψ(r) = 0, (8)
where e = 2mE/�2,A = 2mα/�
2 = (mω/�)2, and B = 2mβ/�2.
We seek eigenfunctions of Eq. (8) within the following Ansatz:
ψ(r) = raGν(γ r−q)φ(r), (9)
that obeys the Dirichlet-type condition: �(0) = 0, andapproaches zero at infinity. In Eq. (9), Gν(x) is a modified Besselfunction of index ν and argument x = γ r−q that satisfies thefamiliar equation[15]
x2G′′ν + xG′
ν − (x2 + ν2)Gν = 0. (10)
Here, the prime indicates the first derivative of the functionrelative to its argument.
Substituting ψ of Ansatz (9) into Eq. (8) and taking Eq. (10)into account, one derives the Schrödinger-type equation for theunknown function φ(r),
d2φ
dr2Gν + dφ
dr
(2a
rGν − 2γ q
rq+1G′
ν
)+ φ
[(e − Ar2 − B
rp
− l(l + 1) − a(a − 1) − ν2q2
r2+ γ 2q2
r2(q+1)
)Gν − γ q
2a − 1
rq+2G′
ν
]= 0.
(11)
There appear two ways of choosing parameters a, ν, γ , and q
dedicated largely by a manner of handling the correspondingintegrals. The former way relies on that
q = p − 2
2, γ = 2
√B
p − 2, ν = 1
q
√1
4+ l(l + 1), a = 1
2.
(12)
This converts Eq. (11) into the following,
d2φ
dr2Gν + 1
r
dφ
dr
(Gν − 2
√B
r(p−2)/2G′
ν
)+ φ(e − Ar2)Gν = 0. (13)
Thanks to the Dirichlet condition imposed on ψ(r),Gν(x)
becomes identified with Kν(x) for p > 2 (q > 0). Using Eq.(9.6.26) of Ref. [15], one obtains
Kν − 2√B
rp−2
2
K ′ν = 2
√B
rp−2
2
Kν+1 +(
1 − 2
√l(l + 1) + 1
4
)Kν (14)
and then rewrites Eq. (13) as
r2 d2φ
dr2+ r
dφ
dr
[(1 −
√l(l + 1) + 1
4+ 2
√B
rp−2
2
Kν+1
Kν
]
+ φ(e − Ar2)r2 = 0, (15)
recalling at this moment that Kν(x) has no real zeros (see, e.g.,Ref. [15]).
Another choice of parameters is predetermined by a value ofν. Let ν = 1/2. Hence, from Eq. (11) it follows
q = p − 2
2, γ = 2
√B
p − 2, a = 1
2(1 + [1 − q2 + 4l(l + 1)]1/2).
(16)
Equation (11) becomes then rewritten in such form (p > 2),
r2 d2φ
dr2+ r
dφ
dr
[2(a − q
2
)+ 2
√B
rp−2
2
Kν+1
Kν
]
+ φ
[(e − Ar2)r2 − (2a − 1)q
2+ (2a − 1)
√B
rp−2
2
Kν+1
Kν
]= 0.
(17)
2 International Journal of Quantum Chemistry 2012,DOI: 10.1002/qua.24136 http://WWW.CHEMISTRYVIEWS.ORG
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Rotational Eigenstates
For a given choice of parameters in Eq. (9) and l = 0, one obtainsν = 1/(p − 2) and simplifies Eq. (12),
r2 d2φ
dr2+ r
dφ
dr
2
√B
rp−2
2
K p−1p−2
K 1p−2
+ (e − Ar2)r2φ = 0. (18)
Let us solve Eq. (18) perturbatively. If B is small enough, the term
Zp(r) = − 2√B
rp−2
2
K p−1p−2
(γ r−q)
K 1p−2
(γ r−q)(19)
can be reasonably approximated, at least for the ground andlower-lying excited eigenstates of Hp, by its value at the minimumof the potential Up(r). Introducing xo = γ (r
(p)o )−q and using only
the first term in the familiar expansion series of Kν(x) (see Ref.[15]), one finds
Zp(r) ≈ Zp(r(p)o
) ≈ Z [o]p
(r(p)o
) =
=2√B sin
(p−1p−2 π
)(r(p)o
) p−22 sin
(π
p−2
) ·[xo2
]−(1+ν)/�(−ν)[
xo2
]−ν/�(1 − ν)
= −2(p − 2) sin
(p−1p−2 π
)�(p−3p−2
)sin(
πp−2
)�(− p−1
p−2
) = −2, (20)
which is independent on p. Involving of the next term for Kν(x)
gives
Z [1]p
(r(p)o
) = −2
1 + B
2p(p−1)
p2−4 π(p−2)
p−4p−2 A
p−1p+2
sin(
πp−2
)(p−1)
(p2
) p−1p+2 �2
(1
p−2
)
1 − B
2pp2−4 π(p−2)
p−4p−2 A
1p+2
sin(
πp−2
)(p2
) 1p+2 �2
(1
p−2
). (21)
For p = 4 one derives from Eq. (21) that
Z[1]4
(r(4)o
) = −21 + B
3
(A
2
)1/2
1 − B1/3(A
2
)1/6 ≈ Z[1]4 = −2
[1 + B1/3
(A
2
)1/6]
.
(22)
One may easily find that in this case of p = 4, Z [1]4 given by Eq.
(22) is precisely equal to the exact value Z[1]4 (r
(4)o ).
