A semiclassical theory of the chemical potential for the Atomic Elements
AA semiclassical theory of the chemical potential for theAtomic Elements
Bernard J. Laurenzi Department of ChemistryUAlbany, The State University of New YorkFebuary 4, 2020
Abstract
The chemical potiential for the ground states of the atomic elements have been calculatedwithin the semiclassical approximation The present work closely follows Schwinger and En-glert’s semiclassical treatment of atomic structure. µ of the Atomic Elements For an atomic element containing N electrons with nuclear charge Z , the electronic chemical po-tentials µ for neutral atomic species (which are in their electronic ground states Ψ with energy E )are defined as µ = (cid:18) ∂E∂N (cid:19) Z .Here N and Z will be regarded as continuous variables and µ can then be calculated by noting that dE ( Z, N ) = (cid:18) ∂E∂Z (cid:19) N dZ + (cid:18) ∂E∂N (cid:19) Z dN, together with the well-known relation (cid:18) ∂E∂Z (cid:19) N = V ne Z .
In the equation above, V ne is the average value of the nuclear-electronic potential energy (in atomicunits) i.e. V ne Z = − N (cid:88) i =1 (cid:90) | Ψ | r i dτ, and dτ = d r d r d r . . . d r N . We have as a result µ = d E ( Z, Z ) d Z − V ne Z , (1)where dE/dZ is the directional derivative of the electronic energy surface E ( Z, N ) along the curve Z = N. In this work E ( Z, Z ) is taken to be the electronic energy which has been computed withinthe semiclassical approximation by Schwinger and Englert.1 a r X i v : . [ phy s i c s . a t o m - ph ] A p r .1 Properties and estimates of the chemical potential for the elements The physical interpretation or significance of the electronic chemical potential µ is seen as a measureof the propensity of an electron to leave an atom. In this context as the atomic number increases µ gives the stability of an element relative to others in the periodic table.An associated quantity η defined by η = 12 (cid:18) ∂µ∂N (cid:19) Z ,has been called the hardness [1] and has been interpreted as the resistance of an atom to theingress of additional electrons. The higher the hardness the the lower the polarizability of theatom’s electron cloud and the greater the resistance of that atom to add an electron. Variousestimates for the chemical potential and the hardness have been made. March [2] has given anestimate of µ by assuming that the energy can be written as a Taylor series in the variable ( N − Z )i.e. E ( Z, N ) = E ( Z, Z ) + ( N − Z ) (cid:18) ∂E∂N (cid:19) | Z = N + ( N − Z ) (cid:18) ∂ E∂N (cid:19) | Z = N + · · · . In that work he has shown that to third order (from a fifth order polynomial in N − Z ) that thechemical potential can be written as µ = µ + µ , with µ = − ( I + A ) ,µ = (3 I − I + 43 I − I + 18 A ) , and where I n is the n-th ionization potential of the atom and A is it’s electron affinity. Usingthe empirical relation I n ∼ = nI the chemical potential to third order µ = A and we have twoestimates µ = − ( I + A ) , to lowest order and µ = − I − A, which includes µ . It is interesting to note that Mulliken’s electronegativity function χ M which isdefined as χ M ≡ ( I + A ) > , and is interpreted as the ability of an atom to attract electrons is approximately related to − µ .Piris and March [3] using natural orbital functional (NOF) theory have estimated µ and comparedit to − I for neutral atom (H-Kr) as seen in the Fig.(1) below.Their chemical potential values parallel the oscillations in the experimental ionization potentialbut deviate widely in magnitude from − I in case of the rare gases. If one wishes to interpret theelectronic chemical potential as the atomic analogue of the macroscopic thermodynamic chemicalpotential, then µ as define above is an indication of the spontaneity of the escaping tendency of anelectron from an atom. 2igure 1: The Chemical Potential µ vs. Z. µ Within the “semiclassical approximation,” Schwinger and Englert [4] (SE) have given an expressionfor E ( Z, Z ). In that work, the authors have shown that the total energy is made up of thesemiclassial i.e. Thomas-Fermi (TF) energy [5] and a quantum oscillating part i.e. E ( Z, Z ) = E T F ( Z, Z ) + E osc ( Z ) . Furthermore, the well-known value for E T F ( Z, Z ) is given by [6] E T F ( Z, Z ) = (cid:0) π (cid:1) / Φ (cid:48) (0) Z / , (2)with Φ (cid:48) (0) = − . x ) and where the TFpotential V T F is V T F = − Z Φ( x ) /r, and x is the TF scaled distance x = 2 / r/ (3 π ) / The average value of the TF potential energy is[8] V T F = − Z (cid:90) ∞ ρ ( r ) r d r =2 (cid:0) π (cid:1) / Φ (cid:48) (0) Z / . (3)We shall see below that the average value of the nuclear-electronic potential V ne like the totalenergy, can also be written as a sum of a TF term and a quantum oscillating contribution i.e. V ne = V T F + V osc . As a result of (2) and (3) the TF part to the chemical potential is seen to vanishes, µ T F = 0 , Z / and we have as a result µ = dE ( Z, Z ) osc dZ − V ne, osc Z .
The purpose of this work is to give the corresponding semiclassical expression for V ne,osc re-sulting in a semiclassical value for the chemical potential for neutral atomic species. As will beseen below the computation of that quantity unfortunately requires a rather elaborate analysis.This investigation does not contain the effects due to the antisymmetry [9] of the system’s wavefunctions or the effects of the tightly bound electrons first taken into account by Scott [10] nor thequantum correction to the wave function due to the kinetic energy [11]. Inclusion of these effectsis problematic and beyond the scope of this work.We begin the analysis of V ne by recognizing that since the terms within V ne are single-particleoperators, integration over the N − N particle wave function reduces V ne to − V ne Z = (cid:90) (cid:37) ( r ) r d r , where (cid:37) is the single-particle electron density function defined as (cid:37) ( r ) = N (cid:90) . . . (cid:90) | Ψ | d r d r . . . d r N . This function could for example be taken to be the Thomas-Fermi electron density. Instead, withinthe semi-classical (WKB) [12], and the Hartree-Fock orbital approximations we take the density tobe (cid:37) ( r ) = 2 ∞ (cid:88) l =0 ∞ (cid:88) n r =0 (2 l + 1) η ( − E l,n r − ζ ) | u l,n r ( r ) / √ πr | . (4)The single particle energies E l,n r associated with the potential V ( r ) are labeled with the radial quan-tum number n r and the angular quantum number l respectively and the quantities u l, n r ( r ) / √ πr are the WKB single-particle semiclassical wave functions where − ζ is the single-particle energy ofthe highest occupied orbital ( ζ ≥ η is the Heaviside function as shown in Fig. (2) below.The latter function having been introduced in order to provide cutoffs in the sums in (4) overthe positive integer quantum numbers l, n r thereby removing energies larger than − ζ . The factorof 2 in Eq.(4) has been included in the sum to account for the spin states. Using (4) we have V ne = 2 ∞ (cid:88) l =0 ∞ (cid:88) n r =0 (2 l + 1) η ( − E l, n r − ς ) V l,n r , (5)with V l , n r = − Z (cid:90) r u r l | u l , n r ( r ) | r dr, and r l and r u are the WKB lower and upper classical turning points which define the classicallyallowed region. The WKB functions u l,n r ( r ) being given by u l,n r ( r ) = A l,n r √ (cid:104) { E l,n r − V ( r ) − ( l +1 / / r } (cid:105) / , (6)4igure 2: The Heaviside function vs. Energy.5here A l,n r are normalization constants and V ( r ) is a central but not necessarily the Coulombicpotential (The r dependencies of the “phase factors” [13] of these functions are being ignored here).The potential V ( r ) represents the interaction of an electron with the nuclear charge as well as withthe other electrons in the atom, an approximate example of which is the TF potential.Within the WKB approximation one also has the relation n r + 1 / π (cid:90) r u r l (cid:113) { E l,n r − V ( r ) − ( l +1 / / r } dr, (7)where r l and r u referred to above are the roots of the quantity within the square root of thatexpression. Following Schwinger and Englert [14] we define the quantities λ, ν, ε λ,ν , and υ λ,ν as λ = l + 1 / ,ν = n r + 1 / ,ε λ,ν = E l , n r ,υ λ,ν = V l , n r , and regard them as continuous variables in the equations below. Then (7) becomes ν = 1 π (cid:90) r u r l (cid:113) { ε λ,ν − V ( r ) − λ / r } dr. (8)Rewriting (5), we note that the sums now extend over the negative as well as the positive values of l and n r . The former values however, do not contribute to the sums and we get V ne = 4 (cid:90) ∞ λdλ ∞ (cid:88) l = −∞ δ ( l + 1 / − λ ) (cid:90) ∞ dν ∞ (cid:88) n r = −∞ δ ( n r + 1 / − ν ) υ λ,ν η ( − ε λ,ν − ς ) , where δ ( z ) is the Dirac delta function. Using the Poisson identities ∞ (cid:88) l = −∞ δ ( l + 1 / − λ ) = ∞ (cid:88) k = −∞ ( − k exp(2 πikλ ) , ∞ (cid:88) n r = −∞ δ ( n r + 1 / − ν ) = ∞ (cid:88) j = −∞ ( − j exp(2 πijν ) , the expression for V ne becomes V ne = 4 ∞ (cid:88) k = −∞ ∞ (cid:88) j = −∞ ( − k + j (cid:90) ∞ λ exp(2 πikλ ) dλ (cid:90) ∞ exp(2 πijν ) υ λ,ν η ( − ε λ,ν − ς ) dν. (9) λ , ν And ε ( λ , ν ) Before proceeding, it is useful to examine the relations among the quantities ν , λ and ε ( λ, ν ) . Fora given V ( r ) and energy ε, and for a range of values of ν which will be discussed below, the rootsof the relation 2 r [ ε λ,ν − V ( r )] − λ ( r ) = 0 , (10)6igure 3: The Effective potential vs. ri.e. r l and r u , define the classical turning points for the system. We note that it follows from Eq.(4) that when ν = 0, that these roots coalesce to the single value of r ε . This behavior can be seengraphically in Fig. (3) where we have plotted the effective potential V (cid:48) = V ( r ) + λ / r versus r .For a given energy ε ( λ ε ,
