Convergence properties of the multipole expansion of the exchange contribution to the interaction energy

0.27 MB·

by  Piotr Gniewek

January18,2016 MolecularPhysics 1a
To appear in Molecular Physics
Vol. 00, No. 00, Month 2015,1–20
−0.2 Φ∼XR−nϕn
molecular ion n
−0.35
E
6 u
01 E −0.5 Eg −2J(R) hΨhΨ|H|ΨΨii RΦ∇ΦdS hΦ0|VΨi
2

n
a −0.65
J 0 2 4 6 8
5 R J(R)=−2eRe−R(cid:2)1+O(R−1)(cid:3)
1

Figure 1. Graphical abstract: The leading term of exchange energy can be calculated from the
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multipoleexpansionofthewavefunctionusingvariationalprinciple,surfaceintegral,orperturbation
h
theory.
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h Research Article
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Convergence properties of the multipole expansion of the
1
v exchange contribution to the interaction energy
3 †
2
9 Piotr Gnieweka∗ and Bogumil Jeziorskia
3
aFaculty of Chemistry, University of Warsaw, Pasteura 1, 02-093 Warsaw, Poland
0
.
1 (January 18, 2016)
0
6
1 TheconventionalsurfaceintegralformulaJsurf[Φ]andanalternativevolumeintegralformula
: Jvar[Φ]areusedtocomputetheasymptoticexchangesplittingoftheinteractionenergyofthe
v hydrogenatomandaprotonemployingtheprimitivefunctionΦintheformofitstruncated
i multipole expansion. Closed-form formulas are obtained for the asymptotics of Jsurf[ΦN]
X andJvar[ΦN],whereΦN isthe multipoleexpansion ofΦ truncated after the1/RN term,R
beingtheinternuclearseparation.Itisshownthattheobtainedsequencesofapproximations
r
a convergetotheexact resultswiththeratecorrespondingtotheconvergence radiusequal to
2and4whenthe surfaceandthevolumeintegralformulasareused,respectively. When the
multipoleexpansionofatruncated,Kthorderpolarizationfunctionisusedtoapproximatethe
primitivefunctiontheconvergenceradiusbecomesequaltounityinthecaseofJvar[Φ].Atlow
ordertheobservedconvergence ofJvar[ΦN]is,however,geometricandswitchestoharmonic
onlyatcertain value ofN =Nc dependent on K.Anequation forNc isderivedwhich very
well reproduces the observed K-dependent convergence pattern. The results shed new light
on the convergence properties of the conventional SAPT expansion used in applications to
many-electrondiatomics.
† DedicatedtoProfessorAndreasSavinontheoccasionofhis65birthday.
∗Correspondingauthor,e-mail:[email protected]
ISSN:00268976print/ISSN13623028online
(cid:13)c 2015Taylor&Francis
DOI:10.1080/0026897YYxxxxxxxx
http://www.informaworld.com
January18,2016 MolecularPhysics 1a
1. Introduction
Energetic effects of noncovalent molecular interactions are very small, compared to
the energies of noninteracting subsystems,nevertheless they determine the proper-
ties ofcondensed matter [1,2]andinfluencechemical reactivity [3].Thetheoretical
framework for the understanding of the nature and origin of noncovalent interac-
tions is provided by SymmetryAdapted Perturbation Theory(SAPT)[4,5],which
introduces rigorous decomposition of the interaction energy into electrostatic, in-
duction, dispersion and exchange contributions. At large interatomic distances R
the first three of these constituents can be expanded in powers of R−1, leading to
the so-called multipole expansion of interaction energy
E (R) C R−n, (1)
int n
∼
Xn
with the C ’s coefficients referred to as the van der Waals constants. While the
n
expansion of Eq. (1) is divergent for any R [6, 7], it is known to provide the correct
asymptotic representation of E (R) for large R [8, 9]. From the perspective of the
int
large R asymptotics of E , the exchange effects might seem to be insignificant.
int
However, at intermediate and short distances the importance of the exchange part
of E is hard to overestimate, as it is responsible for the splitting of potential
int
energy curves and for the strong repulsion in the case of interaction of closed-shell
atoms when it provides the necessary quenching of the attractive induction and
dispersion components of the interaction energy.