Within the zeroth-order approximation for Zp(r) outlined byEq. (20), Eq. (18) takes the form
r2 d2φ[o]
dr2+ 2r
dφ[o]
dr+ (e[o] − Ar2)r2φ[o] = 0. (23)
(23) coincides with Eq. (2.273.4) of Ref. [16] if b = c = 0, β = 2,a = −3/2, m = 1/4, α = √
A, and k = e[o]/4√A (in the notations
of Ref. [16]; see also Ref. [17, 18]). The eigenvalue problem (23)is thus resolved: the eigenfunctions regular at the origin are
φ[o]n (r) = exp
(−√
Ar2
2
)L
(12
)n (
√Ar2), n = 0, 1, . . . , (24)
where L(α)n (x) is a generalized Laguerre polynomial, with the
associated energy eigenvalues,
e[o]n = √
A(4n + 3), E[o]n = �ω
(2n + 3
2
), n = 0, 1, . . . . (25)
Remark. There exists another branch of eigenfunctions, irregu-lar at the origin,
ψ [o]n (r) = 1
rexp
(−√
Ar2
2
)L
(− 1
2
)n (
√Ar2), (26)
E[o]n = �ω
(2n + 1
2
), n = 0, 1, . . . . (27)
By analogy one straightforwardly obtains the first-order eigen-functions,
φ[1]n,p(r) = exp
(−√
Ar2
2
)L
1+Z
[1]p
2
n (√Ar2), (28)
and first-order energy eigenvalues,
E[1]n,p = �ω
(2n + 1 − Z
[1]p
2
), n = 0, 1, . . . . (29)
There is, however, one remarkable feature of the developed per-turbation treatment. It is related with appearance of −1 in the bthexpansion series of Zp(r).This quantity drastically changes the typ-ical level pattern inherent for 3D harmonic oscillator and makesimpossible to obtain it perturbatively by simple approaching B tozero.This conclusion sounds in accord with the arguments of Refs.[8b,10], however, it does not mean that the ordinary perturbationtheory is not applicable for Up(r) at all. Even contrary, it is the goalof this work to demonstrate that it can be done at the level of φthcomponent of the total wave function, with the initial zero-orderHamiltonian,
h(o)φ = d2
dr2+ 2
r
d
dr− Ar2, (30)
presented in Eq. (23).This section is ended by presenting the zero-and first-order total wave functions,
[0]n,p(r) = C[0]
n,pr−1/2K 1
p−2
(2√B
p − 2r− p−2
2
)exp
(−√
Ar2
2
)L
(12
)n (
√Ar2)
(31)
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and
[1]n,p(r) = C[1]
n,pr−1/2K 1
p−2
(2√B
p − 2r− p−2
2
)
× exp
(−√
Ar2
2
)L
− 1+Z
[1]p
2
n (√Ar2) (32)
where C[1]n,p is normalization constant (s = 0, 1).
Approximate Wave Functions with l = 0 andp = 4
Let begin this section with evaluating
d2
dr2[r1/2Kν(x)φ(x)]
= r1/2
{d2φ
dr2Kν(x) + 2qγ
rq+1Kν+1
dφ
dr+ q2γ 2
r2(q+1)Kνφ
}(33)
Hence, one obtains
d2
dr2[r1/2Kν(x)φ(x)] −
(Ar2 + B
rp
)r1/2Kν(x)φ(r)
= Kν
r3/2
{[r2 d
2φ
dr2− rZ [s]
p
dφ
dr− Ar4φ
]− r[Zp(r) − Z [s]
p (r)]dφ
dr
}(34)
where s = 0, 1. Therefore, the first-order energy of theapproximate wave function
[s]n,p is
[1]E[s]n,p =
⟨
[s]n,p | Hp |
[s]n,p
⟩⟨
[s]n,p |
[s]n,p
⟩= E[s]
n,p + �2
m
(C[s]n,p
)2
{−Z
[s]p
2
∫ ∞
0drK2
ν φ[s]n,p
dφ[s]n,p
dr
−√B
∫ ∞
0dr
KνKν+1
rp−2
2
φ[s]n,p
dφ[s]n,p
dr
}. (35)
If p = 4 and s = 0, we particularly have q = 1, γ = √B, ν = 1
2 ,and
K1/2(√Br−1) =
√π r
2√Be−√
Br−1 ,
K3/2(√Br−1) =
(1 + r√
BK1/2(
√Br−1)
),
[0]0,4(r) = C
[0]0,4
√π
2√B
exp
[−
√A
2r2 − √
Br−1
](36)
and finally
[1]E[s]n,p = 3�ω
2+ �
2
m· f1(2A1/4B1/2)
f2(2A1/4B1/2). (37)
where f1 and f2 are defined as the following integrals (Ref. [1],Section 27.5),
fm(x) =∫ ∞
0dttme−t2−xt−1
. (38)
Making use the Zahn’s expansion series of fm(x) (see also Ref.[15], Section 27.5), one obtains
[1]E[0]0,4 ≈ 3�ω
2+ 2A3/4
�2
√πm
· 1 − 2√
πA1/4B1/2 + 2.5368A1/2B
1 − 4√πA1/4B1/2 + 4A1/2B
B1/2
(39)
= 3�ω
2+ 2�
2A3/4
√πm
B1/2 + O(B1/2). (40)
It is worth to give the approximate expression for
C[0]0,4 ≈
√2
πA3/8B1/4
[√π
4− A1/4B1/2 + √
πA1/2B
]−1/2
≈ 2√
2A3/8
π3/4B1/4. (41)
For the first excited state one similarly derives (see alsoAppendix A),
C[0]1,4 ≈
√2
πA3/8B1/4
[1 − 10
3√
πA1/4B1/2 + 4A1/2B
]−1/2√
8
3√
π
≈ 4A3/8
31/2π3/4B1/4 (42)
and
[1]E[0]0,4 ≈ 7�ω
2+ 3A3/4
�2
√πm
[1 − 10
3√
πA1/4B1/2 + 4A1/2B
] [1 − ε]
(43)
where ε = −28√
π
9A1/4B1/2 − 8
9(4 − 21 · 0.3171)A1/2B
]B1/2
(44)
and [1]E[0]0,4 = 7�ω
2+ 3A3/4
�2
√πm
B1/2 + O(B1/2). (45)
With the particular choice of parameters: � = 1,m = 1/2, andα = 1, one easily arrives at [1]E[0]
0,4 = 3 + 4√πB1/2 and [1]E[0]
1,4 =7 + 9√
πB1/2. It is readily seen that the energy spacing between
these levels becomes B-dependent, namely,
[1]E[0]1,4 − [1]E[0]
0,4 ≈ 2�ω + A3/4�
2
√πm
B1/2 (46)
Furthermore, for the overlap integral one derives
⟨
[0]n,4 |
[0]m,4
⟩ = C[0]n,4C
[0]m,4
π
2B1/2A3/4
n∑p=0
m∑q=0
(−1)p+q
(n + 1
2
n − p
)
×(m + 1
2
m − q
)f2(p+q+1)(2A1/4B1/2)
p!q! , (47)
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and in particular,
⟨
[0]0,4 |
[0]1,4
⟩≈ −1
2A1/4B1/2 1 − √
πA1/4B1/2[1 − 4√
πA1/4B1/2 + 4A1/2B
]1/2 [1 + 4A1/2B]≈ −1
2A1/4B1/2. (48)
The absence of B-independent term in the RHS of Eq. (48) istrivially explained by orthogonality of Laguerre polynomials. Ingeneral, one readily derives that, to the first-order w.r.t. B,
〈 [0]n,4 |
[0]m,4〉 ≈ C
[0]n,4C
[0]m,4
π
4B1/2A3/4
[∫ ∞
0dxx1/2L
(12
)n (x)L
(12
)n (x)e−x
−2A1/4B1/2
∫ x
0dxL
(12
)n (x)L
(12
)n (x)e−x + 2A1/2B
×∫ x
0dxx−1/2L
(12
)n (x)L
(12
)n (x)e−x
]
= πC[0]n,4C
[0]m,4
4B1/2A3/4
[�(n + 3
2
)n! δnm
−2
(12
)n
(32
)m
n!m! 3F2
(−m, 1,
1
2;
3
2,
1
2− n; 1
)A1/4B1/2
+2π1/2A1/2(
32
)m
m! B
](49)
where Eq. (B1) of Appendix B has been used. Equation (49) givesus straightforwardly that
C[0]n,4 ≈ 2A3/8B1/4
√π
[�(n + 3
2
)n! δnm
−2A1/4B1/4
(12
)n
(32
)m
n!m! 3F2
(−m, 1,
1
2;
3
2,
1
2− n; 1
)π1/2
+2π1/2A1/2(
32
)n
n! B
]
whose substitution into the RHS of Eq. (27) results in the finalformula for the overlap integral,
〈 [0]n,4 |
[0]m,4〉 ≈ 2A1/4B1/2
[−(
12
)n
(32
)m
n!m! 3F2
(−m, 1,
1
2;
3
2,
1
2− n; 1
)
+(
32
)m
m! A1/4π1/2B1/2
]{[�(m + 3
2
)m! −
−2A1/4B1/2
(12
)m
(32
)m
(m!)2 3F2
(−m, 1,
1
2;
3
2,
1
2− m; 1
)
+2
(32
)m
m! A1/2π1/2B
][�(n + 3
2
)n!