0) the curve V (cid:48) ( r ) is seen to have a single turning point denoted by r ε . However, the curve V (cid:48) ( r ) for any ε ( λ, ν ) > ε ( λ ε ,
0) has two turning points r l , r u which are lessthan and greater than r ε . Recalling that these turning points are the roots of Eq. (8) we will seein what follows that r ε is the distance at which λ has it maximum value i.e. λ ε . This value is givenby λ ε ( r ε ) = (cid:112) r ε { ε ( λ ε , − V ( r ε ) } . Furthermore, in the case where the single-particle energy has its absolute highest value referred toabove as − ζ and hereafter as ε , the corresponding largest of the maximum values of λ ε is heredenoted by λ and satisfies the relation λ ( r ) = (cid:113) r { ε − V ( r ) } , where r is the distance at which λ has the largest maximum value λ and is the min point in the V (cid:48) curve corresponding to the energy ε .To demonstrate the behavior of λ we take as an example the case of the classical turning pointsthe TF potential [15] i.e. − Z Φ( x ) /r . In Fig. (4) we have (for a given ε ) plotted the scaled quantity (cid:101) λ = λ ( r ) / √ aZ / in terms of the scaled energy (cid:101) (cid:15) = − a Z / ε and the scaled distance x = Z / r/a where a = ( π ) / and | (cid:101) (cid:15) | < | (cid:101) (cid:15) (cid:48) | , then (cid:101) λ = (cid:112) x { Φ( x ) − x (cid:101) (cid:15) } . λ vs r for the TF potential.In Fig. (3) we see that at every energy ε for a given λ there are two turning points except atthe maximum value of λ, i.e. λ ε ( r ε ) occurring at r ε . The corresponding range of physical values of x being 0 ≤ x ≤ x max where x max is determined by the roots of the equation Φ( x max ) − x max (cid:101) (cid:15) = 0 . The quantity r ε which allows calculation of λ ε ( r ε ) can be obtained from the equation d (cid:101) λd x | x = x ε = 0 , or d { x Φ( x ) } dx | x = x ε = 2 x ε (cid:101) (cid:15). This behavior is shown in Fig. (4)As a further example of the behavior of λ, consider the case of the Coulomb potential where wehave Z r l = 1 ± √ − (cid:15) λ (cid:15) , where (cid:15) = − ε/Z , which yield two turning points except when ν = 0 and where λ ε ( r ) = Z √ | ε | at which r ε = Z | ε | . InFig. (5) we have plotted λ (cid:15) = (cid:112) r [1 − r (cid:15) ]as a function of the scaled distance r = Z r and thescaled energy (cid:15)
We see in the case of the Coulomb potential for a given ε with λ ε = 0, there is a set of classicalturning points which occur at 0 and Z | ε | , whereas the maximum values of λ ε are λ ε, max ( r ) = Z √ | ε | ,and the set of single turning points occur at r ε, max = Z | ε | . Furthermore, for a given ε ( λ, ν ) = const the variables λ and ν ( λ | ε ) are related as shown in the Fig. (6) below.8igure 5: λ vs r for the Coulomb potential9igure 6: ν vs. λ for the TF potential.We expect the shape of the curves in Fig. 6 to show concave curvature as is the case of the TFpotential. Along these curves the energy is constant ( curves of degeneracy) . In the case of theCoulombic potential, where the energy is ε = − Z / λ + ν ) the curves in the plot of ν ( λ | ε ) versus λ for different ε consists of a family of straight lines whereas in the case of a general potential weexpect these lines to be curved as shown above [16]. In addition we denote the maximum value of ν i.e. ν (0 | ε ) = ν ε . From this we see that for a given ε the quantities λ and ν are restricted to theranges 0 ≤ λ ≤ λ ε ( r ) , ≤ ν ≤ ν ( λ | ε ) . The domain of integration in λ, ν space is seen to consist of all λ, ν values below the curves ofdegeneracy ν ( λ | ε ) corresponding to ε = ε . With this in mind we can for a given ε rewrite theaverage potential V ne as V ne = 4 lim ε −→ ε ∞ (cid:88) k = −∞ ∞ (cid:88) j = −∞ ( − k + j (cid:90) λ ε ( r )0 λ exp(2 πikλ ) dλ (cid:90) ν ( λ | ε )0 exp(2 πijν ) υ λ,ν dν, (11)with ν ( λ | ε ) denoting the curves of degeneracy. In the work to follow it is useful to define the integrals N j ( λ ε , ε λ,ν ) as N j ( λ, ε ) = (cid:90) ν ( λ | ε )0 cos(2 πj ν ) υ λ,ν dν. k, j space. k, j Space
In order to make progress in evaluating the terms in the double sum in Eq. (11) for V ne it is usefulto divide k, j index space (note that λ and ν are associated with the indices k, and j respectively)into the regions shown in the diagram below. These regions shown in Fig. (7) correspond roughlyto those chosen by SE in their evaluation the energy of the system.The TF region consists of the single point j = 0 , k = 0the l TF region consists of the points on the vertical line j = 0 , ≤ k ≤ ∞ , the λ region is given by k = 0 , ≤ j ≤ ∞ ,and the λ, ν region consists of the points covered by the ranges1 ≤ j ≤ ∞ , ≤ k ≤ ∞ . V ne in terms of these regions we have V ne = lim ε → ε (cid:8) V ne,T F ( ε ) + V ne,lT F ( ε ) + V ne, λ ( ε ) + V ne, λ,ν ( ε ) (cid:9) , (12)where V ne,T F ( ε ) = 4 (cid:90) λ ε λ N ( λ, ε ) dλ, (13a) V ne, l T F ( ε ) = 8 ∞ (cid:88) k =1 ( − k (cid:90) λ ε λ cos(2 πkλ ) N ( λ, ε ) dλ, (13b) V ne, λ ( ε ) = 8 ∞ (cid:88) j =1 ( − j (cid:90) λ ε λ N j ( λ, ε ) dλ (13c) V ne, λ,ν ( ε ) = 16 ∞ (cid:88) k =1 ∞ (cid:88) j =1 ( − k + j (cid:90) λ ε λ cos(2 πkλ ) N j ( λ, ε ) dλ . (13d)The numerical factors appearing in Eqs (13) result from the contributions from the second, thirdand fourth quadrants of Fig. (7). We will see below that the first term shown in (13) i.e. V ne,T F produces a non-oscillatory, semiclassical expression for the average nuclear electronic potential. Thesum V ne, l T F, over the second region produces an oscillatory semiclassical expression which we willcall the ‘ l − quantized’ semiclassical average potential. Oscillatory terms in the remaining regionswill be called the ‘ λ , and the λ, ν - quantized’ contributions to the average potential respectively. N j ( λ , ε λ,ν ) = (cid:82) ν ( λ | ε )0 cos(2 π jν ) υ λ,ν dν The program for the evaluation of the expression for V ne in Eq. (12), is best carried out byinvestigating its various parts in a stepwise fashion in order to simplify the exposition of the work.We begin by noting that (recall υ λ,ν = V l , n r , ) − υ λ,ν Z = (cid:90) r u r l | u l,n r ( r ) | r dr = | A λ,ν | (cid:90) r u r l drr (cid:112) { ε λ,ν − V ( r ) − λ / r } , (14)and (cid:90) ∞ | u l,n r ( r ) | dr = 1 = | A λ,ν | (cid:90) r u r l dr (cid:112) { ε λ,ν − V ( r ) − λ / r } . Differentiation of ν with respect to ε in Eq. (8) gives (cid:18) ∂ν∂ε λ,ν (cid:19) | λ = 1 π (cid:90) r u r l dr (cid:112) { ε λ,ν − V ( r ) − λ / r } , and thus | A λ,ν | π (cid:18) ∂ε λ,ν ∂ν (cid:19) | λ . As a result we may rewrite Eq. (14) as (valid for all λ and ν ) − υ λ,ν Z = 1 π (cid:18) ∂ε λ,ν ∂ν (cid:19) λ (cid:90) r u r l dr (cid:112) r { ε λ,ν − V ( r ) } − λ . N j ( λ, ε ) over the variable ν then becomes (for a given λ the energy ν can vary with ε ) N j ( λ, ε ) = − Zπ (cid:90) ν ( λ | ε )0 dν cos(2 πjν ) (cid:18) ∂ε λ,ν ∂ν (cid:19) λ (cid:90) r u r l dr (cid:112) r { ε λ,ν − V ( r ) } − λ . (15)Interchanging the order of integration we get N j ( λ, ε ) = − Zπ (cid:90) r u r l dr (cid:90) ν ( λ | ε )0 dν (cid:18) ∂ε λ,ν ∂ν (cid:19) λ cos(2 πjν ) (cid:112) r { ε λ,ν − V ( r ) } − λ . (16)If the relation( ∂ε λ,ν ∂ν ) λ cos(2 πjν ) (cid:112) r { ε λ,ν − V ( r ) } − λ = 2 πj sin(2 πjν ) √ r { ε λ,ν − V ( r ) }− λ r + ∂∂ν [cos(2 πjν ) √ r { ε λ,ν − V ( r ) }− λ r ] λ , is used, we get an expression for N j ( λ, ε ) which is partially integrable i.e. N j ( λ, ε )= − Zπ (cid:90) r u r l dr (cid:90) ν ( λ | ε )0 { πj sin(2 πjν ) √ r { ε λ,ν − V ( r ) }− λ r + ∂∂ν [cos(2 πjν ) √ r { ε λ,ν − V ( r ) }− λ r ] λ } dν . And finally we obtain N j ( λ, ε ) = − Zπ (cid:90) r u r l drr cos(2 πjν ( λ | ε )) (cid:113) r { ε λ,ν ( λ | ε ) − V ( r ) } − λ − Z j (cid:90) r u r l drr (cid:90) ν ( λ | ε )0 sin(2 πjν ) (cid:113) r { ε λ,ν − V ( r ) } − λ dν, an expression which will be useful in the evaluation of the integrals in Eq. (12). r , λ , ω , ν (1)0 , ν (2)0 , K ( r ) At this juncture it is useful to review Thomas-Fermi (TF) theory and to compute some of theparameters which will be needed in the final calculation of the chemical potential. In TF theorythe potential V ( r ) is defined by the equations ∇ V = − π(cid:37) T F ( r ) ,(cid:37) T F ( r ) = 13 π (2 { ε − V ( r ) } ) / . (17)In the case of a spherically symmetric system this equation can be rewritten as d Φ d x = Φ( x ) / √ x , with Φ(0) = 1 , Φ( ∞ ) = 0 ,