The exchange effects are caused by the resonant tunneling of one or more elec-
trons between the interacting systems and are considered to be non-perturbative,
that is, an application of the straightforward Rayleigh-Schro¨dinger perturbation
treatment is unable to provide even qualitative description of the exchange contri-
bution to E [10, 11]. This problem can becircumvented by thesymmetry forcing
int
procedure[12],inwhichthewave functioniscorrectly symmetrizedineachorderof
perturbation theory. Different variants of symmetry forcing lead to different SAPT
expansions [12, 13], of which the most important is the Symmetrized Rayleigh-
Schro¨dinger (SRS) perturbation approach [10, 14]. SRS is nowadays routinely ap-
plied in studies of noncovalent interactions of many-electron systems [15–18].
The success of SRS approach is somewhat bewildering, as it is based on the so-
called polarization approximation (PA), which is the straightforward application
of the Rayleigh-Schro¨dinger (RS) perturbation theory to molecular interactions.
PA is known to diverge when at least one of the interacting subsystems has more
than two electrons [19–22]. On the other hand, PA converges for H+ and H to the
2 2
ground state wave function and energy [10, 11, 23]. For large R this convergence is
very slow, as the convergence radiusof PA exceeds oneby a quantity exponentially
decaying with R [11, 23–25]. Irrespectively of the ultimate convergence or diver-
gence, PA provides the asymptotic approximation to the primitive function [19] Φ
of the interacting system, in the sense that [8, 26]
Ψ = Φ(K)+O(R−κ(K+1)), (2)
ν ν
A
where Ψ is the wave function of the state ν, is the symmetry operator asso-
ν ν
A
ciated with this state, Φ(K) is the PA wave function computed through the Kth
order, and κ equals two if at least one of the interacting subsystems is charged and
three if both are neutral.
From the practical point of view it is important to know the limits of validity
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January18,2016 MolecularPhysics 1a
and convergence properties of various approaches to calculation of the exchange
contribution to E . In this paper we present exact results concerning the rate
int
of convergence of the exchange energy J(R) of the hydrogen molecular ion H+
2
when computed using the multipole expansion of the primitive wave function. The
exchange energy is defined in this context as
J(R) = 1 E (R) E (R) , (3)
2 g − u
(cid:2) (cid:3)
where E and E are the Born-Oppenheimer energies of the lowest gerade and
g u
ungerade states, respectively. The H+ ion is particularly suitable for studies of the
2
convergence of exchange effects for two reasons:(a)theanalysis is simplifiedby the
fact that there are no electron correlation effects involved and (b) the asymptotic
expansion of J(R) is known exactly for this system [7]. H+ is one of the most
2
important models of the theory of molecular interactions and there is a wealth of
analytical results available for this species [7, 13, 27–29]. H+ is also a prototypical
2
double-well system [30, 31]. The asymptotic expansion of exchange energy of H+
2
can be conveniently written in the form
2
J(R) = Re−R(j +j R−1+j R−2+...), (4)
0 1 2
e
where j = 1,j = 1, and j = 25. The higher j coefficients can be found in
0 − 1 −2 2 − 8 n
Ref. [7].
The most widely used method of calculating J(R) employs the surface-integral
formula J [Φ] given by [32–35]
surf
Φ ΦdS
J [Φ]= M ∇ , (5)
surf Φ ΦR 2 Φ2dV
h | i− right
R
where the primitive function Φ is assumed to be localized in the “left” half of
space and M is the median plane dividing the space into “left” and “right” halves.
Atomic units (~ = e = m = 1) are used in Eq. (5) and throughout the paper.
e
Alternatively, the exchange energy can be calculated from the SAPT formula
[25]
ϕ VPΦ ϕ Φ ϕ VΦ ϕ PΦ
0 0 0 0
J [Φ] = h | ih | i−h | ih | i, (6)
SAPT ϕ Φ 2 ϕ PΦ 2
0 0
h | i −h | i
where V is the interaction operator, ϕ is the unperturbed wave function, and P
0
is the operator inverting the electronic coordinates with respect to the midpoint of
the internuclear axis. When Φ is approximated by the polarization function Φ(K)
andther.h.s.of Eq.(6)is expandedin powersof V,oneobtains theSRSexpansion
of the exchange energy through the Kth order.