−2A1/4B1/2
(12
)n
(32
)n
(n!)2 3F2
(−n, 1,
1
2;
3
2,
1
2− n; 1
)
+2
(32
)n
n! A1/2π1/2B
]}−1/2
. (50)
The first-order energy expression for an arbitrary, say, nthapproximate eigenstate takes the form,
[1]E[0]n,4 ≈ �ω
(2n + 3
2
)− π(C
[0]n,4)
2
2√B
· �2√B
m
×∫ ∞
0dr exp[−√
Ar2 − 2√Br−1]
×√
Ar(L
(12
)n (
√Ar2))2 + 2n
r
(L
(12
)n (
√Ar2)
)2
− 2(n + 1
2
)r
L
(12
)n−1 (
√Ar2)L
(12
)n (
√Ar2)
= �ω
(2n + 3
2
)+ π
(C
[0]n,4
)2
2√B
· �2√B
m
×1
2
∫ ∞
0dx
(L
(12
)n (x)
)2
exp[−x − 2A1/4B1/2x−1/2]
+∫ ∞
0dxL
(32
)n−1 (x)
[L
(32
)n (x) − L
(32
)n−1 (x)
]
× exp[−x − 2A1/4B1/2x−1/2] (51)
where we use Eq. (22.7.30) of Ref. [15] and that
nL
(12
)n−1 (x) −
(n + 1
2L
(12
)n−1 (x) = −xL
(32
)n−1 (x) (52)
The final expression of [1]E[0]n,4 is for n ≥ 1
[1]E[0]n,4 ≈ �ω
(2n + 3
2
)
+ 2A3/4B1/2�
2
m
{[(12
)n
(32
)n
(n!)2 3F2
(−n, 1,
1
2;
3
2,
1
2− n, 1
)
+2
(52
)n−1
(32
)n
(n − 1)!n! 3F2
(−n + 1, 1, −1
2;
5
2, −1
2− n; 1
)
−2
(32
)n−1
(52
)n−1
[(n − 1)!]2 3F2
(−n + 1, 1, −1
2;
5
2,
1
2− n; 1
)]
− π1/2A1/4B1/2
[(32
)n
n! + 2
(52
)n(2)n−1
(n − 1)!n!
× 3F2
(−n + 1,
1
2; −1;
5
2, −n; 1
)− 2
(52
)n−1
(2)n−1
[(n − 1)!]2
× 3F2
(−n + 1,
1
2; −1;
5
2, −n; 1
)]}
×[
�(n + 3
2
)n! − 2π1/2A1/4B1/2
(12
)n
(32
)n
(n!)2
×3F2
(−n, 1,
1
2;
3
2,
1
2− n; 1
)+ 2π1/2A1/2
(32
)n
n! B
]−1
.