13s seen above in the case of the Coulombic potential the value of r at which λ is a maximum fora given ε had been given as r ε = Z | ε | . Here using the TF potential we give the value of r corresponding to the maximum of λ in the casewhere ε → ε . Using Eq. (10) we have d λ ( r ) d r = 0 = dd r {− r V ( r ) } , which is tantamount to d { x Φ( x ) } d x = 0 . The maximum of the quantity x Φ( x ) occurs at x = 2 . x ) = 0 . r being 2 . a/Z / [17]. The largest of the maximum values of λ ε i.e. λ isthen λ ( r ) = (cid:112) a x Φ( x ) Z / = 0 . Z / . In the work that follows a collection of quantities (which have been computed by Schwinger andEnglert ) for the TF potential is given below and will prove useful in the estimation of the Z dependence of the remaining terms in V ne . We have x = 2 . ,r = 1 . Z − / ,ω ε = − d λ ε ( r ) d r | r = r ε ω = 0 . Z / ,λ ( r ) = 0 . Z / , • λ = ∂λ ε ( r ) ∂ε | ε =0 = r /λ ( r ) = 3 . Z λ ( r ) • λ = (cid:16) λ ( r ) r (cid:17) = 0 . Z / ν (1)0 = − ∂ν ( λ ) ∂λ | λ = λ = √ λ ( r ) ω r = 1 . ,ν (2)0 = − ∂ ν ( λ ) ∂λ | λ = λ = ν (1)0 λ ( r ) [( ν (1)0 ) − −
15 + 23( ν (1)0 ) − ν (1)0 ) ] = 0 . /Z / , ω λ ( r ) = 0 . Z K ( r ) = λ ( r ) Z r = 0 . , λ ( x ) ω r = 0 . Z / , .7 The TF Term V ne,T F In the case where j = 0 we have for N ( λ, ε λ,ν ( λ | ε ) ) N ( λ, ε λ,ν ( λ | ε ) ) = − Zπ (cid:90) r u r l drr (cid:113) r { ε λ,ν ( λ | ε ) − V ( r ) } − λ , (18)the corresponding value of V ne,T F then becomes V ne,T F ( ε ) = − Zπ (cid:90) λ ε λdλ (cid:90) r u r l (cid:113) r { ε λ,ν ( λ | ε ) − V ( r ) } − λ dr/r . Interchanging the order of integration results in ( r l ≤ r ≤ r u ) V ne,T F ( ε ) = − Zπ (cid:90) r u r l dr/r (cid:90) λ ε λdλ (cid:113) r { ε λ,ν ( λ | ε ) − V ( r ) } − λ , and we note that as λ varies over its range from 0 to λ ε in the limit as ε → ε → r l = 0 and r u = ∞ these being the roots of 2 r { ε − V ( r ) } − λ = 0 . Integration over λ gives the result V ne,T F = − Z π (cid:90) ∞ (cid:0) r { ε − V ( r ) } (cid:1) / dr/r . (19)Remarkably, if V ( r ) is taken to be the Thomas-Fermi potential, the corresponding particle density (cid:37) T F ( r ) with the form (cid:37) T F ( r ) = 13 π (2 { ε − V ( r ) } ) / , then V ne,T F in Eq. (19) rewritten as a integral over 3 dimensional space is just V ne,T F = − Z (cid:90) (cid:37) T F ( r ) r d r = V T F
We see that the leading term in the expression for V ne is the non-oscillatory Thomas-Fermi averagevalue of the nuclear-electronic interaction V T F . The remaining terms in the sums in Eq. (12)represent the semi-classical and oscillatory contributions to the nuclear-electronic interaction. V ne , l T F
The l TF Oscillations
The sum representing V ne , l T F can be rewritten as V ne , l T F ( ε ) = 8 ∞ (cid:88) k =1 ( − k (cid:90) λ ε dλ λ cos(2 πikλ ) (cid:90) ν ε ( λ )0 υ λ ,ν dνV ne , l T F = lim ε −→ ε V ne , l T F ( ε )15nce again interchange of the order of integration in the integrals above and use of the procedureto obtained V T F (with the expression for N ( λ, ε λ,ν ( λ | ε ) ) in Eq. (18) V ne , lT F ( ε ) can then berewritten as V ne , l T F ( ε ) = − Zπ ∞ (cid:88) k =1 ( − k (cid:90) r u r l drr (cid:90) λ ε dλ · λ cos(2 πikλ ) (cid:113) r { ε λ ε , ν ( λ | ε ) − V ( r ) } − λ . Now we write λ ε = 2 r { ε λ ε , ν ( λ | ε ) − V ( r ) } , and λ as λ = λ ε ( r ) cos θ, ≤ θ ≤ π/ . The expression for V ne , l T F ( ε ) with this change of variable becomes V ne , l T F ( ε ) = − Zπ ∞ (cid:88) k =1 ( − k (cid:90) r u r l λ ε ( r ) r dr (cid:90) π/ sin θ cos θ cos(2 πkλ ε ( r ) cos θ ) dθ. The angular integral appearing in the equation above is well-known and we have (cid:90) π/ sin θ cos θ cos(2 πkλ ε cos θ ) dθ = 13 − π H (2 πkλ ε )4 π k λ ε , where H ( z ) are the Struve functions [18] of order 2. The required sum is then V ne , l T F ( ε ) = − Zπ ∞ (cid:88) k =1 ( − k (cid:90) r u r l λ ε ( r ) r (cid:20) − π H (2 πkλ ε ( r ))4 π k λ ε ( r ) (cid:21) dr. In the case of integer order, the Struve functions H k ( z ) are related to the Weber functions E k ( z )[19] . In this case 13 − π z H ( z ) = π z E ( z ) , and we can write within the semiclassical approximation an exact expression for V ne , l T F ( ε ) as V ne , l T F ( ε ) = − Zπ ∞ (cid:88) k =1 ( − k k (cid:90) r u r l λ ε ( r ) r E (2 πkλ ε ( r )) dr. For the Thomas-Fermi function Φ( x ) where ε = − ς = 0 we note that λ ( r ) = Z / (cid:112) a x Φ( x ) , and see that for large Z, that λ ( r ) is large. For large Z, using the asymptotic expansion for theleading and next to leading terms [20] for E ( z ) i.e. π z E ( z ) ∼ (cid:114) π z / (cid:26) − cos( z + π/
4) + 158 z sin( z + π/ (cid:27) , (cid:90) r u r l λ ε ( r ) r E (2 πkλ ε ( r )) dr = − πk / (cid:90) ∞ λ / ε ( r ) r (cid:26) cos(2 πkλ ε ( r ) + π/ − πkλ ε ( r ) sin(2 πkλ ε ( r ) + π/ (cid:27) dr. As stated in the SE paper we are not interested in the detailed content in this quantity, but insteadonly in the leading oscillatory contributions to it. Evaluation of this integral for large λ ε can beobtained using the ‘ stationary phase approximation .’[21] Recalling that λ ε ( r ) has a maximum atthe point r ε expansion of that function around r ε gives λ ε ( r ) = λ ε ( r ε ) − ω ε λ ε ( r ε ) ( r − r ε ) + . . . , where ω /λ ( r ) is proportional to Z and is large. In the limit as ε → ε and within thatapproximation the leading oscillatory terms for the average potential V ne, l T F is − V ne , l T F Z = 2 √ λ ( r ) ω r (cid:34) ∞ (cid:88) k =1 ( − k ( πk ) cos(2 πkλ ( r )) − λ ∞ (cid:88) k =1 ( − k ( πk ) sin(2 πkλ ( r )) (cid:35) . In the equation appearing above, sums of the kind S n ( z ) = ∞ (cid:88) k =1 ( − k ( πk ) n +1 sin(2 πk z ) ,C n ( z ) = ∞ (cid:88) k =1 ( − k ( πk ) n cos(2 πk z ) , occur. These infinite sums can be rewritten [22] in closed-form in terms of the periodic function (cid:104) z (cid:105) def ined by (cid:104) z (cid:105) = z − (cid:98) z + 1 / (cid:99) , − ≤ (cid:104) z (cid:105) < , where (cid:98) z (cid:99) is the floor function. The f irst few of these sums are given here as S ( z ) = − (cid:104) z (cid:105) ,C ( z ) = (cid:104) z (cid:105) − ,S ( z ) = (cid:104) z (cid:105) [ (cid:104) z (cid:105) − ] ,C ( z ) = − [ (cid:104) z (cid:105) − ] . The leading terms in the potential energy V ne, l T F /Z written in terms of the closed-form ex-pressions becomes − V ne, l T F Z = 2 √ λ ( r ) ω r (cid:20) C ( λ ) − λ ( r ) S ( λ ) (cid:21) , (20)with λ ( r ) ω r = 0 . Z / . The l T F contribution is seen to contain terms of orders Z / and Z / .17igure 8: The k vs. j regions of summation V ne,λ and V ne,λ ,ν Oscillations
The terms V ne,λ and V ne,λ,ν are more complex in nature. In those cases the integrals N j ( λ, ε λ,ν )with j > ν and trigonometric terms therebyrequiring a more elaborate analysis. Using the bounds defined by the appropriate regions ofintegration for λ, ν we have V ne, λ ( ε ) = 8 ∞ (cid:88) j =1 ( − j (cid:90) λ ε λ N j ( λ, ε λ,ν ) dλ (21a) V ne,λ,ν ( ε ) = 16 ∞ (cid:88) k =1 ∞ (cid:88) j =1 ( − k + j (cid:90) λ ε λ cos(2 πkλ ) N j ( λ, ε λ,ν ) dλ , (21b)where the λ, and the λ, ν region has been divided into the two, subregions def ined by k = 0 , ≤ j ≤ ∞ and 1 ≤ j ≤ ∞ , ≤ k ≤ ∞ , respectively.18 .9.1 The Term N j ( λ, ε λ,ν ) For j ≥ The λ Oscillations
We have seen that N j ( λ, ε λ,ν ) has been partially integrated with respect to ν i.e. N j ( λ, ε λ,ν ) = − Zπ (cid:90) r u r l drr cos(2 πjν ( λ | ε )) I ( ν ( λ | ε ) , r, λ ) − Z j (cid:90) r u r l drr (cid:90) ν ( λ | ε )0 sin(2 πjν ) I ( ν, r, λ ) dν, where I ( ν, r, λ ) = (cid:113) r { ε λ,ν − V ( r ) } − λ . The quantity I ( ν, λ, r ) evaluated at ν ( λ | ε ) simplifies and we get I ( ν ( λ | ε ) , λ, r ) = (cid:113) r { ε λ,ν ( λ | ε ) − V ( r ) } − λ = (cid:112) λ ε ( r ) − λ ,I ( ν ( λ | ε ) , λ, r ) = λ ε ( r ) sin θ. Integration by parts of the integral with respect to ν gives N j ( λ, ε λ,ν ) = − Zπ { (cid:90) r u r l drr cos(2 πjν ( λ | ε )) I ( ν ( λ | ε ) , r, λ )+2 πj (cid:90) r u r l drr [sin(2 πjν ( λ | ε ) I ( ν ( λ | ε ) , λ, r ) − πj (cid:90) ν ( λ | ε )0 cos(2 πjν ( λ | ε )) I ( ν ( λ | ε ) , λ, r ) dν ] } , where I ( ν ( λ | ε ) , λ, r ) = (cid:90) ν ( λ | ε )0 I ( ν ( λ | ε ) , λ, r ) dν. This process can be continued and the integral with respect to ν in the equation above can beintegrated by parts once more to yield N j ( λ, ε λ,ν ) = − Zπ { (cid:90) r u r l drr cos(2 πjν ( λ | ε )) I ( ν ( λ | ε ) , λ, r )+ (2 πj ) (cid:90) r u r l drr sin(2 πjν ( λ | ε )) I ( ν ( λ | ε ) , λ, r ) − (2 πj ) (cid:90) r u r l drr cos(2 πjν ( λ | ε )) I ( ν ( λ | ε ) , λ, r ) − (2 πj ) (cid:90) r u r l drr (cid:90) ν ( λ | ε )0 sin(2 πjν ) I ( ν, λ, r ) dν } , where I ( ν ( λ | ε ) , λ, r ) = (cid:90) ν ( λ | ε )0 I ( ν ( λ | ε ) , λ, r ) dν. The process introduced above can in principle be continued indefinitely however, it suff ices toterminate the expression for N j ( λ, ε λ,ν ) at order j .19he integrals I ( ν ( λ | ε ) , λ, r ) and I ( ν ( λ | ε ) , λ, r ) have been approximately evaluated in appendixA and are given here by I ( ν ( λ | ε ) , λ, r ) Zr = λ ε ( r ) Zr (cid:112) λ ε − λ − arctan( λ ε ( r ) Zr (cid:112) λ ε ( r ) − λ ) , or in terms of the angular variable θI ( ν ( λ | ε ) , λ, r ) Zr = K ε sin θ − arctan( K ε sin θ ) , = [ K ε sin θ ] I ( K ε sin θ ) , where I ( z ) is I ( z ) = ∞ (cid:88) κ =0 ( − κ z κ (2 κ + 3) , | z | ≤ I ( z ) = 13 − z z · · · , and K is the unit less constant K = λ ( r ) Z r = 0 . . Then I ( ν ( λ | ε ) , λ, r ) Zr = 13 [ K ε sin θ ] −