In our recent work [36] we showed that the exchange energy can be very accu-
rately calculated from the variational principle based formula J [Φ]
var
Φ HPΦ Φ Φ Φ HΦ Φ PΦ
J [Φ]= h | ih | i−h | ih | i, (7)
var Φ Φ 2 Φ PΦ 2
h | i −h | i
where H is the full Hamiltonian of the interacting system. Numerical results il-
lustrating the convergence of J [Φ], J [Φ], and J [Φ] with respect to the
surf SAPT var
order of multipole expansion used to approximate the primitive wave function Φ
were presented in our previous work [36]. By analyzing these numerical results we
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January18,2016 MolecularPhysics 1a
have found that the convergence rate of the leading term, j , of the asymptotic
0
expansion of Eq. (4) corresponds to the convergence radius equal to 2, 1, and 4 for
the surface-integral, SAPT, and variational formula, respectively.
In this communication we shall present a rigorous mathematical derivation of
the discovered convergence rates and the corresponding convergence radii (with
respect to the order of multipole expansion of Φ) for j calculated from the surface
0
J [Φ]andvariationalJ [Φ]expression(inthecaseofJ [Φ]suchanalysishas
surf var SAPT
already been presented in Ref. [36]). Furthermore we derive closed-form formulas
for partial sums of the series approximating j and show that these partial sums
0
converge tothecorrectvaluej = 1.Weshallalsoexplainwhyinthecalculations
0
−
of j using the primitive function approximated by a finite polarization expansion
0
Φ(K) the convergence is initially geometric but switches to harmonic when the
multipole expansion of Φ(K) is carried out to sufficiently high order.
2. Asymptotic expansion of the polarization wave function
Thenon-relativisticBorn-OppenheimerHamiltonianforaninteractingsystemcon-
sisting of two molecules A and B decomposes naturally as H = H + V, where
0
H = H + H is the unperturbed part (H and H being the Hamiltonians
0 A B A B
of isolated monomers A and B, respectively) and V is the perturbation account-
ing for the Coulomb interactions between charged particles belonging to different
monomers. In the case of H+ the pertinent partitioning of H is
2
1 1 1 1
H = 2 , V = , (8)
0
−2∇ − r R − r
a b
where r and r are the distances of the electron from the nuclei a and b, respec-
a b
tively.
Polarization approximation is obtained when one applies the usual RS perturba-
tion theory to the above definition of H and V, with the the unperturbed wave
0
function ϕ taken as the product of the ground state wave functions of isolated
0
subsystems (in case of H+ the unperturbedwave function is the ground state wave
2
function of a hydrogen atom a, ϕ = 1s ). The polarization corrections to energy
0 a
are given by
E(k) = ϕ Vϕ(k−1) , (9)
0
h | i
whereas the wave function corrections are defined recursively by equations
k
(H E )ϕ(k) = Vϕ(k−1) + E(m)ϕ(k−m) (10)
0 0
− −
mX=1
and the intermediate normalization condition ϕ(k) ϕ = 0 holding for k > 0. For
0
h | i
k = 0wehaveϕ(0) = ϕ .Thewavefunctioncorrectionscanbewritteninaproduct
0
form, separating the zeroth-order wave function,ϕ(k) = ϕ f(k), wheref(0) = 1 and
0
for k 1 the factors f(k) satisfy the equation [37, 38]
≥
∂f(k) k
1 2f(k)+ = Vf(k−1)+ E(m)f(k−m). (11)
− 2∇ ∂r −
a mX=1
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January18,2016 MolecularPhysics 1a
When the perturbation V in Eq. (11) is represented by its multipole expansion
V V R−2+V R−3+V R−4+..., (12)
2 3 4
∼
V = rn−1P (cosθ ), (13)
n − a n−1 a
one obtains the asymptotic expansion for f(k)
f(k) f(k)(r ,θ )R−n, (14)
∼ n a a
Xn
where θ is the inclination angle of the electron in a coordinate system centered at
a
the nucleus a and P (x) is the Legendre polynomial. Using Eq. (11) one can verify
l
(k)
that the factors f (r ,θ ) are finite order polynomials in r and cosθ
n a a a a
f(k)(r ,θ )= rm t(k) P (cosθ ). (15)
n a a a nml l a
Xm Xl=0
such that m n and l m. Eq. (15) may be viewed as an expansion of Eq. (28) of
≤ ≤
Ref. [36] in powers of the perturbation V [the function f in Eq. (28) of Ref. [36]
n
is defined via φ = f φ ; this definition has been erroneously omitted in the line
n n 0
preceding Eq. (28)].