(53)
Within the first-order perturbation treatment based on the useof Ansatz (28), one obtains, to the first-order of B,
〈 [0]n,4 |
[0]m,4〉 ≈ C
[1]n,4C
[1]m,4
{√π
2·(
32 + λ
)m
(λ)n
n!m!× 3F2
(−m,
3
2; 1 − λ;
3
2+ λ, 1 − λ − n; 1
)
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− 2A1/4B1/2
(32 + λ
)m
(12 + λ
)n
n!m!× 3F2
(−m, 1,
1
2;
3
2+ λ,
1
2− λ − n; 1
)
+ 2π1/2A1/2B
(32 + λ
)m
(λ)n
n!m!×3F2
(−m,
1
2; −λ;
3
2+ λ, −λ − n; 1
)}(54)
where λ = (A/2)1/6B1/3 and
C[1]n,4 ≈ 2A3/8B1/4
√π
[√π
2·(
32 + λ
)n(λ)n
(n!)2
× 3F2
(−n,
3
2; 1 − λ;
3
2+ λ, 1 − λ − n; 1
)
− 2A1/4B1/2
(32 + λ
)n
(12 + λ
)n
(n!)2
× 3F2
(−n, 1,
1
2− λ;
3
2+ λ,
1
2− λ − n; 1
)
+ 2π1/2A1/2B
(32 + λ
)n(λ)n
(n!)2
× 3F2
(−n,
1
2, −λ;
3
2+ λ, −λ − n; 1
)]−1/2
. (55)
One sees from Eq. (55), which with regard to the normalizationconstant, λ contributes only to excited states. To this contributionof λ, let consider the first-excited and ground-state energieswithin Ansatz (28),
[1]E[1]0,4 = �ω
(3
2+ λ
)+ �
2(C
[1]0,4
)2
m
×{(1 + λ)
∫ ∞
0drr2 exp[−√
Ar2 − 2√Br−1]
−B1/2
∫ ∞
0dr
(1 + r√
B
)exp[−√
Ar2 − 2√Br−1]
}π
√A
2√B
≈ [1]E[0]0,4 + 4�
2A3/4
m√
π
(A
2
)1/6
B5/6. (56)
Equation (56) demonstrates that within Ansatz (28), the second-order term appears to be of the form of B5/6, contrary to B inthe case of Ansatz (25). The approximate excited-state energiestake on the following appearance,
[1]E[1]n,4 = �ω
(2n + 3
2+ �ωλ + �
2A3/4
m
×{B1/2
[(32 + λ
)n
(12 + λ
)n
(n!)2
× 3F2
(−n, 1,
1
2− λ;
3
2+ λ,
1
2− λ − n; 1
)
+ 2
(52 + λ
)n−1
(12 + λ
)n
(n − 1)!n!× 3F2
(−n + 1, 1,
1
2− λ;
5
2+ λ,
1
2− λ − n; 1
)]
× − λ√
π
2A1/4
(32 + λ
)n(λ)n
(n!)2
× 3F2
(−n,
3
2; 1 − λ;
3
2+ λ, 1 − λ − n; 1
)
+ 2
(52 + λ
)n−1
(λ)n
(n − 1)!n!× 3F2
(−n + 1,
3
2, 1 − λ;
5
2+ λ, 1 − λ − n; 1
)]
+ 2λB1/2
[(32 + λ
)n
(12 + λ
)n
(n!)2
× 3F2
(−n, 1,
1
2− λ;
3
2+ λ,
1
2− λ − n; 1
)
+ 2
(52 + λ
)n−1
(12 + λ
)n
(n − 1)!n!× 3F2
(−n + 1, 1,
1
2− λ;
5
2+ λ,
1
2− λ − n; 1
)]}
+{√
π
2
(32 + λ
)n(λ)n
(n!)2
× 3F2
(−n,
3
2, 1 − λ;
3
2+ λ, 1 − λ − n; 1
)
− 2A1/4B1/2
(32 + λ
)n
(12 + λ
)n
(n!)2
× 3F2
(−n, 1,
1
2− λ;
3
2+ λ,
1
2− λ − n; 1
)}−1
. (57)
Expanding the hypergeometric functions 3F2(a, b, c; d, e, f ) interms of ψ functions by means of Eq. (7.4.1) (p. 497) of [19],one may arrive at such couple of conclusions. First. The free,B-independent term
�ω
[2n + 3
2+ 4(−1)n
(2n + 1)n!]
(58)
includes an extra term, comparing with the zero-order energy.This term changes a sign depending on odd or even state isunder consideration, and it approaches zero, as n tends to infinity.Second. Contrary to the expression for [1]E[0]
n,4 which leadingcontribution originates from B1/2 in case of sufficiently small B,and contrary to the same behavior of the ground-state energy[1]E[0]
0,4 as well, the first-order excited-state energy is dominantlyprovided by the terms of B1/6 and B1/3.
General Case: Small Enough B
Let us consider Eq. (15) for small B with an arbitrary p. UsingEqs. (19) and (20), one converts (15) into the following
r2 d2φ
dr2− 2(l − 1)
dφ
dr+ φ(e − Ar2)r2 = 0. (59)
Upon comparing (59) with Eq. (2.273.4) of Ref. [4], one readilyobtains the values of parameters (in the notations of Ref. [4]),
a = l − 3
2, α = √
A, m =∣∣l − 1
2
∣∣2
, β = 2, k = e/4√A
(60)
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and the general form of the corresponding eigenfunctions,
φ1(r) = c[1]1 r2l−1e
−√A
2 r21F1
(l + 1
2
2− e
4√A
, l + 1
2,√Ar2
)
+ c[1]2 e
−√A
2 r21F1
(−l + 3
2
2− e
4√A
, −l + 3
2,√Ar2
),
φ2(r) = c[2]1 e
−√A
2 r21F1
(3
4− e
4√A
,3
2,√Ar2
)
+ c[2]2 r−1e
−√A
2 r21F1
(1
4− e
4√A
,1
2,√Ar2
), (61)
where the upper one corresponds to l = 0 and the lower tointeger l.
Hence, for an arbitrary l, there exist two branches of admissibleeigenfunctions decaying properly at infinity. They are
i.
l = 0 :
φ[0]n0 (r) = Cn0e
−√A
2 r21F1
(−n,
3
2,√Ar2
)
= Cn0e−
√A
2 r2L
(12
)n (
√Ar2) (62)
with e[1]n0 = √
A(4n + 3), and (63)
ψ[0]n0 (r) = Cn0r
−1e−
√A
2 r21F1
(−n,
1
2,√Ar2
)
= Cn0r−1e
−√A
2 r2L
(− 1
2
)n (
√Ar2) (64)
with e[2]n0 = √
A(4n + 1), (65)
ii.
l �= 0 :
φ[0]nl
(r) = Cnlr2l−1e
−√A
2 r21F1
(−n, l + 1
2,√Ar2
)
= Cnlr2l−1e
−√A
2 r2L
(l− 1
2
)n (
√Ar2) (66)
with e[1]n0 = √
A(4n + 3), and (67)
ψ[0]nl
(r) = Cnle−
√A
2 r21F1
(−n, −l + 3
2,√Ar2
)
= Cn0e−
√A
2 r2L
(−l+ 1
2
)n (
√Ar2) (68)
with e[2]nl
= √A(4n − 2l + 3). (69)
One should notice that due to the restriction imposed on α forL(α)n (x) (see Eq. (22.3.9), Ref. [1]), l takes only the value l = 1 in
Eqs. (68)–(69). The corresponding total radial wavefunctions andenergy eigenvalues take then the form, to zero order,
i.
[0]n0,p(r) = C
[0]n0,pr
−1/2e−
√A
2 r2L
(12
)n (
√Ar2)K 1
p−2
(2√B
(p − 2)rp−2
2
)
(70)
E[1]n0 = �ω
(2n + 3
2
), (71)
and
[0]nl,p(r) = C
[0]nl,pr
2l− 32 e
−√A
2 r2L
(l− 1
2
)n (
√Ar2)K 2l+1
p−2
(2√B
(p − 2)rp−2
2
)
(72)
E[1]nl
= �ω
(2n + l + 1
2
), (73)
ii.