15 [ K ε sin θ ] + · · · Similarly we have I ( ν ( λ | ε ) , λ, r )2 Zrλ ε ( r ) = λ ε ( r ) Zr (cid:112) λ ε ( r ) − λ − λ ε ( r ) Zr (cid:112) λ ε ( r ) − λ )+ arcsin h ( λ ε ( r ) Zr (cid:112) λ ε ( r ) − λ ) (cid:114) (cid:16) λ ε ( r ) Zr (cid:17) ( λ ε ( r ) − λ ) , or in terms of θ we have I ( ν ( λ | ε ) , λ, r )2 rZ λ ε ( r ) = K ε sin( θ ) − K ε sin θ ) + arcsin h ( K ε sin θ ) (cid:113) K ε sin θ , = [ K ε sin θ ] I ( K ε sin θ ) , where I ( z ) is I ( z ) = ∞ (cid:88) κ =0 [Γ( κ + 5 / − √ π Γ( κ + 3)] ( − k z κ Γ( κ +7 / , | z | ≤ I ( z ) = − z + z − z + · · · . I ( ν ( λ | ε ) , λ, r )2 rZλ ε ( r ) = [ K ε ( r ε ) sin θ ] − [ K ε ( r ε ) sin θ ] + · · · . We note that the integral I is small compared to I and will be dropped.The integrals in V ne, λ ( ε ) and V ne, λ,ν ( ε ) can then be written as V neλ ; λ,ν = lim ε → (cid:2) V ne, λ ( ε ) + V ne, λ,ν ( ε ) (cid:3) = ∞ (cid:88) j =1 ( − j V ( j ) + 16 ∞ (cid:88) k =1 ∞ (cid:88) j =1 ( − k + j V ( j, k ) . (22)The V ( j ) integrals contain the λ oscillation terms and the V ( j, k ) terms contain the mixed λ, ν oscillation terms. Now we get the expressions V ( j ) = − Zπ { (cid:90) r u r l drr (cid:90) λ ε λ I ( ν ( λ | ε )) cos(2 πj ν ( λ | ε )) dλ + 2 πj (cid:90) r u r l drr (cid:90) λ ε λ I ( ν ( λ | ε )) sin(2 πj ν ( λ | ε )) dλ } , V ( j, k ) = − Zπ { (cid:90) r u r l drr (cid:90) λ ε λ I ( ν ( λ | ε )) cos(2 πkλ ) cos(2 πj ν ( λ | ε )) dλ + 2 πj (cid:90) r u r l drr (cid:90) λ ε λI ( ν ( λ | ε )) cos(2 πkλ ) sin(2 πj ν ( λ | ε )) dλ } . In the equations above we have interchange the order of integration over r and λ and used λ = λ ε cos θ where 0 ≤ θ ≤ π/
2. to give V ( j ) = − Zπ { (cid:90) r u r l λ ε ( r ) r (cid:90) π/ sin θ cos θ cos(2 πj ν ( λ | ε )) dθdr (23)+ 2 πjZ (cid:90) r u r l λ ε ( r ) r (cid:90) π/ sin θ cos θ sin(2 πj ν ( λ | ε )) I ( K ε sin θ ) dθdr } , V ( j, k ) = − Zπ { (cid:90) r u r l λ ε ( r ) r (cid:90) π/ sin θ cos θ cos(2 πkλ ε cos θ ) cos(2 πj ν ( λ | ε )) dθdr (24b)+ 2 πjZ (cid:90) r u r l λ ε ( r ) r (cid:90) π/ sin θ cos θ cos(2 πkλ ε cos θ ) sin(2 πj ν ( λ | ε )) I ( K ε sin θ ) dθdr } . V ( j ) = − Zπ (cid:26)(cid:90) r u r l λ ε ( r ) r C (1)0 ( b, z, (cid:37) ) dr + 2 πjZ (cid:90) r u r l λ ε ( r ) r C (2)0 ( K ε , b, z, (cid:37) ) dr (cid:27) , V ( j, k ) = − Zπ (cid:26)(cid:90) r u r l λ ε ( r ) r C (1)1 ( b, z, z (cid:48) , (cid:37) ) dr + 2 πjZ (cid:90) r u r l λ ε ( r ) r C ( 2)1 ( K ε , b, z, z (cid:48) , (cid:37) )) dr (cid:27) , where the angular integrals are defined by C (1)0 ( b, z, (cid:37) ) = (cid:90) π/ sin θ cos θ cos(2 πj ν ( λ | ε )) dθ,C (1)1 ( b, z, z (cid:48) , (cid:37) ) = (cid:90) π/ sin θ cos θ cos(2 πkλ ε cos θ ) cos(2 πj ν ( λ | ε )) dθ,C (2)0 ( K ε , b, z, (cid:37) ) = (cid:90) π/ sin θ cos θ sin(2 πj ν ( λ | ε )) I ( K ε sin θ ) dθ,C ( 2)1 ( K ε , b, z, z (cid:48) , (cid:37) ) = (cid:90) π/ sin θ cos θ cos(2 πkλ ε cos θ ) sin(2 πj ν ( λ | ε )) I ( K ε sin θ ) dθ, As will be seen below these angular integrals are complicated in that they contain the functionscos(2 πj ν ( λ | ε )), and cos(2 πkλ ε cos θ ) cos(2 πj ν ( λ | ε )) as well as their sin counterparts, quantitieswhich are functions of θ as well as λ ε ( r ). In order to make progress in evaluating these integralswhich contain the term ν ( λ | ε ), we will expand that quantity as follows. Taking into account the factthat the ν ( λ | ε ) versus λ curves show curvature, the Thomas Fermi lines of nonlinear degeneracy willbe replaced by a quadratic polynomial with the choice of parameters used by SE (in SE’s notation ν (1) ε = ν (cid:48) ε , and ν (2) ε = ν (cid:48)(cid:48) ε ) that is we write ν ( λ | ε ) = ν (1) ε [ λ ε ( r ) − λ ] − ν (2) ε [ λ ε ( r ) − λ ] , with (we use SE’s values for ν (1) ε , ν (2) ε and assume that they are constants independent of r ) ν (1) ε = √ λ ε /ω ε r ε ,ν (2) ε = ( ν (1) ε − /λ ε ( r ε ) . Expressing λ in terms of the angle θ we write2 πjν ( λ | ε ) = b + z cos θ + (cid:37) cos θ, (24)where using the values given above we have 22 = 2 πjλ ε ( r ) [ ν (1) ε − λ ε ( r ) ν (2) ε ] b = πj λ ε ( r ) [ ν (1) ε + 1] ,(cid:37) = − πjλ ε ( r ) ν (2) ε (cid:37) = − πjλ ε ( r ) [ ν (1) ε − ,z = − πjλ ε ( r ) [ ν (1) ε − λ ε ( r ) ν (2) ε ] z = − πjλ ε ( r ) ,z (cid:48) = 2 πkλ ε ( r ) . Recalling that λ ε ( r ) = λ ε ( r ε ) − ω ε λ ε ( r ε ) ( r − r ε ) + . . . , and within the limit as ε approaches zero b = lim ε → πj λ ε ( r ε ) [ ν (1) ε + 1] = 2 . π j Z / , | z | = lim ε → πj λ ε ( r ε ) = 1 . π j Z / ,z (cid:48) = lim ε → πk λ ε ( r ε ) = 1 . π k Z / , | (cid:37) | = lim ε → πj λ ε ( r ε ) [ ν (1) ε −
1] = 0 . π j Z / . With z = − πjλ ε ( r ε ) , z (cid:48) = 2 πkλ ε ( r ε ) and (cid:37) = − πjλ ε ( r ε )[ v (1)0 −
1] we have upon expanding thecos(2 πj ν ( λ | ε )) and sin(2 πj ν ( λ | ε )) terms and obtained the trigonometric expressionscos(2 πj ν ( λ | ε )) = cos( b ) (cid:8) cos( z cos θ ) cos( (cid:37) cos θ ) − sin( z cos θ ) sin( (cid:37) cos θ ) (cid:9) (T1) − sin( b ) (cid:8) sin( z cos θ ) cos( (cid:37) cos θ ) + cos( z cos θ ) sin( (cid:37) cos θ ) (cid:9) , and sin(2 πj ν ( λ | ε )) = sin( b ) (cid:8) cos( z cos θ ) cos( (cid:37) cos θ ) − sin( z cos θ ) sin( (cid:37) cos θ ) (cid:9) (T2)+ cos( b ) (cid:8) sin( z cos θ ) cos( (cid:37) cos θ ) + cos( z cos θ ) sin( (cid:37) cos θ ) (cid:9) . λ ε ( r ε ), and ν ( λ | ε )cos(2 πkλ ε ( r ε )) cos(2 πj ν ( λ | ε )) = cos( b )[cos([ z − z (cid:48) ] cos θ ) cos( (cid:37) cos θ ) (T3)+ cos([ z + z (cid:48) ] cos θ ) cos( (cid:37) cos θ ) − sin([ z − z (cid:48) ] cos θ ) sin( (cid:37) cos θ ) − sin([ z + z (cid:48) ] cos θ ) sin( (cid:37) cos θ )] − sin( b ) { sin([ z − z (cid:48) ] cos θ ) cos( (cid:37) cos θ )+ sin([ z + z (cid:48) ] cos θ ) cos( (cid:37) cos θ )+ cos([ z − z (cid:48) ] cos θ ) sin( (cid:37) cos θ )+ cos([ z + z (cid:48) ] cos θ ) sin( (cid:37) cos θ ) } , and cos(2 πkλ ε ( r ε )) sin(2 πj ν ( λ | ε )) = cos( b )[sin([ z − z (cid:48) ] cos θ ) cos( (cid:37) cos θ ) (T4)+ sin([ z + z (cid:48) ] cos θ ) cos( (cid:37) cos θ )+ cos([ z − z (cid:48) ] cos θ ) sin( (cid:37) cos θ )+ cos([ z + z (cid:48) ] cos θ ) sin( (cid:37) cos θ ) sin( b )[cos([ z − z (cid:48) ] cos θ ) cos( (cid:37) cos θ )+ cos([ z + z (cid:48) ] cos θ ) cos( (cid:37) cos θ ) − sin([ z − z (cid:48) ] cos θ ) sin( (cid:37) cos θ ) − sin([ z + z (cid:48) ] cos θ ) sin( (cid:37) cos θ )] , Using the ν ( λ | ε ) expressions and (T1) the integrals C (1)0 becomes C (1)0 ( b, z, (cid:37) ) = cos( b ) {C cc (0 , z, (cid:37) ) − S ss (0 , z, (cid:37) ) } − sin( b ) {C cs (0 , z, (cid:37) ) + S sc (0 , z, (cid:37) ) } . then using (T3) C (1)1 can be written as C (1)1 ( b, z, z (cid:48) , (cid:37) ) = cos( b )[ C cc (0 , z − z (cid:48) , (cid:37) ) − S ss (0 , z − z (cid:48) , (cid:37) ) + C cc (0 , z + z (cid:48) , (cid:37) ) − S ss (0 , z + z (cid:48) , (cid:37) )] − sin( b )[ C cs (0 , z − z (cid:48) , (cid:37) ) + S sc (0 , z − z (cid:48) , (cid:37) ) + C cs (0 , z + z (cid:48) , (cid:37) ) + S sc (0 , z + z (cid:48) , (cid:37) )] , where the primitive angular integrals C cc ( κ, z, ζ ) , S ss ( κ, z, ζ ) , C cs ( κ, z, ζ ) , S sc ( κ, z, ζ ) are defined as C cc ( κ, z, (cid:37) ) = (cid:90) π/ sin κ +2 θ cos θ cos( z cos θ ) cos( (cid:37) cos θ ) dθ, (26a) S ss ( κ, z, (cid:37) ) = (cid:90) π/ sin κ +2 θ cos θ sin( z cos θ ) sin( (cid:37) cos θ ) dθ, (26b) C cs ( κ, z, (cid:37) ) = (cid:90) π/ sin κ +2 θ cos θ cos( z cos θ ) sin( (cid:37) cos θ ) dθ, (26c) S sc ( κ, z, (cid:37) ) = (cid:90) π/ sin κ +2 θ cos θ sin( z cos θ ) cos( (cid:37) cos θ ) dθ. (26d)24ollecting terms in C (1)1 ( b, z, z (cid:48) , (cid:37) ) with argument z + z (cid:48) and z − z (cid:48) we get C (1)1 ( b, z, z (cid:48) , (cid:37) ) = C (1)0 ( b, z + z (cid:48) , (cid:37) ) + C (1)0 ( b, z − z (cid:48) , (cid:37) ) , The integrals C (2)0 becomes with (T2) C (2)0 ( K ε , b, z, (cid:37) ) = cos( b ) { T cs ( K ε , z, (cid:37) ) + T sc ( K ε , z, (cid:37) ) } + sin( b ) { T cc ( K ε , z, (cid:37) ) − T ss ( K ε , z, (cid:37) ) } , then we can also write for C (2)1 ( K ε , b, z, z (cid:48) , (cid:37) ) using (T4) C (2)1 ( K ε , b, z, z (cid:48) , (cid:37) ) = cos( b )[ T cs ( K ε , z − z (cid:48) , (cid:37) ) + T sc ( K ε , z − z (cid:48) , (cid:37) )]+ sin( b )[ T cc ( K ε , z − z (cid:48) , (cid:37) ) − T cs ( K ε , z − z (cid:48) , (cid:37) )]+ cos( b )[ T sc ( K ε , z + z (cid:48) , (cid:37) ) + T cs ( K ε , z + z (cid:48) , (cid:37) )]+ sin( b )[ T cc ( K ε , z + z (cid:48) , (cid:37) ) − T cs ( K ε , z + z (cid:48) , (cid:37) )] , where the angular integrals T cc ( K ε , z, ζ ) , T ss ( K ε , z, ζ ) , T cs ( K ε , z, ζ ) , T sc ( K ε , z, ζ ) have been definedas T cc ( K ε , z, (cid:37) ) = (cid:90) π/ I ( K ε sin θ ) sin θ cos θ cos( z cos θ ) cos( (cid:37) cos θ ) dθ, (27a) T ss ( K ε , z, (cid:37) ) = (cid:90) π/ I ( K ε sin θ ) sin θ cos θ sin( z cos θ ) sin( (cid:37) cos θ ) dθ, (27b) T cs ( K ε , z, (cid:37) ) = (cid:90) π/ I ( K ε sin θ ) sin θ cos θ cos( z cos θ ) sin( (cid:37) cos θ ) dθ, (27c) T sc ( K ε , z, (cid:37) ) = (cid:90) π/ I ( K ε sin θ ) sin θ cos θ sin( z cos θ ) cos( (cid:37) cos θ ) dθ. (27d)Using the expressions above we get C (2)1 ( K ε , b, z, z (cid:48) , (cid:37) ) = C (2)0 ( K ε , b, z + z (cid:48) , (cid:37) ) + C (2)0 ( K ε , b, z − z (cid:48) , (cid:37) ) . We note that all of the C ( k ) i integrals appearing above can be written in terms of the singlequantity F ( b, κ, z, (cid:37) ) defined by F ( b, κ, z, (cid:37) ) = cos( b ) { C cc ( κ, z, (cid:37) ) − S ss ( κ, z, (cid:37) ) } − sin( b ) { C cs ( κ, z, (cid:37) ) + S sc ( κ, z, (cid:37) ) } . Observing that F ( b + π/ , κ, z, (cid:37) ) = − sin( b ) { C cc ( κ, z, (cid:37) ) − S ss ( κ, z, (cid:37) ) } − cos( b ) { C cs ( κ, z, (cid:37) ) + S sc ( κ, z, (cid:37) ) } , (27)25he C ( k ) i integrals can be rewritten in the compact forms C (1)0 ( b, z, (cid:37) ) = F ( b, , z, (cid:37) ) , (29a) C (1)1 ( b, z, z (cid:48) , (cid:37) ) = {F ( b, , z + z (cid:48) , (cid:37) ) + F ( b, , z − z (cid:48) , (cid:37) ) } , (29b) C (2)0 ( K ε , b, z, (cid:37) ) = ∞ (cid:88) κ =1 ( − κ K κ − ε (2 κ + 1) F ( b + π/ , κ, z, (cid:37) ) , (29c) C (2)1 ( K ε , b, z, z (cid:48) , (cid:37) ) = ∞ (cid:88) κ =1 ( − κ K κ − ε (2 κ + 1) {F ( b + π/ , κ, z + z (cid:48) , (cid:37) ) + F ( b + π/ , κ, z − z (cid:48) , (cid:37) ) } . (29d)Recalling that z < z ± z (cid:48) must also be less than zero in (29b) and (29d). Thelatter restrictions imply that j > k when sums containing z ± z (cid:48) are evaluated.For the radial integrals in (24a) and (24b) i.e. those performed over the region 0 ≤ r ≤ ∞ wedefine the integrals V ( k ) i V (1)0 ( j ) = (cid:90) ∞ λ ( r ) r C (1)0 ( b, z, (cid:37) ) dr, (30a) V (1)1 ( j, k ) = (cid:90) ∞ λ ( r ) r C (1)1 ( b, z, z (cid:48) , (cid:37) ) dr, (30b) V (2)0 ( j ) = (cid:90) ∞ λ ( r ) r C (2)0 ( K ε , b, z, (cid:37) ) dr, , (30c) V (2)1 ( j, k ) = (cid:90) ∞ λ ( r ) r C ( 2)1 ( K ε , b, z, z (cid:48) , (cid:37) ) dr, (30d)or written in terms of the F functions V (1)0 ( j ) = (cid:90) ∞ λ ( r ) F ( b, , z, (cid:37) ) r dr, (31a) V (1)1 ( j, k ) = (cid:90) ∞ λ ( r ) r {F ( b, , z + z (cid:48) , (cid:37) ) + F ( b, , z − z (cid:48) , (cid:37) ) } dr, V (2)0 ( j ) = ∞ (cid:88) κ =1 ( − κ K κ − ε (2 κ + 1) (cid:90) ∞ λ ( r ) r F ( b + π/ , κ, z, (cid:37) ) dr, (31b) V (2)1 ( j, k ) = ∞ (cid:88) κ =1 ( − κ K κ − ε (2 κ + 1) ·· (cid:90) ∞ λ ( r ) r {F ( b + π/ , κ, z + z (cid:48) , (cid:37) ) + F ( b + π/ , κ, z − z (cid:48) , (cid:37) ) } . V ( j ) = − Zπ [ V (1)0 ( j ) + (cid:18) πjZ (cid:19) V (2)0 ( j )] , (32a) V ( j, k ) = − Zπ [ V (1)1 ( j, k ) + (cid:18) πjZ (cid:19) V (2)1 ( j, k )] . (32b)Because of the rapidly oscillating terms contained in the cos( b ) and sin( b ) factors in the V and V integrals, the stationary phase approximation [21] will be used to evaluate these integrals. Since z and (cid:37) are negative quantities and are arguments of the Bessel and Struve functions which areinvolved in the calculations indicated above, the required asymptotic expansions for those functionsare ( z <
0) taken to be J ( z ) ∼ (cid:115) π | z | cos( | z | − π/
4) + · · · , (32) J ( z ) ∼ − (cid:115) π | z | sin( | z | − π/
4) + · · · ,H ( z ) ∼ − (cid:115) π | z | sin( | z | − π/ − π | z | · · · ,H ( z ) ∼ − (cid:115) π | z | cos( | z | − π/
4) + 2 π + · · · , as a result we have F ( b, κ, z, (cid:37) ) = 1 (cid:112) | z (cid:37) | sin( b + | (cid:37) | − | z | )[ F ( κ, z, (cid:37) ) − F ( κ, z, (cid:37) )]+ cos( b + | (cid:37) | − | z | )[ F (+)7 ( κ, z, (cid:37) ) + F (+)8 ( κ, z, (cid:37) )] − sin( b − | (cid:37) | − | z | )[ F ( − )7 ( κ, z, (cid:37) ) − F ( − )8 ( κ, z, (cid:37) )] − cos( b − | (cid:37) | − | z | )[ F (+)5 ( κ, z, (cid:37) ) + F ( κ, z, (cid:37) )] (33)+ (cid:115) π | (cid:37) | cos( b + | (cid:37) | − π/ (cid:34) F (+)4 ( κ, z, (cid:37) ) + F ( − )5 ( κ, z, (cid:37) )2 π j λ ε ( r ) (cid:35) + sin( b + | (cid:37) | − π/ (cid:34) F ( − )6 ( κ, z, (cid:37) ) + F (+)7 ( κ, z, (cid:37) )2 π j λ ε ( r ) (cid:35) − sin( b − | (cid:37) | − π/ (cid:34) F ( − )4 ( κ, z, (cid:37) ) + F (+)5 ( κ, z, (cid:37) )2 π j λ ε ( r ) (cid:35) + cos( b − | (cid:37) | − π/ (cid:34) F (+)6 ( κ, z, (cid:37) ) + F ( − )7 ( κ, z, (cid:37) )2 π j λ ε ( r ) (cid:35) , F i ( κ, z, (cid:37) ) quantities appearing in (34) are defined as F ( κ, z, (cid:37) ) = − (cid:98) P (2) c ( κ, , z, (cid:37) ) + Q (2) s ( κ, , z, (cid:37) ) , (34) F ( κ, z, (cid:37) ) = − P (1) s ( κ, , z, (cid:37) ) − (cid:98) Q (1) c ( κ, , z, (cid:37) ) , F ( ± )4 ( κ, z, (cid:37) ) = − [ (cid:98) Q (2) c ( κ, , z, (cid:37) ) + (cid:98) R (2) c ( κ, , z, (cid:37) )] ± [ Q (1) s ( κ, , z, (cid:37) ) + R (1) s ( κ, , z, (cid:37) )] , F ( ± )5 ( κ, z, (cid:37) ) = (cid:98) P (2) c ( κ, , z, (cid:37) ) ± P (1) s ( κ, , z, (cid:37) ) , F ( ± )6 ( κ, z, (cid:37) ) = ± [ (cid:99) Q (1) c ( κ, , z, (cid:37) ) + (cid:98) R (1) c ( κ, , z, (cid:37) )] − [ Q (2) s ( κ, , z, (cid:37) ) + R (2) s ( κ, , z, (cid:37) )] , F ( ± )7 ( κ, z, (cid:37) ) = ± (cid:98) P (1) c ( κ, , z, (cid:37) ) + P (2) s ( κ, , z, (cid:37) ) , F ( ± )8 ( κ, z, (cid:37) ) = ± (cid:99) Q (2) c ( κ, , z, (cid:37) ) − Q (1) s ( κ, , z, (cid:37) ) , F ( κ, z, (cid:37) ) = (cid:99) Q (1) c ( κ, , z, (cid:37) ) − Q (2) s ( κ, , z, (cid:37) ) , and where the P , Q , and R , quantities are polynomials in 1 /z and 1 /(cid:37) and have been given inAppendix B.The leading terms for the F i ( κ, z, (cid:37) ) appearing above are (Cf. Appendix B) F (2 κ, z, (cid:37) ) ∼ ( − κ z κ +1 (4 κ + 1)!! , F (2 κ + 1 , z, (cid:37) ) ∼ O (1 /z κ +3 ) , F (2 κ, z, (cid:37) ) ∼ ( − κ z κ +1 (4 κ + 1)!! , F (2 κ + 1 , z, (cid:37) ) ∼ O (1 /z κ +3 ) , F ( ± )4 (2 κ, z, (cid:37) ) ∼ O (1 /z κ +2 ) , F ( ± )4 (2 κ + 1 , z, (cid:37) ) ∼ O (1 /z κ +3 ) , F ( ± )5 (2 κ, z, (cid:37) ) ∼ ( − κ z κ +1 (4 κ + 1)!! , F ( ± )5 (2 κ + 1 , z, (cid:37) ) ∼ O (1 /z κ +3 ) , F ( ± )6 (2 κ, z, (cid:37) ) ∼ O (1 /(cid:37)z κ +2 ) , F ( ± )6 (2 κ + 1 , z, (cid:37) ) ∼ O (1 /(cid:37)z κ +2 ) , F ( ± )7 (2 κ, z, (cid:37) ) ∼ O (1 /(cid:37)z κ +1 ) , F ( ± )7 (2 κ + 1 , z, (cid:37) ) ∼ O (1 /(cid:37)z κ +3 ) , F ( ± )8 (2 κ, z, (cid:37) ) ∼ O (1 /z κ +2 ) , F ( ± )8 (2 κ + 1 , z, (cid:37) ) ∼ O (1 /z κ +2 ) F (2 κ, z, (cid:37) ) ∼ O (1 /(cid:37)z κ +2 ) , F (2 κ + 1 , z, (cid:37) ) ∼ O (1 /(cid:37)z κ +2 ) . (35)The argument b − | (cid:37) | − | z | = 0 in the trigonometric expressions above causes the sin of thatargument to vanish and the cos of the same argument to produce a non-oscillating terms which areof no interest here and has been dropped.The arguments of the trigonometric functions in the remaining forms occurring in (34) withinthe F ( b, κ, z, (cid:37) ) function when written in explicit terms are b + | (cid:37) | − | z | + δπ/ πj [ ν (1) ε − λ ε ( r ) + δπ/ ,b ± | (cid:37) | − π/ δπ/ πj N ± λ ε ( r ) + ( δ − / π// , where δ = 0 or 1 ,N + = ν (1) ε , N − = 1.28e have F ( b + δπ/ , κ, z, (cid:37) ) = (36) π j λ ε ( r ) (cid:113) ν (1) ε − sin(2 πj [ ν (1) ε − λ ε ( r ) + δπ/ · [ F ( κ, z, (cid:37) ) − F ( κ, z, (cid:37) )]+ cos(2 πj [ ν (1) ε − λ ε ( r ) + δπ/ · [ F (+)7 ( κ, z, (cid:37) ) + F (+)8 ( κ, z, (cid:37) )] + π (cid:113) j λ ε ( r )[ ν (1) ε − cos(2 π j ν (1) ε λ ε ( r ) + ( δ − / π/ · (cid:104) F (+)4 ( κ, z, (cid:37) ) + F ( − )5 ( κ, z, (cid:37) ) / π j λ ε ( r ) (cid:105) + sin(2 π j ν (1) ε λ ε ( r ) + ( δ − / π/ · (cid:104) F ( − )6 ( κ, z, (cid:37) ) + F (+)7 ( κ, z, (cid:37) ) / π j λ ε ( r ) (cid:105) − sin(2 π j λ ε ( r ) + ( δ − / π/ · (cid:104) F ( − )4 ( κ, z, (cid:37) ) + F (+)5 ( κ, z, (cid:37) ) / π j λ ε ( r ) (cid:105) + cos(2 π j λ ε ( r ) + ( δ − / π/ · (cid:104) F (+)6 ( κ, z, (cid:37) ) + F ( − )7 ( κ, z, (cid:37) ) / π j λ ε ( r ) (cid:105) , an expression which will occur in its most general form within the terms V (2) i as will be seen below. F ( b + δπ/ , κ, z, (cid:37) ) and G ( b + δπ/ , κ, z, (cid:37) ) The radial integrals F ( b + δπ/ , κ, z, (cid:37) ) are a generalized form of the V (1) i integrals shown in Eqs.(31a) i.e. F ( b + δπ/ , κ, z, (cid:37) ) = (cid:82) ∞ λ ( r ) r F ( b + δπ/ , κ, z, (cid:37) ) dr. (37)Integrals containing a smooth integrand F ( r ) such as the cases occuring above, are given withinthe stationary state approximation by (cid:90) ∞ λ ( r ) r F ( r ) (cid:40) cos(2 πj [ ν (1) ε − λ ( r ) + δπ/ πj [ ν (1) ε − λ ( r ) + δπ/ dr (38)= (cid:113) j [ ν (1) ε − λ / ( r ) ωr F ( r ) (cid:40) cos(2 πj [ ν (1) ε − λ ( r ) + ( δ − / π/ πj [ ν (1) ε − λ ( r ) + ( δ − / π/ , and (cid:90) ∞ λ ( r ) r F ( r ) (cid:26) cos(2 πjN ± λ ( r ) + ( δ − / π/ πjN ± λ ( r ) + ( δ − / π/ dr (39)= (cid:113) jN ± λ / ( r ) ωr F ( r ) (cid:26) cos(2 πjN ± λ ( r ) + δπ/ πjN ± λ ( r ) + δπ/ , λ ( r ) is taken to be λ ( r ) = λ ( r ) − ω λ ( r ) ( r − r ) . The radial integrals F ( b + δπ/ , κ, z, (cid:37) ) are a generalized form of the V (1) i integrals shown in Eqs.(31a) i.e. F ( b + δπ/ , κ, z, (cid:37) ) = (cid:82) ∞ λ ( r ) r F ( b + δπ/ , κ, z, (cid:37) ) dr. (40)When the integration over r is performed all of the quantities contained in (40) are evaluated withthe constants r , λ , ω , ν (1)0 , ν (2)0 and with K ( r ) = λ ( r ) /Zr = 0 . . We have F ( b + δπ/ , κ, z, (cid:37) ) = λ / ( r ) πj / ω r [ ν (1)0 − (cid:40) sin(2 πj [ ν (1)0 − λ ( r ) + ( δ − / π/ F ( κ, z, (cid:37) ) − F ( κ, z, (cid:37) ) ]cos(2 πj [ ν (1)0 − λ ( r ) + ( δ − / π/