Since Φ Φ = 1 + O(R−4) and Φ HΦ = E + O(R−4) the asymptotically
0
h | i h | i
leading term of J [Φ], defined by the j coefficient in Eq. (4), can be obtained by
var 0
considering a simplified formula
J∗ [Φ]= Φ (H E )PΦ . (16)
var h | − 0 i
Evaluation of Eq. (16) with the multipole expansion of the polarization function
truncated after the Kth term,
K
Φ(K) ϕ R−n f(k), (17)
∼ 0 n
Xn Xk=0
leads to a linear combination of integrals
1
π Z e−ra−rbramrbm′Pl(cosθa)Pl′(cosθb)d3r =
(18)
(m+1)!(m′+1)!
= 2e−R Rm+m′+2 1+O(R−1) .
(m+m′+3)!
(cid:2) (cid:3)
The asymptotic formula (18) shows that only the terms with m = n in Eq. (15)
contribute to j of the expansion (4). For this reason we will further consider only
0
(k) (k)
the dominant, m = n part of f , denoted f , given by
n n
e
f(k) = rn t(k)P (cosθ ), (19)
n a nl l a
Xl=0
e
(k) (k)
where t has been denoted by t . Inserting Eqs. (14) and (15) into Eq. (11) and
nnl nl
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January18,2016 MolecularPhysics 1a
comparing coefficients at terms proportional to rn−1R−n one obtains the relation
a
n−1 n−2
(k) (k−1)
n t P (cosθ ) = t P (cosθ )P (cosθ ). (20)
nl l a jl l a n−j−1 a
Xl=0 Xj=0Xl=0
In deriving Eq. (20) we took advantage of the fact that 2f(k) and the last term
∇
in Eq. (11) do not contain terms proportional to rn−1R−n.
a
SincetheasymptoticvalueoftheintegralofEq.(18)isindependentoflandl′,the
angular dependenceoff(k) doesnotaffect theasymptotic estimateof Eq.(16)and,
therefore, has no influence on j . In other words, the value of the asymptotically
0
e
leading term of J [Φ] depends only on the values of Φ on the line joining the
var
nuclei a and b, where Pl(cosθa) = Pl′(cosθb) = 1, similarly as it was observed
for J [Φ] in Ref. [36]. Thus, in the analysis of the convergence properties of
SAPT
(k)
approximations to j it is sufficient to calculate the values of the functions f on
0 n
the line joining the nuclei,
e
f(k) = t(k)rn, (21)
n θa=0 n a
(cid:12)
e (cid:12)
(k)
where t is defined as the sum
n
t(k) = t(k). (22)
n nl
Xl=0
(k)
Substituting θ = 0 in Eq. (20) we obtain a simple recurrence relation for the t
a n
coefficients
n−2
1
t(k) = t(k−1) (23)
n n j
j=X2k−2
(0) (0)
with the initial, k = 0 values given by t = 1 and t = 0 for n > 0. In setting
0 n
(k)
the lower summation limit in Eq. (23) we used the fact that t = 0 for n < 2k.
n
In Eq. (23) and throughout the paper we use the convention that when the lower
summation limit is greater than the upper one the sum is zero. Using Eq. (23) it
is straightforward to show that
n 1 1
t(k) = − t(k) + t(k−1), n > 1, (24)
n n n−1 n n−2
(k)
which furnishes a fast algorithm for the calculation of t .
n
The values of the multipole expansion of Φ(K), Eq. (17), truncated after R−N
(k)
termandcomputedonthelinejoiningthenucleiwithf replacedbyitsdominant
n
(k)
part f , Eq. (19), are given by
n
e
N
Φ(K) = ϕ d(K)rnR−n, (25)
N θa=0 0 n a
(cid:12) nX=0
e (cid:12)
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January18,2016 MolecularPhysics 1a
where
K
d(K) = t(k). (26)
n n
Xk=0
The multipole expansion of the primitive function Φ truncated after R−N, denoted
by Φ , is
N
N [n/2]
Φ = lim Φ(K) = ϕ R−n f(k), (27)
N N 0 n
K→∞
nX=0 Xk=0
where [n/2] denotes the integral part of n/2. The finite upper limit of the k sum-
(k)
mation in Eq. (27) is a consequence of the fact that t = 0 for k > n/2. The
n
dominant part of Φ , denoted by Φ , is defined in the same way as Φ except
N N N
(k) (k)
that f is replaced by f . Onthe linejoining thenuclei the function Φ is given
n n e N
by Eq. (25) with superscripts (K) omitted and d defined by
n
e e
[n/2]
d = lim d(K) = t(k). (28)
n n n
K→∞
X
k=0
We shall now derive a closed-form formula for d . This can be done via a recur-
n
rence for differences of consecutive d ’s with n 2, which using Eqs. (23) and (28)
n
≥
takes the form
∞ n−2 n−3
1 1
(k−1) (k−1)
d d = t t , (29)
n − n−1 (cid:18)n j − n 1 j (cid:19)
Xk=1 j=X2k−2 − j=X2k−2
where the lower limit of the k summation was set equal to 1 since for n 1 the
≥
k = 0 term does not contribute to thesummation in Eq.(28). Thesummation over
(k)
k is actually finite and limited by the condition t = 0 for k > n/2.