�[0]n0,p(r) = C
[0]n0,pr
−3/2e−
√A
2 r2L
(− 1
2
)n (
√Ar2)K 1
p−2
(2√B
(p − 2)rp−2
2
)
(74)
E[1]n0 = �ω
(2n + 1
2
), (75)
and
�[0]nl,p(r) = C
[0]nl,pr
− 12 e
−√A
2 r2L
(− 1
2
)n (
√Ar2)K 2l+1
p−2
(2√B
(p − 2)rp−2
2
)
(76)
E[1]nl
= �ω
(2n + 1
2
), (77)
To check an appropriateness of the radial wavefunctions listedin Eqs. (70)–(76), let us analyze the potential
U(l)p = αr2 + β
rp+ �
2l(l + 1)
2mr2. (78)
It reaches extremum at ro that obeys the algebraic equation
2αr2o = pβ
rpo
+ �2l(l + 1)
2mr2o
. (79)
This equation takes on the appearance for p = 4,
x3 = �2l(l + 1)
2αmx − 2β
α= 0, (80)
where x = r2o . Equation (80) is an “incomplete" cubic equation
x3 + Px + Q = 0 with coefficients P = −�2l(l + 1)/(2αm) =
−l(l+1)/A,Q = −2α/β = −2A/B. Define D = (P/3)3 +(Q/2)2 =(l(l + 1)/3A)3 + (B/A)2. If D > 0, the potential U(4)
p (r) possessesa single extremum (minimum). Non-negativity of D implies that
B > A−1/2[l(l + 1)/3]3/2. (81)
This inequality (81) shows that for l �= 0, the potential U(4)p (r)
gains a double-well character for small enough B. Therefore, thepresent perturbation treatment developed for the harmonic partof U(4)
p (r) becomes inadequate. One may expect the analogoussituation for any p > 2. For this reason, we will deal furtherwith rotationless eigenstates of Hp for any p > 2 describedapproximately by Eq. (70) only.
At the first step, let us evaluate the overlap integral
⟨
[0]n0,p |
[0]m0,p
⟩ = Fnm√B + GnmB
1p−2
√HnnHmm
+ O(B1/2, B1/p−2) (82)
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where
Fnm =16aν−A
p−28(
32
)m
(p−2
4
)n�(
32 − p−2
4
)bν−(p − 2)n!m!
× 3F2
(−m,
3
2− p − 2
4, 1 − p − 2
4;
3
2, 1 − n − p − 2
4; 1
)(83)
Gnm = 2αν−A14 2
2p−2(
32
)m
(12
)n
αν+(p − 2)2
p−2 n!m!3F2
(−m, 1,
1
2;
3
2, 1 − n; 1
)(84)
and
Hnn = 1 + Fnn
√B + GnnB
1p−2 . (85)
If p > 4, the last with B1
p−2 dominates; if 4 > p > 2,B-dependenceof the overlap integral is determined by
√B.
Upon deriving the overlap integral (82), we have used therepresentation of Kν(z) in terms of Whittaker function (see Eqs.(9.6.48), (13.1.34), and (13.1.32) of Ref. [1]),
Kν(z) =( π
2z
)1/2W0,ν(2z)
=√
π
2e−z[αν
−M(aν
+, bν+, 2z
)zν + αν
+M(aν
−, bν−, 2z
)z−ν]
(86)
with
αν± = �(±2ν)2
12 ∓ν
�(
12 ± ν
) , aν± = 1
2± ν, bν
± = 1 ± 2ν. (87)
It follows from Eq. (82) that in the overlap integral, in the lead-ing term, B occurs in the power 1/2, independent on p. Thenormalization constant
C[0]n,p =
√√√√ B1
p−2
πHnn
(2
p − 2
) 1p−2 + O
(B
1p−2
)(88)
Let us now evaluate the first-order energies. Equation (35) isagain the key formula hereof. Defining the intermediate quantity,
εn,p =[
[1]E[0]n,p − �ω
(2n + 3
2
)] πmHnn(αν+)2(p−2
2
) 2p−2
�2B1
p−2 A3/4(89)
one straightforwardly derives
εn,p =∫ ∞
0drKνφ
[0]n,p
dφ[0]n,p
dr
(Kν −
√B
r− p−2
2
Kν+1
). (90)
We now thoroughly consider the term in curved brackets onthe RHS of Eq. (90),
Kν
(2√B
p − 2r− p−2
2
)−
√B
r− p−2
2
Kν+1
(2√B
p − 2r− p−2
2
)
= Kν(z) − p − 2
2zKν(z)
× z
2ν
[2ν
zKν(z) − Kν+1(z)
]= z
2ν
[Kν+1(z) − Kν−1(z) − Kν+1(z)
]= − z
2ν[Kν−1(z) (91)
where
z = 2√B
p − 2r− p−2
2 , ν = 1
p − 2, (92)
and where Eq. (9.6.26) of Ref. [1] is used. From Eq. (86) one derives
Kν(z) ≈√
π
2e−z
αν
−
(2√B
p − 2
) 1p−2
r− 1
2 + αν+
(2√B
p − 2
)− 1p−2
r12
+αν+aν−bν−
(2√B
p − 2
)− 1p−2 4
√B
p − 2r
12 − p−2
2
(93)
and
Kν−(z) ≈√
π
2e−z
αν−1
−
(2√B
p − 2
) 1p−2 −1
r− 1
2 + p−22
+ αν−1+
(2√B
p − 2
)1− 1p−2
r12 − p−2
2
+αν+a
ν−1−
bν−1−
(2√B
p − 2
)1− 1p−2 4
√B
p − 2r
12 −p−2
. (94)
With the use of Eqs. (54c), (55a), and (55b), Eq. (90) becomesrewritten as follows
εn,p = −√B
∫ ∞
0
dr
rp−2
2
φ[0]n,p
dφ[0]n,p
dr
× Kν
(2√B
p − 2r− 1
p−2
)Kν−1
(2√B
p − 2r− 1
p−2
)(95)
εn,p ≈ √AB