2) + [ F (+)7 ( κ, z, (cid:37) ) + F (+)8 ( κ, z, (cid:37) )] (cid:41) (41)+ λ ( r ) πjω r (cid:113) N + [ ν (1)0 − (cid:40) cos(2 πjN + λ ( r ) + δπ/ F (+)4 ( κ, z, (cid:37) ) + F ( − )5 ( κ, z, (cid:37) ) / πjλ ( r )]+ sin(2 πjN + λ ( r ) + δπ/ F ( − )6 ( κ, z, (cid:37) ) + F (+)7 ( κ, z, (cid:37) ) / πjλ ( r )] (cid:41) + λ ( r ) πjω r (cid:113) N − [ ν (1)0 − (cid:40) cos(2 πjN − λ ( r ) + δπ/ F ( − )4 ( κ, z, (cid:37) ) + F (+)5 ( κ, z, (cid:37) ) / πjλ ( r )] − sin(2 πjN − λ ( r ) + δπ/ F (+)6 ( κ, z, (cid:37) ) + F ( − )7 ( κ, z, (cid:37) ) / πjλ ( r )] (cid:41) . The radial integrals which contain higher-order powers of λ ( r ) i.e. G ( b + δπ/ , κ, z, (cid:37) ) are ageneralized form of the V (2) i integrals occurring in Eqs. (31b) and are defined by G ( b + δπ/ , κ, z, (cid:37) ) = (cid:82) ∞ λ ( r ) r F ( b + δπ/ , κ, z, (cid:37) ) dr . (42)Then we have G ( b + δπ/ , κ, z, (cid:37) )= λ / ( r ) πj / ω r (cid:113) ν (1)0 − (cid:40) sin(2 πj [ ν (1)0 − λ ( r ) + δπ/ F ( κ, z, (cid:37) ) − F ( κ, z, (cid:37) ) ]cos(2 πj [ ν (1)0 − λ ( r ) + δπ/ F (+)7 ( κ, z, (cid:37) ) + F (+)8 ( κ, z, (cid:37) ) ] (cid:41) + λ ( r ) πjω r (cid:113) N + [ ν (1)0 − (cid:40) cos(2 πjN + λ ( r ) + δπ/ F (+)4 ( κ, z, (cid:37) ) + F ( − )5 ( κ, z, (cid:37) ) / πjλ ( r )]+ sin(2 πjN + λ ( r ) + δπ/ F ( − )6 ( κ, z, (cid:37) ) + F (+)7 ( κ, z, (cid:37) ) / πjλ ( r )] (cid:41) + λ ( r ) πjω r (cid:113) N − [ ν (1)0 − (cid:40) cos(2 πjN − λ ( r ) + δπ/ F ( − )4 ( κ, z, (cid:37) ) + F (+)5 ( κ, z, (cid:37) ) / πjλ ( r )] − sin(2 πjN − λ ( r ) + δπ/ F (+)6 ( κ, z, (cid:37) ) + F ( − )7 ( κ, z, (cid:37) ) / πjλ ( r )] (cid:41) . V (1)0 ( j ) = F ( b, , z, (cid:37) ) , V (1)1 ( j, k ) = { F ( b, , z + z (cid:48) , (cid:37) ) + F ( b, , z − z (cid:48) , (cid:37) ) } , V (2)0 ( j ) = ∞ (cid:88) κ =1 ( − κ K κ − ε (2 κ + 1) G ( b + π/ , κ, z, (cid:37) ) , V (2)1 ( j, k ) = ∞ (cid:88) κ =1 ( − κ K κ − ε (2 κ + 1) { G ( b + π/ , κ, z + z (cid:48) , (cid:37) ) + G ( b + π/ , κ, z − z (cid:48) , (cid:37) ) } . (43)The terms needed in the average value V ne, being V ( j ) = − Zπ F ( b, , z, (cid:37) ) − jZ ∞ (cid:88) κ =1 ( − κ K κ − ε (2 κ + 1) G ( b + π/ , κ, z, (cid:37) ) , (44a) V ( j, k ) = − Z π { F ( b, , z + z (cid:48) , (cid:37) ) + F ( b, , z − z (cid:48) , (cid:37) ) }− jZ ∞ (cid:88) κ =1 ( − κ K κ − ε (2 κ + 1) { G ( b + π/ , κ, z + z (cid:48) , (cid:37) ) + G ( b + π/ , κ, z − z (cid:48) , (cid:37) ) } . (44b)Retaining only the leading terms in the quantities F i ( κ, z, (cid:37) ) we find that only the difference F ( κ, z, (cid:37) ) − F ( κ, z, (cid:37) ) = ( − κ πjλ ε ( r )] κ +1 , (45)survives whereas all of the others terms are small and have been dropped. Then Z F ( b, , z, (cid:37) ) = − Zλ ( r ) / sin(2 πj [ ν (1) ε − λ ε ( r ) − π/ π [ ν (1)0 − ω r j / z , (46)then Z F ( b, , z, (cid:37) ) = Zλ ( r ) / sin(2 πj [ ν (1) ε − λ ε ( r ) − π/ π [ ν (1)0 − ω r j / , and Z { F ( b, , z + z (cid:48) , (cid:37) ) + F ( b, , z − z (cid:48) , (cid:37) ) } = Zλ ( r ) / sin(2 πj [ ν (1) ε − λ ε ( r ) − π/ π [ ν (1)0 − ω r j / [ j − k ] , (47)terms which are on the order of Z / . Similarly one find that G ( b + π/ , κ, z, (cid:37) ) /Z ∼ O ( Z − / ) , a quantity which is small and has been dropped.31 .11 The sum over V ( j ) The first part of the average V ne,λ i.e. ∞ (cid:88) j =1 ( − j V ( j ) = ∞ (cid:88) j =1 ( − j +1 { Zπ F ( b, , z, (cid:37) ) + 2 jZ ∞ (cid:88) κ =1 ( − κ K κ − ε (2 κ + 1) G ( b + π/ , κ, z, (cid:37) ) } , (48)reduces to ∞ (cid:88) j =1 ( − j V ( j ) = 14 π Z λ ( r ) / (cid:113) [ ν (1)0 − ω r ∞ (cid:88) j =1 ( − j sin(2 πj [ ν (1)0 − λ ( r ) − π/ j / , (49)when only the leading terms in Z are kept. We have after performing the sum in (49) − V ne,λ Z = − Z ∞ (cid:88) j =1 ( − j V ( j ) (50)= √ π λ ( r ) / [ ν (1)0 − ω r (cid:40) Re[ Li / ( − exp( − πi [ ν (1)0 − λ ( r ))]+ Im[ Li / ( − exp( − πi [ ν (1)0 − λ ( r ))] (cid:41) , a quantity which is of order Z / . V ( j, k ) The second part of the average V ne, λν i.e. ∞ (cid:88) j =1 ∞ (cid:88) k =1 ( − j + k V ( j, k ) (51)= − Z π ∞ (cid:88) j =1 ∞ (cid:88) k =1 ( − j + k { F ( b, , z + z (cid:48) , (cid:37) ) + F ( b, , z − z (cid:48) , (cid:37) ) }− jZ ∞ (cid:88) j =1 ∞ (cid:88) k =1 ( − j + k ∞ (cid:88) κ =1 ( − κ K κ − ε (2 κ + 1) { G ( b + π/ , κ, z + z (cid:48) , (cid:37) ) + G ( b + π/ , κ, z − z (cid:48) , (cid:37) ) } . The triple sum leads to terms too small to consider here and has been dropped. In the case of thedouble sum, whenever the argument z + z (cid:48) = 2 π ( k − j ) λ ε ( r ) occurs we note that in the expressionsfor V ne where 1 ≤ j ≤ ∞ , and whenever j has been replaced by | k − j | that 1 ≤ | k − j | ≤ ∞ .Since j > k the region to be summed over is shown below.The sum of interest taking into account the restriction j > k is ∞ (cid:88) j =1 ∞ (cid:88) k =1 ( − j + k V ( j, k ) = ∞ (cid:88) k =1 ∞ (cid:88) j = k +1 ( − j + k V ( j, k ) , (52)= ∞ (cid:88) j =2 j − (cid:88) k =1 ( − j + k V ( j, k ) . k vs. j region of summationFor the latter form of the double sum (52) (where the order of summation has been reversed) wehave shown the allowed region of summation in Fig. (9).We get ∞ (cid:88) j =1 ∞ (cid:88) k =1 ( − j + k V ( j, k ) = − Z π λ ( r ) / (cid:113) [ ν (1)0 − ω r ∞ (cid:88) j =2 j − (cid:88) k =1 ( − j + k sin(2 πj [ ν (1)0 − λ ( r ) − π/ j / [ j − k ] . (53)Then the required expression for V ne,λ,ν /Z becomes − V ne,λ,ν Z = π √ λ ( r ) / (cid:113) [ ν (1)0 − ω r { Re S ( x ) + Im S ( x ) } , (54)with the complex quantity x has been defined by x = exp { πi [ ν (1)0 − λ ( r ) } , and S ( x ) = ∞ (cid:88) j =2 j − (cid:88) k =1 ( − j + k x j j / [ j − k ] . The double sum S ( x ) can be reduced to the terms (cf. Appendix B) S ( x ) = −
12 [ { − ln(2) } x + ln(2) Li / ( x ) + Li / ( − x )] + 14 ∞ (cid:88) j =2 x j j / [ ψ ( j ) − ψ ( j + 1 / , (55)33 closed-form expression for this expression is not known, however the infinite sum converges rapidlyand may be safely truncated to contain eight terms to produce five figure accuracy.Finally we have for V ne,λ + V ne,λ,ν the expression − V ne,λ + V ne,λ,ν Z = √ π λ ( r ) / (cid:113) [ ν (1)0 − ω r (cid:8) Re[ Li / ( − x )] + Im[ Li / ( − x )] (cid:9) (56)+ √ π λ ( r ) / (cid:113) [ ν (1)0 − ω r { Re S ( x ) + Im S ( x ) } , with √ π λ ( r ) / (cid:113) [ ν (1)0 − ω r = 0 . Z / . As seen above the non-oscillating part of the chemical potential is zero, here we wish to evaluatethe derivative of the SE energy in order to complete the calculation of the oscillating part of thechemical potential. µ osc = dE ( Z, Z ) osc dZ − V ne,osc Z ,
In the work by SE the j, k plane is divided into regions within which the various contributions tothe energy E ( Z, Z ) have been computed. They write the energy as E osc ( Z, Z ) = ∞ (cid:88) k =1 E ,k + ∞ (cid:88) j =1 E j, + ∞ (cid:88) j =1 ∞ (cid:88) k =1 E j,k , (57)with ∞ (cid:88) k =1 E ,k = E l T F ( Z ) , ∞ (cid:88) j =1 E j, + ∞ (cid:88) j =1 ∞ (cid:88) k =1 E j,k = E λ − osc ( Z ) + E ν − osc ( Z ) + E ν,λ − osc ( Z ) . In that work the energy associated with the ν oscillation was small and the λ, ν oscillationswhere found to be negligible compared to that of the λ oscillations. In the latter case the E ν,λ − osc contribution is on the order of 0 . Z / and has been dropped. As a result we write E ( Z, Z ) = E l T F ( Z ) + E λ − osc ( Z ) + E ν − osc ( Z ) . (58)Schwinger and Englert have given E l T F ( Z ) as E l T F ( Z ) = λ r [ 12 K C ( λ ) − λ ν (1) S ( λ ) ] , (59)where the terms C n ( λ ) and S n ( λ ) are sums defined below and K is a constant i.e. K = − ν (1) − ν (1)0 ) + ( ν (1) ) = − . S n ( z ) = ∞ (cid:88) k =1 ( − k sin(2 πkz )( πk ) n +1 ,C n ( z ) = ∞ (cid:88) k =1 ( − k cos(2 πkz )( πk ) n ,dS n ( z ) dz = 2 C n ( z ) ,dC n +1 ( z ) dz = − S n ( z ) , and dλ dZ = 13 (cid:18) λ Z (cid:19) ,ddz (cid:18) λ ω r (cid:19) = 43 Z (cid:18) λ ω r (cid:19) ,ddz (cid:18) λ r (cid:19) = 1 Z (cid:18) λ r (cid:19) , we have dE l T F ( Z ) dZ = 2 λ Zr [ 32 K C ( λ ) + λ { ν (1) − K } S ( λ ) + λ ν (1) C ( λ )] , (60)and λ Zr = 0 . Z , Z / , Z / respectively.The SE energy of the λ oscillations is given by − ( r /λ ) E λ − osc ( Z ) = (61) S (cid:48) ( λ ) − S (2 λ ) + K S (2 λ ) − λ / [ K Im Li / ( − exp( − πiλ ))]+ 1 Z / (cid:40) K (cid:110) C (cid:48) ( λ ) + (cid:101) C (2 λ ) (cid:111) − K C (cid:48) ( λ ) + K (cid:101) C (2 λ ) + K (cid:101) C (2 λ )+ λ / [ K Re Li / ( − exp( − πiλ ))] + λ / [ K Re Li / ( − exp( − πiλ ))] (cid:41) , and ( λ r ) = 0 . Z / , S (cid:48) n C (cid:48) n (cid:101) S n (cid:101) C n are sums given below and the numerical constants K i are K = [2 − ν (1) ] / ν (1) = 0 . , K = √ (cid:113) ν (1) [( ν (1) ) − −
15 + 23( ν (1) ) − ν (1) ) ] π = 0 . , K = [3 − ν (1) ) ] / (2 π ) = − . , K = [1 − ν (1) ] / ν (1) ) = − . , K = [34 − ν (1) + 31( ν (1) ) ] /