n
Extracting the j = n 2 term form the left inner sum and combining the re-
−
maining part with the other sum gives
∞ n−3
1 1
(k−1) (k−1)
d d = t t . (30)
n− n−1 n (cid:18) n−2 − n 1 j (cid:19)
Xk=1 − j=X2k−2
Comparison with Eqs. (23) and (26) allows to rewrite Eq. (30) as
1
d d = (d d ), (31)
n n−1 n−1 n−2
− −n −
which together with the fact that d = 1 and d = 0 gives
0 1
n ( 1)m
d = − . (32)
n
m!
mX=0
Eq. (32) agrees with Eq. (36) of Ref. [36] obtained using a different derivation, not
related to the polarization expansion of the wave function.
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January18,2016 MolecularPhysics 1a
(k)
One may note that after multiplication by n! the coefficients d and t have a
n n
combinatorial interpretation: n!d is equal to the so-called derangement number
n
D (see Ref. [39] p. 59, Ref. [40] p. 180, and Ref. [41] entry A000166) defined as
n
the number of permutations with no fixed points (ie. permutations in which all
(k)
elements change places), while n!t is equal to the associated Stirling number
n
d(n,k), defined as the number of derengements of n elements having exactly k
permutation cycles (Ref. [40] p. 256, Ref. [39] p. 73, and Ref. [41] entry A008306).
3. Closed-from expressions for the partial sums and the convergence rates of
the multipole expansion of the exchange energy j0
Approximations to j obtained from J [Ψ] and J [Ψ], where Ψ is any approxi-
0 surf var
mation to the primitive function Φ, will be denoted by jsurf[Ψ] and jvar[Ψ], respec-
0 0
tively. Theconvergence of theseapproximations resultingwhenΨ is substitutedby
the multipole expanded polarization function can be characterized with the help
of the increment ratios defined as
jsurf[Φ(K)] jsurf[Φ(K) ]
ρ(K)(jsurf) = 0 N − 0 N−1 , (33)
N 0 jsurf[Φ(K) ] jsurf[Φ(K)]
0 N+1 − 0 N
where Φ(K) is the multipole expansion of Φ(K) truncated after the R−N term, cf.
N
Eq. (17). Increment ratios ρ(K)(jvar) are defined analogously. The convergence of
N 0
results obtained when Ψ is substituted by Φ will be characterized by the incre-
N
ment rations ρ (jsurf) and ρ (jvar), defined as in Eq. (33), but with superscripts
N 0 N 0
(K) omitted. Similar convention will be applied to other quantities as well —
symbols without the superscript (K) will denote the K limits of the corre-
→ ∞
sponding expressions carrying this superscript. It should be noted that the r.h.s.
(K) (K)
of Eq. (33) remains unchanged if the functions Φ or Φ are replaced by Φ
N N N
or Φ , respectively.
N e
e
3.1. Surface-integral formula
IthasbeenshowninRef.[38]thatthelargeRasymptotics ofJ [Φ]isdetermined
surf
by the value of Φ at the mid-point of the line joining the nuclei. Specifically, if
Φ = φ f(r ,θ ) then
0 a a
1
J [Φ] = Re−R[f(R/2,0)]2 +O(e−R). (34)
surf
−2
(K)
Thus, in view of Eq. (25) the asymptotics of J [Φ ] is determined by the
surf N
constant
e
jsurf[Φ(K)] = jsurf[Φ(K)]= r(K) 2, (35)
0 N 0 N −4 N
(cid:0) (cid:1)
e
where
N (K)
d
(K) n
r = . (36)
N 2n
Xn=0
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January18,2016 MolecularPhysics 1a
(K)
Using mathematical induction one can easily prove that the K limit of r ,
→ ∞ N
denoted by r , is given by
N
r = 2e 1 2−Nd , (37)
N N − 2 − N
(cid:0) (cid:1)
where e (x) denotes the exponential sum function
n
n xm
e (x) = . (38)
n
m!
mX=0
Combining Eqs. (35) and (37) one obtains a simple closed-form formula for
jsurf[Φ ], that is for the exchange energy determined by partial sums of the mul-
0 N
tipole expansion of the primitive function:
e
jsurf[Φ ]= 2e ( 1) 2−Nd 2. (39)
0 N −4 N −2 − N
(cid:2) (cid:3)
Taking the N limit we see that the leading, j term in the expansion of Eq.