παν+αν−−(
2p−2
) 2p−2
2(
2√B
p−2
) 2p−2 √
B
B1
p−2
∫ ∞
0dre−√
Ar2L
(12
)n (
√Ar2)
×
1, n = 0
L
(12
)n (
√Ar2) + L
(32
)n− (
√Ar2), n > 0
εn,p =παν+αν−
−(
2p−2
) 2p−2
4(
2√B
p−2
) 2p−2 √
B
B1
p−2
{δn0 + (1 − δn0)
×[(
12
)n
(32
)n
(n!)2 3F2
(−n, 1,
1
2;
3
2,
1
2− n; 1
)
+2(
52
)n−1
(12
)n
(n − 1)!n! 3F2
(−n + 1, 1,
1
2;
5
2,
1
2− n; 1
)]}. (96)
The final result for the first-order energy is
[1]E[0]n,p ≈ �ω
2n + 3
2+(
2p−2
) 2p−2 A1/4�
(12 + 1
p−2
)�(
2p−2 − 2
)2Hnn�
(− 12 + 1
p−2
)�(
2p−2
) B1
p−2
×[δn0 + (1 − δn0)(
(12
)n
(32
)n
(n!)2 3F2
(−n, 1,
1
2;
3
2,
1
2− n; 1
)
+2(
52
)n−1
(12
)n
(n − 1)!n! 3F2
(−n + 1, 1,
1
2;
5
2,
1
2− n; 1
))]). (97)
For the particular case p = 6 see Appendix C and Ref. [13]. It isworth noticing that the basic formula (86) underlying all above
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derivations is invalid for integer ν. The latter case takes placeparticularly for p = 3 and p = 5/2. For the spiked potentialwith p = 5/2 see Refs. [10, 15, 16]. When ν becomes integer, sayν = N, one has instead of Eq. (86),
KN(z) = π1/22Ne−zzN
[− 1
(2N)!� ( 12 − N
)×{M
(N + 1
2, 2N + 1, 2z
)ln 2z
+∞∑k=0
2k(N + 1
2
)k
k!(2N + 1)k
[ψ
(N + k + 1
2
)− ψ(k + 1)
− ψ(2N + k + 1)
]zk}
+ (2N − 1)!�(N + 1
2
) 2N−1∑k=0
2k−2N(
12 − N
)k
k!(1 − 2N)kzk−2N
]. (98)
Hence, for z = 2N√Br−1/2N , one obtains
KN
(2N
√B
r1
2N
)= π1/2(4N
√B)Ne−2N
√B/r−1/2N
× r−1/2
[− 1
(2N)!� ( 12 − N
)]
×{M
(N + 1
2, 2N + 1,
4N√B
r1
2N
)ln
(4N
√B
r1
2N
)}
+∞∑k=0
(4N√B)k(N + 1
2N
)k
k!(2N + 1)k
×[ψ
(N + k + 1
2
)− ψ(k + 1) − ψ(2N + k + 1)
]
× r−k/2n + (2N − 1)!�(N + 1
2
)×
2N−1∑k=0
(4N√B)(k−2N)
(12 − N
)k
k!(1 − 2N)kr1−k/2N
](99)
Therefore, the overlap integral takes the following expression,
〈 n0,N | l0,N〉 = C[0]n,NC
[0]l,N B
Nα2N
×∫ ∞
0dr exp
[−√
Ar2 − 2N√B
r1
2N
][L
(12
)n (
√Ar2)
]2
×(
β2N
{M2
(N + 1
2, 2N + 1,
4N√B
r1
2N
)
× ln2
(4N
√B
r1
2N
)+ 2M
(N + 1
2, 2N + 1,
4N√B
r1
2N
)
× ln
(4N
√B
r1
2N
) ∞∑k=0
Bk/2γ(N)
kr−k/2n
+∞∑
t,s=0
B(t+s)/2γ(N)t r−(t+s)/2N
}
+2N−1∑u,v=0
δ(N)u δ(N)
v B−2N+(u+v)/2r2−(u+v)/2N
× ln
(4N
√B
r1
2N
)+
∞∑k=0
Bk/2γ(N)
kr−k/2n
]
×2N−1∑u=0
δ(N)u B−N+u/2r1−u/2N
)(100)
where the following notations
αk = π1/2(4N)k , βk = − 1
(2k)!� ( 12 − k
) ,
γ(u)
k= (4N)k
(u + 1
2
)k
k!(2u + 1)k
[ψ
(u + k + 1
2
)
− ψ(k + 1) − ψ(2u + k + 1)
]
δ(u)
k= (2u − 1)!(4N)k−2u
(12 − u
)k
�(
12 + u
)k!(1 − 2u)k
(101)
are introduced. The leading contribution to the overlap integral(102) at sufficiently small B is then as follows
〈 n0,N | l0,N〉 ≈ C[0]n,NC
[0]l,N B
Nα2N
×∫ ∞
0dr exp
[−√
Ar2 − 2N√B
r1
2N
][L
(12
)n (
√Ar2)
]2
× [(δ(N)o
)2B−2Nr2 + 2B−2N
(δ
(N)1 δ(N)
o B1/2)r2−1/2N
]= C
[0]n,NC
[0]l,Nα2
N
(δ
(N)o
)2
2A3/4BN
[∫ x
0dxx−1/2e−xL
(12
)n (x)L
(12
)l
(x)
+ 2δ(N)1 B1/2
δ(N)o
A1/8N
×∫ x
0dxx−1/2−1/4Ne−xL
(12
)n (x)L
(12
)l
(x)
]
+ C[0]n,NC
[0]l,Nα2
N
(δ
(N)o
)2
2A3/4BN
×[δln + 2δ
(N)1 A1/8NB1/2
(1
4N
)n�(
32 − 1
4N
)δ
(N)o l!n!
× 3F2
(−l,
3
2− 1
4N, 1 − 1
4N;
3
2, 1 − n − 1
4N; 1
)].
(102)
Defining
An =: 1 + 2δ(N)1 A1/8NB1/2
(1
4N
)n�(
32 − 1
4N
)δ
(N)o (n!)2
3F2
(−l,
3
2− 1
4N, 1 − 1
4N;
3
2, 1 − n − 1
4N; 1
), (103)
one readily derives the expressions for the normalizationconstant,
C[0]n,N =
√2A3/8
δ(N)o αNA
1/2n
BN/2 + O(BN/2), (104)
http://onlinelibrary.wiley.com International Journal of Quantum Chemistry 2012,DOI: 10.1002/qua.24136 9
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and the overlap integral,
〈 n0,N | l0,N〉 = 2δ(N)1 A1/8N
(32
)l
(1
4N
)n�(
32 − 1
4N
)δ
(N)o l!n!√AnAl
× 3F2
(−l,
3
2− 1
4N, 1 − 1
4N;
3
2, 1 − n − 1
4N; 1
)B1/2 + O(B1/2).
(105)
Hence,
[1]E[0]0,1 ≈ �ω
(3
2+ 32�
(34
)A3/8
4 − 3�(
54
)A1/8B1/2
B1/2
),
[1]E[0]0,1 ≈ �ω
(2n + 3
2+ 8�2
(34
) (34
)n
Ann!