576 = 0 . , K = 7[92 − ν (1) + 150( ν (1) ) − ν (1) ) + 15( ν (1) ) ] / − . , K = [2( ν (1)0 ) − π ) (cid:114) ν (1)0 − = 0 . , K = (cid:18) ν (1)0 − (cid:19) / [2 − ν (1) − ν (1) ) + 12( ν (1) ) − ν (1) ) − ν (1) ) − ( ν (1) ) ] / (4 π ) = − . (cid:101) S ( z ) = ∞ (cid:88) k =1 sin(2 πk z )( πk ) = − (cid:104) z − / (cid:105) , (cid:101) C ( z ) = ∞ (cid:88) k =1 cos(2 πk z ) = − / , (cid:101) S ( z ) = ∞ (cid:88) k =1 sin(2 πk z )( πk ) = (cid:104) z − / (cid:105) [ (cid:104) z − / (cid:105) − ] , (cid:101) C ( z ) = ∞ (cid:88) k =1 cos(2 πk z )( πk ) = (cid:104) z − / (cid:105) − , (cid:101) S ( z ) = ∞ (cid:88) k =1 sin(2 πk z )( πk ) = − (cid:104) z − / (cid:105) [ (cid:104) z − / (cid:105) − (cid:104) z − / (cid:105) + ] , (cid:101) C ( z ) = ∞ (cid:88) k =1 cos(2 πk z )( πk ) = − [ (cid:104) z − / (cid:105) − ] . S (cid:48) ( z ) = ∞ (cid:88) k =0 ( − k sin(2 π [2 k + 1] z ) = 0 , C (cid:48) ( z ) = ∞ (cid:88) k =0 ( − k cos(2 π [2 k + 1] z )[ π (2 k + 1)] = ( − (cid:98) z +1 / (cid:99) , S (cid:48) ( z ) = ∞ (cid:88) k =0 ( − k sin(2 π [2 k + 1] z )[ π (2 k + 1)] = [ < z + > − < z − > ] , C (cid:48) ( z ) = ∞ (cid:88) k =0 ( − k cos(2 π [2 k + 1] z )[ π (2 k + 1)] , = [ < z + > { − < z + > }− < z − > { − < z − > } ] . The derivative of E λ − osc ( Z ) is given by − Z ( r /λ ) dE λ − osc ( Z ) dZ = (62)4 S (cid:48) ( λ ) − S (2 λ ) + 4 K S (2 λ ) − λ / [ K Im Li / ( − exp( − πiλ ))]+ λ [2 C (cid:48) ( λ ) − C (2 λ )] + λ / [4 π K Re Li / ( − exp( − πiλ ))]+ 1 Z / K [ C (cid:48) ( λ ) + (cid:101) C (2 λ )] − K C (cid:48) ( λ ) + 3 K (cid:101) C (2 λ ) + 3 K (cid:101) C (2 λ )+ λ / [ K Re Li / ( − exp( − πiλ ))] − λ [ K { S (cid:48) ( λ ) + (cid:101) S (2 λ ) } − K S (cid:48) ( λ ) + 2 K (cid:101) S (2 λ )]+ λ / [ K Re Li / ( − exp( − πiλ )) + 4 π K Im Li / ( − exp( − πiλ ))]+ λ / [4 π K Im Li / ( − exp( − πiλ ))] . , with 1 Z ( λ r ) = 0 . Z / . The E ν − osc contribution to the energy is E ν − osc = 2( λ /r ) K (cid:101) S ([1 + ν (cid:48) ] λ / , where K is K = 1[( ν (cid:48) ) − ν (cid:48) − ( ν (cid:48) ) − ( ν (cid:48) ) + ( ν (cid:48) ) ] = 0 . , with E ν − osc being on the order of 0 . Z / which is small compared to the λ oscillations. Thederivative of E ν − osc is dE ν − osc dZ = 23 Z ( λ r ) K [4 (cid:101) S ([1 + ν (cid:48) ] λ /
2) + [1 + ν (cid:48) ] λ (cid:101) C ([1 + ν (cid:48) ] λ / Z contributions to dE ν − osc /dZ of order Z / and Z / and has been neglected.37 .14 Numerical Calculations As a result of the analysis given above, the chemical potential is given by µ osc = dE lT F ( Z ) dZ + dE λ − osc ( Z ) dZ − V ne,lT F Z − V ne,λ + V ne,λ,ν Z . (63)The final calculation of the chemical potential can now proceed by combining the eqations above.The results of these calculations are shown in Figs. (10)Figure 10 the oscillating part of the chemical potential vs. ZThe figure above shows the oscillating part of the chemical potential and therefore cannot bedirectly compared with the chemical potential in Fig. 1. Furthermore the results are only valid forlarge Z. We see that in the semiclassical approximation to the potential energy that the fine scaleoscillations disappointingly have been smoothed out. In the case of very large Z relativistic effectsare important but these have not been included in these calculations.38ppendix A .0.1 Evaluation of the integrals I ( ν, λ ε , r ) and I ( ν, λ ε , r )Here we give details of the calculation of I and I . In order for I to be evaluated, i.e. I ( ν, λ ε , r ) = (cid:90) (cid:113) r { ε λ,ν − V } − λ dν, a relationship must be established between ε λ,ν and ν which will allow us to give an approximationfor this integral. We chose the Coulombic potential where this relationship is ε λ,ν = − Z λ + ν ) . Then the integral I ( ν ) can be written as I ( ν, λ ε , r ) = − Zr (cid:90) (cid:115) ε λ,ν − V − λ / r − ε λ,ν d ε λ,ν , which immediately produces I ( ν, λ ε , r ) = Zr (cid:34)(cid:115) r { ε λ,ν − V } − λ − r ε λ,ν − arctan (cid:115) r { ε λ,ν − V } − λ − r ε λ,ν (cid:35) . (A1)The value of that integral along the curves of degeneracy is then I ( ν, λ ε , r ) /Zr = λ ε ( r ) Zr (cid:112) λ ε ( r ) − λ − arctan( λ ε ( r ) Zr (cid:112) λ ε ( r ) − λ ) , where we have use the relations ε λ,ν ( λ | ε ) = ε λ ε , = − Z / λ ε , and λ ε ( r ) = 2 r (cid:8) ε λ ε ,ν ( λ | ε ) − V (cid:9) , or in terms of the angle θ (A1) becomes I ( ν, λ ε , r ) Zr = K ε ( r ) sin( θ ) − arctan( K ε ( r ) sin θ ) , where K ε ( r ) = λ ε /rZ . Since K ε ( r ) sin θ is less than unity we have the final form of the integral I ( ν, λ ε , r ) Zr = [ K ε ( r ) sin( θ )] − [ K ε ( r ) sin( θ )] + · · · (A2)In a similar way, the integral for I is 39 ( ν, λ ε , r ) = √ Z r (cid:90) (cid:34)(cid:114) r { ε λ,ν − V }− λ − r ε λ,ν − arctan (cid:114) r { ε λ,ν − V }− λ − r ε λ,ν (cid:35) dε λ,ν ( − ε λ,ν ) / , which gives I ( ν, λ, r ) = −√ Z r − √ r { ε λ,ν − V }− λ √ rε λ,ν + √ − ε λ,ν arctan (cid:113) r { ε λ,ν − V }− λ − r ε λ,ν − √ r √ r { ε λ,ν − V }− λ − r ε λ,ν ln( (cid:113) r { ε λ,ν − V }− λ − r ε λ,ν − (cid:113) r { ε λ,ν − V }− λ − r ε λ,ν − r ε λ,ν ) . Then I ( ν, λ ε , r ) = 2 λ ε (cid:112) λ ε − λ − Zrλ ε arctan( λ ε (cid:112) λ ε − λ rZ )+ 2 r Z λ ε (cid:112) r Z + λ ε ( λ ε − λ ) ln( λ ε (cid:112) λ ε − λ rZ − (cid:112) λ ε ( λ ε − λ ) + r Z rZ )or with λ = λ ε cos θ we have I ( ν, λ ε , r ) / rZλ ε = λ ε rZ sin θ − λ ε rZ sin θ )+ 1 (cid:113) λ ε rZ sin θ ] ln( λ ε rZ sin θ − (cid:114) [ λ ε rZ sin θ ] + 1] ) , In the more compact form I ( ν, λ ε , r ) becomes I ( ν, λ ε , r )2 rZλ ε = K ε ( r ) sin θ − K ε ( r ) sin θ ) + arcsin h ( K ε ( r ) sin θ ) (cid:112) K ε ( r ε ) sin θ ] . (A3)For small argument K ε ( r ) sin θ we have I ( ν, λ ε , r )2 rZλ ε = [ K ε ( r ) sin θ ] − [ K ε ( r ) sin θ ] + · · · . (A4)40ppendix BThe double sum S ( x ) = ∞ (cid:88) j =2 j − (cid:88) k =1 ( − j + k x j j / [ j − k ] , with x = exp( iθ ) and θ = 2 π [ [ ν (1)0 − λ ( r ) ] can be simplified as follows. The sum over k canbe written as j − (cid:88) k =1 ( − k [ j − k ] = − j − ( − j j {
12 ln(2) + 14 ψ ( j + 1 / − ψ ( j ) } , where ψ is the digamma function. Then S ( x )is given by S ( x ) = −
12 [ { − ln(2) } x + ln(2) Li / ( x ) + Li / ( − x )] + 14 ∞ (cid:88) j =2 x j j / [ ψ ( j ) − ψ ( j + 1 / , = −
12 [ { − ln(2) } x + ln(2) Li / ( x ) − Li / ( x ) + 12 / Li / ( x )]+ 14 ∞ (cid:88) j =2 x j j / [ ψ ( j ) − ψ ( j + 1 / , The remaining sum in the equation above is rapidly convergent, eight terms being sufficient for sixfigure accuracy. 41ppendix C .1 The Primitive Integrals C ss , S ss , C cs , S sc The key primitive integrals C ss , S ss , C cs , S sc which appear in the text above and where z and (cid:37) are < C cc ( κ, z, (cid:37) ) = (cid:90) π/ sin κ +2 θ cos θ cos( z cos θ ) cos( (cid:37) cos θ ) dθ, (C1) S ss ( κ, z, (cid:37) ) = (cid:90) π/ sin κ +2 θ cos θ sin( z cos θ ) sin( (cid:37) cos θ ) dθ, C cs ( κ, z, (cid:37) ) = (cid:90) π/ sin κ +2 θ cos θ cos( z cos θ ) sin( (cid:37) cos θ ) dθ, S sc ( κ, z, (cid:37) ) = (cid:90) π/ sin κ +2 θ cos θ sin( z cos θ ) cos( (cid:37) cos θ ) dθ. can be expressed in terms of the integrals J ( κ, j, a, z ) , and H ( κ, j, a, z ) which have closed formsand are defined as J ( κ, j, a, z ) = (cid:90) π/ sin κ +2 θ cos j +2 a +1 θ sin( z cos θ ) dθ, H ( κ, j, a, z ) = (cid:90) π/ sin κ +2 θ cos j +2 a +1 θ cos( z cos θ ) dθ. The latter integrals are given by Maple as J ( κ, j, a, z ) = z B (2 j + a + 3 / , κ + 3 / F (cid:0) j + a +3 / − z / / , κ +2 j + a +3 (cid:1) , H ( κ, j, a, z ) = B (2 j + a + 1 , κ + 3 / F (cid:0) j + a +1 ; − z / / , κ +2 j + a +5 / (cid:1) , where B ( α, β ) is the Beta function and F (cid:0) a ; zb , b (cid:1) is a generalized hypergeometic function. As willbe seen below, these integrals are also expressible in terms of quantities which involve only productsof the Bessel functions J ( z ), and J ( z ) or the Struve functions H ( z ) , H ( z ), and polynomials witharguments 1 /z. That is to say J ( κ, j, a, z ) (C2)= ( − a +1 π [ J ( z ) ℘ ( κ, j + 2 a + 1 , z ) + J ( z ) Q ( κ, j + 2 a + 1 , z )] , H ( κ, j, a, z ) (C3)= ( − a π [ H ( z ) ℘ ( κ, j + 2 a + 1 , z )+ H ( z ) Q ( κ, j + 2 a + 1 , z )+ π R ( κ, j + 2 a + 1 , z )] , where the quantities ℘, Q , R are related to the Lommel polynomials. In the latter forms theoscillatory behavior of these integrals and those of the key integrals mentioned above is mademanifest. 42 J ( κ, j, a, z ) and H ( κ, j, a, z ) expressions Using the integral representations for Bessel functions of the first kind J ν ( z ) and the Struve functions H ν ( z ) [25] i.e. √ π Γ( ν +1 / J ν ( z )( z/ ν = (cid:90) π/ sin ν θ cos( z cos θ ) dθ, √ π Γ( ν +1 / H ν ( z )( z/ ν = (cid:90) π/ sin ν θ sin( z cos θ ) dθ, and differentiating these expressions with respect to z, j + 2 a + 1 times gives J ( κ, j, a, z ) = ( − a +1 π (2 κ + 1)!! d j +2 a +1 [ J κ +1 ( z ) /z κ +1 ] d z j +2 a +1 , (C4) H ( κ, j, a, z ) = ( − a π (2 κ + 1)!! d j +2 a +1 [ H κ +1 ( z ) /z κ +1 ] d z j +2 a +1 , (C5)where (2 κ + 1)!! is the double factorial function i.e.(2 κ + 1)!! = κ (cid:89) i =1 (2 i + 1) = 2 κ +1 √ π Γ( κ + 3 / . We have by direct observation [26]that for ν ≥ , (2 κ + 1)!! d ν d z ν [ J κ +1 ( z ) z κ +1 ] = ℘ ( κ, ν, z ) J ( z ) + Q ( κ, ν, z ) J ( z ) , (C6)(2 κ + 1)!! d ν d z ν [ H κ +1 ( z ) z κ +1 ] = ℘ ( κ, ν, z ) H ( z ) + Q ( κ, ν, z ) H ( z ) + π R ( κ, ν, z ) , (C7)Using (C4) and (C5) together with (C6) and (C7) we get (C2) and (C3). .2 Properties of the Polynomials ℘ ( κ, ν, z ) , Q ( κ, ν, z ) , and R ( κ, ν, z ) The polynomials ℘ ( κ, ν, z ), Q ( κ, ν, z ) , and R ( κ, ν, z ) are interrelated by differential recurrence re-lations (which follow from the expressions for d ν +1 d z ν +1 [ J κ +1 ( z ) z κ +1 ] and d ν +1 d z ν +1 [ H κ +1 ( z ) z κ +1 ] and the linearindependence of the Bessel J , J and Struve functions H , H ), we have ℘ ( κ, ν + 1 , z ) = d℘ ( κ, ν, z ) dz + Q ( κ, ν, z ) , (C8) Q ( κ, ν + 1 , z ) = d Q ( κ, ν, z ) dz − ℘ ( κ, ν, z ) − Q ( κ, ν, z ) z , R ( κ, ν + 1 , z ) = d R ( κ, ν, z ) dz + ℘ ( κ, ν, z ) . with ℘ (0 , , z ) = 1 /z, Q (0 , , z ) = − /z , R (0 , , z ) = 0 .