0
→ ∞
(4) is correctly obtained in this way, i.e,
lim jsurf[Φ ]= 1, (40)
N→∞ 0 N −
in agreement with the results of Ref. [36].
Equation (39) can be compared with the formula for j obtained by Tang et al.
0
[38] usingthe surface-integral expression and the polarization function Φ(K). Their
Eq. (7.25) can be recast in our notation as follows:
∞ ∞
1 1 1
jsurf[Φ(K)]= 1+ +
0 −2(cid:20) mX1=2m12m1 m1X,m2=2m1(m1+m2)2m1+m2
(41)
∞ 2
1
+ +... ,
m1,mX2,m3=2m1(m1+m2)(m1+m2+m3)2m1+m2+m3 (cid:21)
where the terms in the square brackets involving ktuple summation originate from
the kth order polarization function φ(k). The last term (not explicitly written)
involves theKtuplesummationandoriginatesfromφ(K).Eq.(41)isasumofterms
of increasingly high order in V, each of which is an infinite sum corresponding to
the expansion in powers of R−1. On the other hand, Eq. (39) contains only single
sums over terms of different order in R−1, each of which is a closed-form sum of
contributions of different orders in V.
Knowing the expression for r , the increment ratio ρ (jsurf), defined by the
N N 0
K limit of Eq. (33), can be written in the form
→ ∞
d r 2−N−1d
ρ (jsurf) = 2 N N − N . (42)
N 0 d r +2−N−2d
N+1 N N+1
When N the ratio d /d is equal to 1 with an error of the order of
N N+1
→ ∞
1/(N +1)! Estimating the remaining factor using Eq. (37) one finds that
3
ρ (jsurf)= 2 2−N +O(4−N), (43)
N 0 − 4√e
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January18,2016 MolecularPhysics 1a
which is the result discovered numerically in Ref. [36]. We see that the sequence of
approximations to j obtained usingthe surfaceintegral formula and themultipole
0
expansion of the wave function converges like a series with the convergence radius
equal to 2. In Sec. 4.1 we show that for finite K the convergence radius corre-
sponding to the sequence ρ(K)(jsurf) remains also equal to 2, although the rate of
N 0
convergence becomes somewhat slower in this case.
ConcludingthisSubsectionitmay beremarkedthattheauthorsofRef.[38]were
only concerned with the K limit of Eq. (41) and did not consider the con-
→ ∞
vergence rate of the obtained series expansions. In fact, this rate cannot be simply
inferred from Eq. (41). On the other hand, the convergence rate corresponding to
Eq. (39) can be easily obtained and is given by Eq. (43).
3.2. Variational formula
In the case of the variational volume-integral formula it is easier to analyse the
convergence rate than to obtain a closed-form expression for the partial sum of the
corresponding expansion. We therefore start by considering the increment ratio of
Eq. (33) first.
3.2.1. Convergence rate of the variational formula
Employing Eqs. (18) and (25) one obtains
N
2
jvar[Φ(K)]= d(K)d(K)G , (44)
e 0 N m1 m2 m1,m2
m1X,m2=0
where
m !m !(m m [m +m +4] 2)
1 2 1 2 1 2
G = 2 − . (45)
m1,m2 (m +m +3)!
1 2
The numerator in Eq. (33) multiplied by 2/e can now be written as
(K) (K) (K) (K) 2
q = 2d p + d G , (46)
N N N N N,N
(cid:0) (cid:1)
where
N−1
p(K) = d(K)G . (47)
N m m,N
mX=0
(K)
A closed-form formula for the the K limits of p , denoted by p , can be
→ ∞ N N
obtained using the summation by parts formula,
n n−1
u w = u W u u W , (48)
j j n n j+1 j j
− −
Xj=0 Xj=0(cid:0) (cid:1)
where
j
W = w . (49)
j m
mX=0
10