×[(
32
)n
n! 3F2
(−n,
3
4, 1;
3
2,
1
4− n; 1
)
+2(
52
)n−1
(n − 1)! 3F2
(−n + 1,
3
4, 1;
5
2,
1
4− n; 1
)]A3/8B1/2
)
(106)
And finally, let us consider
(ii)p = 5
2,N = 2,
Ao = 1 + 8�
(11
8
)A1/16B1/2
An = 1 +(
8(
32
)n
(18
)n�(
118
)(n!)2
× 3F2
(−n,
11
8,
9
8;
3
2,
9
8− n; 1
))A1/16B1/2
)
[1]E[0]0,2 ≈ �ω
(3
2+ 8�
(98
)A3/16
1 + 8�(
118
)A1/16B1/2
B1/2
),
[1]E[0]n,1 ≈ �ω
(2n + 3
2+ 8�2
(98
) (38
)n
Ann!
×[(
32
)n
n! 3F2
(−n,
11
8, 1;
3
2,
5
8− n; 1
)
+2(
52
)n−1
(n − 1)! 3F2
(−n + 1,
11
8, 1;
5
2,
5
8− n; 1
)]A3/16B1/2
)
(107)
Some Thoughts Around Equations:Molto Allegro
Well, let us think around the equations obtained in the previoussections. We are dealing there with the branch of approximateeigenwavefunctions taken in the form ζ(r)φ(r), where φ(r) isthe solution of the unperturbed problem. It is likely the correctassumption for the ground state (see also Ref. [11]). However,to what extent it is a truth for the excited states? Or, in theequivalent form, how does the Klauder phenomenon, if it is reallyholds, manidest itself?
Proposal: Klauder’s phenomenon is the appearance of a newbranch of approximate wave functions absolutely absent for theunperturbed problem. This branch for p = 4 is given by Eq. (26),and for an arbitrary p by Eq. (74) at small enough B �= 0.
Because this branch is absent for the unperturbed problem,taking a limit of B approaching 0 in this case is incorrect. Whatwe need is just to pick some value of B, say Bo �= 0, small enoughto justify our approach developed in “The Heydays: MolecularInteractions” section. We restrict ourselves by treating the casep = 4 only. The zero-order total wave function takes the form
[1]� [0]n,p(r) = C[0]
n,pr−3/2K 1
p−2
(2√B
p − 2r− p−2
2
)e
−√A
2 r2L
(− 1
2
)n (
√Ar2)
(108)
[1]E[0]n,p = �ω
(2n + 1
2
), (109)
and for p = 4,
[1]� [0]n,4(r) = C
[0]n,4
√π
2√Bo
r−1L
(− 1
2
)n (
√Ar2)e−
√A
2 r2−√Br−1
(110)
Bo is fixed and sufficiently small. Let us now evaluate the overlapintegral
〈 n,4 | m,4〉 = π C[0]n,4C
[0]m,4
4A1/4B1/2o
∫ ∞
0dxx−1/2
× exp[−x − αx−1/2]L(− 1
2
)n (x)L
(− 1
2
)m (x)
= π C[0]n,4C
[0]m,4
4A1/4B1/2o
[(n − 1
2
n
)(m − 1
2
m
)F−1/2(α)
−((
n − 12
n − 1
)(m − 1
2
m
)
+(n − 1
2
n
)(m − 1
2
m
))F1/2(α)
+n∑
p,q=1
(−1)p+q
p1q!
(n − 1
2
n − p
)(m − 1
2
m − q
)Fp+q−1/2(α)
(111)
where
Fp(x) =∫ ∞
0dyype−y−xy−1/2
. (112)
Using the differential equation
dmFp(x)
dxm= (−1)mF
p− 12 m
(x) (113)
firstly derived by Zahn (Ref. [27.13] in [15]) and his expressionfor Fo(x), one obtains
F−1/2(x) ≈ √π + 2x ln x − 0.2684x
F−1(x) ≈ 1.7316 − 2 ln x
F−2(x) ≈ 0.7321 + 2
x2+ 1.99992lnx,
F1/2(x) ≈√
π
2− x and Fn+1/2(x) = �
(n + 3
2
)− n!x. (114)
Taking into account the orthonormalization of Laguerre polyno-
mials L(− 1
2 )
n (x), one finally arrives at the formula,
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〈 n,4 | m,4〉 = π C[0]n,4C
[0]m,4
4A1/4B1/2o
×[δnm +
(n − 1
2
n
)(m − 1
2
m
)(2αlnα + 0.7316α)
− α
n∑p,q=1
(−1)p+q�(p + q)
p1q!
×(n − 1
2
n − p
)(m − 1
2
m − q
)], α = 2A1/4B1/2
o . (115)
Introducing
Gnm =(n − 1
2
n − p
)(m − 1
2
m − q
)[0.7316 + 2ln(2A1/4B1/2
o )]
−n∑
p,q=1
(−1)p+q�(p + q)
p1q!
(n − 1
2
n − p
)(m − 1
2
m − q
), (116)
one obtains
C[0]n,4 ≈ 2π−1/2A1/8[1 + 2A1/4B1/2
o Gnn]−1/2B1/4o (117)
and for n �= m,
〈 n,4 | m,4〉 ≈ 2A1/4GnmB1/2o√
GnnGmm
(118)
In particular,
Goo = 0.7316 + 2 ln(2A1/4B1/2
o
). (119)
Let us turn to evaluation of the first-order energies. From Eq. (35)one derives
[1]E[0]0,4 = π�
2A1/2(C
[0]0,4
)2
4m
[F−2
(2A1/4B1/2
o
)+ F−1
(2A1/4B1/2
o
)]+ �ω
2
≈ �ω
{1
2+ 1
1 + 2A1/4B1/2o Goo
[1
1 + 2A1/4B1/2o
− A1/4B1/2o (0.9995 + 0.00008 ln(2A1/4B1/2
o ))
]}(120)
and for n 1,
[1]E[0]0,4 = �ω
(2n + 1
2
)+ π�
2A1/2(C
[0]0,4
)2
4m
×∫ ∞
0dx exp
[− x − 2A1/4B1/2o x−1/2
]
× L
(− 1
2
)n (x)
L
(− 1
2
)n (x)
x2+ L
(− 1
2
)n (x) + 2L
(12
)n (x)
x
≈ �ω
2n + 1
2+ 1
1 + 2A1/4B1/2o Gnn
×
[(n − 1
2
n
)]2
2A1/4B1/2o
+ A1/4B1/2o
−0.00008
[(n − 1
2
n
)]2
× ln(2A1/4B1/2
o
)− 0.99995
[(n − 1
2
n
)]2
+[(
n − 12
n − 1
)]2
− 2
(n − 1
2
n − 1
)(n − 1
2
n
)
−(n − 1
2
n
)(n − 1
2
n − 2
)+ 2
(n − 1
2
n − 1
)(n − 1
2
n − 2
)
+n∑
p,q=1
(−1)p+q(p + q)(p + q − 2)!p!q!
(n − 1
2
n − p
)(n − 1
2
n − q
)
+ 2n∑
p=1
n−1∑q=1
(−1)p+q(p + q − 1)!p!q!
(n − 1
2
n − p
)(n − 1
2
n − q
) .
(121)
One sees easily from Eqs. (123) and (124) that at sufficientlysmall B, these energy expressions are well approximated by
[1]E[0]0,4 = �ω
2
[1 + A−1/4B−1/2
o
](122)
[1]E[0]n,4 = �ω
2
4n + 1 +
[(n − 1
2
n − 1
)]2
A1/4B1/2o
. (123)
We may conclude that this essentially novel branch of approxi-mate wave functions contributes to the excite spectrum of theHamiltonian with a spiked potential.
Reminiscence and Summary
With scientific rigor and sense of wonder, delving into the reasons
things happen in nature.
Author‘s Conjecture: Kaplan‘s Motto in his Life in Physics
Since 1973 when I first met him at the Fock Winter Schoolorganized by A.V.Tulub, Ilya Kaplan played the very important rolein my life. This work is a reflection of my grateful appreciation tohim. Actually, what has been done above is the following. Despitethe common belief[8d] that the traditional perturbation theoryis not applicable to study the class of potentials combining theharmonic oscillator with a singular potential of the type λr−α ,[9h]
my intention was to propose and develop such treatment forα = 4 on the basis of the appropriate choice of the zero-order Hamiltonian Ho = −(�2/(2m))∇2
r + U2, which is exactlysolvable. The presented approach will be further extended tothe class of potentials that describe the process of low-energyelectron scattering by polar molecules where the long-rangedipole interaction is important.[20]
Appendix A
f4(x) = 3√
π
8− 1
2x +
√π
8+ O(x2), (A1)
f4(x) = 1 − 3√
π
8+ 1
4x2 + O(x2), (A2)
f4(x) = 15√
π
16− x + 3
√π
16x2 + O(x2). (A3)
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Appendix B
∫ ∞
0dxxα−1e−xL(γ )
m (x)L(λ)m (x) = (1 + γ )m(λ − α + 1)n�(α)
m!n!× 3F2(−m, α, α − λ; γ + 1, α − λ − n; 1). (B1)
Appendix C
Consider the case: p = 6, α = 1, m = 12 , � = 1, A = 1, ω = 2, and
β = B = 9/64 to compare with Ref. [13]. Our formalism givesfor the ground state
[0]0,6 = C
[0]0,6r
−1/2e− 1
2 r2K 1
4
(3
16r2
). (C1)
Due to that
α14± =
21/4√π
�1, +
− 21/4√π
�2, −,
(C2)
where �1 = �(3/4) = 1.2254167024, �2 = �(1/4) =3.6256099082, and
a14± =
{34 , +
− 14 , −,
)b
14± =
{32 +
− 12 −,
}
one obtains
K 14
(3
16r2
)=√
π
2e
− 316r2
[(3
16
)1/4
r−1/2 21/4√π
�1M
(3
4,
3
2,
3
8r2
)
−(
16
3
)1/4
r1/2 21/4√π
�2M
(1
4,
1
2,
3
8r2
)]. (C3)
Substitution of (C2) into the RHS of Eq. (C1) gives the totalwave function that approximates the ground state and whichexponential part exactly coincides with that given in Ref. [8c].Factor rλ is different for these representations. In terms of Weberfunctions, one has
[0]0,6(r) = πC
[0]0,6e
− 12 r2
21/231/4
E
(1)−1/2
(√3
2r
)23/4�1
−23/2E
(0)−1/2
(√3
2r
)�2
(C4)
To proceed further, we evaluate Foo = 2√
π ,Goo = 25/4�1/�2,and
[1]E[0]0,6 ≈ 3 − �1�2
4π√
6(
1 + 31/2�121/4�2
+ 3√
π
4
) (C5)
that is below 3 and is not equal to 4 at all, the ground-stateapproximate energy found in Ref. [8c].
Keywords: molecular interaction • potential • spiked potential
• perturbation treatment • Klauder phenomenon
How to cite this article: E. S. Kryachko, Int. J. Quantum Chem.
2012, DOI: 10.1002/qua.24136
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[8] The name “spiked” of the potential Up(r) is provided by the existenceof the pronounced peak at the origin if β > 0. See, for example: (a)W. M. Frank, D. J. Land, R. M. Spector, Rev.Mod.Phys. 1971, 43, 36; (b) R.L. Hall, N. Saad, A. B. von Keviczky, J.Math. Phys. 2002, 43, 94; (c) E. M.Harrell, Ann. Phys. (N.Y.) 1977, 105, 379; (d) V. C. Aguilera-Navarro, A. L.Coelho, n. Ullah, Phys.Rev.A 1994, 49, 1477; (e) Y. P. Varshni, Phys. Lett.A1993, 183, 9; (f ) E. S. Kryachko, Int. J. Quantum Chem. 2011, 111, 1792;(g) A. de S. Dutra, Phys. Rev. A 1993, 47, R2435; (h) W. Kirsch, B. Simon,Ann. Phys. (N.Y.) 1988, 183, 122; (i) E. Papp, Europhys. Lett. 1989, 9, 309;(k) Y. P. Varshni, Europhys.Lett. 1992, 20, 295; (j) N. Kawakami, J.Phys.Soc.Jpn. 1993, 62, 4163; (k) K. Hikami, M. Wadati, J. Phys. Soc. Jpn. 1993, 62,4203 and references therein.
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[11] J. R. Klauder, Phys. Lett. 1973, B47, 523.
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Received: 1 March 2012
Revised: 1 March 2012
Accepted: 20 March 2012
Published online on Wiley Online Library
12 International Journal of Quantum Chemistry 2012,DOI: 10.1002/qua.24136 http://WWW.CHEMISTRYVIEWS.ORG