43n addition it follows from the differential equations defining J κ +1 ( z ) and H κ +1 ( z ) that the functions J κ +1 ( z ) /z κ +1 and H κ +1 ( z ) /z κ +1 satisfy the differential equations z d (cid:2) J κ +1 ( z ) /z κ +1 (cid:3) d z + (2 κ + 3) d (cid:2) J κ +1 ( z ) /z κ +1 (cid:3) d z + z [ J κ +1 ( z ) /z κ +1 ] = 0 ,z d (cid:2) H κ +1 ( z ) /z κ +1 (cid:3) d z + (2 κ + 3) d (cid:2) H κ +1 ( z ) /z κ +1 (cid:3) d z + z [ H κ +1 ( z ) /z κ +1 ] = 2 π (2 κ + 1)!! . Repeated differentiation of these relations gives for ν ≥ ,z d ν (cid:61) ( κ, z ) d z ν + (2 κ + ν + 1) d ν − (cid:61) ( κ, z ) d z ν − + z d ν − (cid:61) ( κ, z ) d z ν − + ( ν − d ν − (cid:61) ( κ, z ) d z ν − = 0 , where (cid:61) ( κ, z ) is either J κ +1 ( z ) /z κ +1 or H κ +1 ( z ) /z κ +1 . Using (C2) and (C3) it follows that thecoefficients of the functions J ( z ), J ( z ) , H ( z ) , and H ( z ) in the resulting relations vanish andwe get the pseudo 4th-order recurrence relations z F ( κ, ν, z ) + (2 κ + ν + 1) F ( κ, ν − , z ) + z F ( κ, ν − , z ) + ( ν − F ( κ, ν − , z ) = 0 , ν ≥ F ( κ, ν, z ) stands for any of the polynomials ℘ ( κ, ν, z ) , Q ( κ, ν, z ) , R ( κ, ν, z ). In the specialcase where ν < R ( κ, ν, z ) polynomials are interrelated by z R ( κ, , z ) + (2 κ + 3) R ( κ, , z ) + z R ( κ, , z ) = 1 . We also note that since(2 κ + 3)!! d ν d z ν (cid:20) J κ +2 ( z ) z κ +2 (cid:21) = (2 κ + 1)!! (cid:26) d ν d z ν (cid:20) J κ +1 ( z ) z κ +1 (cid:21) + d ν +2 d z ν +2 (cid:20) J κ +1 ( z ) z κ +1 (cid:21)(cid:27) , (as well as the corresponding relation for the H κ +1 ( z ) /z κ +1 ) it follows from the definition of thepolynomial ℘ ( κ, ν, z ) , Q ( κ, ν, z ) , R ( κ, ν, z ), that they also satisfy partial difference equations in κ and ν, we have F ( κ + 1 , ν, z ) = F ( κ, ν, z ) + F ( κ, ν + 2 , z ) . In summary, we will see that all of the integrals occurring above in the body of the text canbe expressed in terms of the Bessel and Struve functions of orders zero and one together with thepolynomials ℘ ( κ, ν, z ) , Q ( κ, ν, z ) , R ( κ, ν, z ) or the Lommel polynomials R m,ν ( z ).Below, the polynomials ℘, Q , and R and their relation to the Lommel polynomials [27] R m,ν ( z )is examined. .3 ℘ ( κ, ν, z ) , Q ( κ, ν, z ) , and R ( κ, ν, z ) and the Lommel polynomials As will be seen below, the polynomials ℘ ( κ, ν, z ), Q ( κ, ν, z ) , and R ( κ, ν, z ) can be expressed explic-itly as sums of the Lommel polynomials R m,ν ( z ) the latter being given by R m,ν ( z ) = (cid:100) m (cid:101) (cid:88) n =0 ( − n ( m − n )!Γ( ν + m − n ) n !( m − n )!Γ( ν + n ) (2 /z ) m − n . (C9)44sing (C9), we see that the leading terms for the Lommel polynomials are R κ, µ ( z ) ∼ ( − κ [1 − κ ( κ + 1)( κ + µ − κ + µ ) /z ] + · · · ,R κ +1 , µ ( z ) ∼ ( − κ κ + 1)( κ + µ ) /z + · · · , expressions which will be useful in the sequel in obtaining the leading terms of the ℘ ( κ, ν, z ), Q ( κ, ν, z ) , and R ( κ, ν, z ) polynomials.It is important to note that the R m,ν ( z ) polynomials can also be generated by the Bessel functionrelations [30] first obtained by Lommel i.e. J µ + m ( z ) = J µ ( z ) R m, µ ( z ) − J µ − ( z ) R m − , µ +1 ( z ) . (C10)The corresponding relations involving the Struve functions while being more complicated are givenby [28] H µ + m ( z ) = H µ ( z ) R m, µ ( z ) − H µ − ( z ) R m − , µ +1 ( z ) (C11)+ 1 √ π (cid:16) z (cid:17) µ + m − m − (cid:88) j =0 R j, µ + m − j ( z )Γ( µ + m + 1 / − j ) (cid:18) z (cid:19) j . Generalizing differential expressions due to Brychkov [29] we have the relations d ν (cid:2) J κ +1 ( z ) /z κ +1 (cid:3) d z ν = ( − ν ν ! z κ +1 (cid:98) ν (cid:99) (cid:88) i =0 ( − i J κ +1+ ν − i ( z ) i !( ν − i )!(2 z ) i (C12) d ν (cid:2) H κ +1 ( z ) /z κ +1 (cid:3) d z ν = ( − ν ν ! z κ +1 (cid:98) ν (cid:99) (cid:88) i =0 ( − i H κ +1+ ν − i ( z ) i !( ν − i )!(2 z ) i + ( − ν +1 ν !( z/ ν − π κ +1 (cid:98) ν (cid:99) (cid:88) i =0 ( − i i !( ν − i )! ( z ) i ν − − i (cid:88) j =0 Γ( j + 1 / κ + 3 / ν − j − i ) ( 2 z ) j . (C13)Using Eq. (C10) with µ = 1 and κ replaced by κ + ν − i in Eqs. (C12) and (C13) we get usingEqs. (C6) and (C7) the desired expressions for the ℘ ( κ, ν, z ), Q ( κ, ν, z ) , and R ( κ, ν, z ) polynomials.45 computed in Lommelmarch4.mw ) ℘ ( κ, ν, z ) = ( − ν +1 ν !(2 κ + 1)!! z κ +1 (cid:98) ν (cid:99) (cid:88) i =0 ( − i R κ + ν − − i , ( z ) i !( ν − i )!(2 z ) i , Q ( κ, ν, z ) = ( − ν ν !(2 κ + 1)!! z κ +1 (cid:98) ν (cid:99) (cid:88) i =0 ( − i R κ + ν − i , ( z ) i !( ν − i )!(2 z ) i , R ( κ, ν, z ) = ( − z/ ν − ν !(2 κ + 1)!!2 κ +2 [ (cid:98) ν (cid:99) (cid:88) m =0 c ( ν, m, m )Γ( κ + ν + 3 / − m ) ( 2 z ) m + ν − (cid:88) m = (cid:98) ν (cid:99) +1 c ( ν, m, (cid:98) ν (cid:99) )Γ( κ + ν + 3 / − m ) ( 2 z ) m − (cid:98) ν (cid:99) (cid:88) i =0 ( − /z ) i i !( ν − i )! κ + ν − − i (cid:88) j =0 √ πR j, κ + ν +1 − i − j ( z )Γ( κ + ν + 3 / − i − j ) ( 2 z ) j ] , where the coefficients c ( ν, m, N ) are given by c ( ν, m, N ) = N (cid:88) j =0 ( − / j Γ( m + 1 / − j ) j !( ν − j )! . In the case of ℘ ( κ, ν, z ), and Q ( κ, ν, z ) it is interesting to note that only higher powers of 1 /z occur. The leading terms of these polynomials are then given by ℘ (2 κ, µ + 1 , z ) ∼ ( − κ + µ (4 κ + 1)!! /z κ +1 ,℘ (2 κ + 1 , µ + 1 , z ) ∼ ( − κ + µ (4 κ + 3)!! { κ + 2 κ [2 µ + 3] + 7 µ + 4 } /z κ +3 ,℘ (2 κ, µ, z ) ∼ ( − κ + µ (4 κ + 1)!! { κ + 2 κ [2 µ + 1] + 3 µ } /z κ +2 ,℘ (2 κ + 1 , µ, z ) ∼ ( − κ + µ +1 (4 κ + 3)!! /z κ +2 , and Q (2 κ, µ + 1 , z ) ∼ ( − κ + µ +1 (4 κ + 1)!! { κ + 4 κ [ µ + 1] + 3 µ + 2 } /z κ +2 , Q (2 κ + 1 , µ + 1 , z ) ∼ ( − κ + µ (4 κ + 3)!! /z κ +2 , Q (2 κ, µ, z ) ∼ ( − κ + µ (4 κ + 1)!! /z κ +1 , Q (2 κ + 1 , µ, z ) ∼ ( − κ + µ (4 κ + 3)!! { κ + 4 κ [ µ + 1] + 5 µ + 2 } /z κ +3 . In the case of the polynomials R ( κ, µ, z ) the leading terms are more difficult to obtain. Using the46ifferential difference equations for the ℘ ( κ, ν, z ), Q ( κ, ν, z ) , and R ( κ, ν, z ) polynomials we get R ( κ, ν, z ) = d ν − R ( κ, , z ) d z ν − + ν − (cid:88) j =1 d ν − − j ℘ ( κ, j, z ) d z ν − − j , R ( κ, ν, z ) ∼ ( − ν ν ! z ν +1 + ℘ ( κ, ν − , z ) + d ℘ ( κ, ν − , z ) d z + · · · , ν > , R ( κ, ν, z ) ∼ ( − ν ν ! z ν +1 + 2 ℘ ( κ, ν − , z ) − Q ( κ, ν − , z ) . Using the leading term expressions for the ℘ ( κ, ν, z ), and Q ( κ, ν, z ) polynomials we get R (2 κ + 1 , µ, z ) ∼ (cid:26) (2 µ )! /z µ +1 , µ ≤ κ, ( − κ + µ +1 (4 κ + 3)!! { κ + 4 κ ( µ + 1) + 5 µ + 1 } /z κ +3 , µ > κ, , R (2 κ + 1 , µ + 1 , z ) ∼ − (2 µ + 1)! /z µ +2 , µ < κ, − [(4 κ + 3)!! + (2 κ + 1)!] /z κ +2 , µ = κ, ( − κ + µ +1 (4 κ + 3)!! /z κ +2 , µ > κ, , R (2 κ, µ + 1 , z ) ∼ − (2 µ + 1)! /z µ +2 , µ < κ, [(4 κ + 1)!! { κ + 7 κ + 1 } − (2 κ + 1)!] /z κ +2 , µ = κ, ( − κ + µ (4 κ + 1)!! { κ + 4 κ ( µ + 1) + 3 µ + 1 } /z κ +2 , µ > κ, , R (2 κ, µ, z ) ∼ (2 µ )! /z µ +1 , µ < κ, − [(4 κ + 1)!! − (2 κ )!] /z κ +1 , µ = κ, ( − κ + µ +1 (4 κ + 1)!! /z κ +1 , µ > κ, . .3.1 The Integrals C cc , S ss , C cs , S sc and T cc , T ss , T sc , T cs Recalling the integrals C cc , S ss , C cs , S sc defined above .i.e. C cc ( κ, z, (cid:37) ) = (cid:90) π/ sin κ +2 θ cos θ cos( z cos θ ) cos( (cid:37) cos θ ) dθ, (C14) S ss ( κ, z, (cid:37) ) = (cid:90) π/ sin κ +2 θ cos θ sin( z cos θ ) sin( (cid:37) cos θ ) dθ, C cs ( κ, z, (cid:37) ) = (cid:90) π/ sin κ +2 θ cos θ cos( z cos θ ) sin( (cid:37) cos θ ) dθ, S sc ( κ, z, (cid:37) ) = (cid:90) π/ sin κ +2 θ cos θ sin( z cos θ ) cos( (cid:37) cos θ ) dθ,
47e note that the cos( (cid:37) cos θ ) and sin( (cid:37) cos θ ) terms appearing in these integrals can be expandedin terms of Bessel functions [31] using the well known relations i.e.cos( (cid:37) cos φ ) = J ( (cid:37) ) + 2 ∞ (cid:88) κ =1 ( − κ J κ ( (cid:37) ) cos(2 κφ ) , sin( (cid:37) cos φ ) = 2 ∞ (cid:88) κ =0 ( − κ J κ +1 ( (cid:37) ) cos([2 κ + 1] φ ) . The former can be written as powers of the cos θ ascos( (cid:37) cos θ ) = J ( (cid:37) ) + 2 ∞ (cid:88) κ =1 ( − κ J κ ( (cid:37) ) T κ (cos θ ) , sin( (cid:37) cos θ ) = 2 ∞ (cid:88) κ =0 ( − κ J κ +1 ( (cid:37) ) T κ +1 (cos θ ) . where T n ( x ) are the Chebychev polynomials [32] of the first kind and are explicitly given by T ( x ) = 1 ,T n ( x ) = n (cid:98) n (cid:99) (cid:88) j =0 ( − j ( n − j − j !( n − j )! (2 x ) n − j for n ≥ . Using these expressions and Eq. (26) and having interchanged the order of the summations we getcos( (cid:37) cos θ ) = J ( (cid:37) ) + 2 ∞ (cid:88) κ =1 J κ ( (cid:37) )+ ∞ (cid:88) m =1 ( − m m (2 m )! cos m θ (cid:40) ∞ (cid:88) κ = m κ ( κ + m − κ − m )! J κ ( (cid:37) ) (cid:41) , sin( (cid:37) cos θ ) = ∞ (cid:88) m =0 ( − m m +1 (2 m + 1)! cos m +2 θ (cid:40) ∞ (cid:88) κ = m (2 κ + 1)( κ + m )!( κ − m )! J κ +1 ( (cid:37) ) (cid:41) . The sums containing the Bessel functions can be further reduced to terms containing J ( (cid:37) ) and J ( (cid:37) ) and the Lommel polynomials i.e. ∞ (cid:88) κ = m κ ( κ + m − κ − m )! J κ ( (cid:37) ) = J ( (cid:37) ) σ (1) c ( m, (cid:37) ) − J ( (cid:37) ) σ (2) c ( m, (cid:37) ) , ∞ (cid:88) κ = m (2 κ + 1)( κ + m )!( κ − m )! J κ +1 ( (cid:37) ) = J ( (cid:37) ) σ (1) s ( m, (cid:37) ) − J ( (cid:37) ) σ (2) s ( m, (cid:37) ) , and in the sums below as N approaches ∞ they converge rapidly and N can safely be set to 4. Therequired σ polynomials together with their leading term expressions are then given by (for N ≥ m σ (1) c ( m, (cid:37) ) = N (cid:88) κ = m κ ( κ + m − κ − m )! R κ − , ( (cid:37) ) ∼ ζ ( − N +1 [ N ( N + 1) − m ] ( N + m )!( N − m )! ,σ (2) c ( m, (cid:37) ) = N (cid:88) κ = m κ ( κ + m − κ − m )! R κ − , ( (cid:37) ) ∼ ( − N +1 ( N + m )!( N − m )! ,σ (1) s ( m, (cid:37) ) = N (cid:88) κ = m (2 κ +1) ( κ + m )!( κ − m )! R κ, ( (cid:37) ) ∼ ( − N ( N + m + 1)!( N − m )! ,σ (2) s ( m, (cid:37) ) = N (cid:88) κ = m (2 κ +1) ( κ + m )!( κ − m )! R κ − , ( (cid:37) ) ∼ ζ ( − N +1 [ N ( N + 2) − m ] ( N + m +1)!( N − m )! , (cid:98) σ (1) c ( (cid:37) ) = N (cid:88) κ =1 R κ − , ( (cid:37) ) ∼ ζ ( − N +1 N ( N + 1) , (cid:98) σ (2) c ( (cid:37) ) = − / N (cid:88) κ =1 R κ − , ( (cid:37) ) ∼
12 ( − N +1 . Combining the terms above we havecos( (cid:37) cos θ ) = J ( (cid:37) ) (cid:34) (cid:98) σ (1) c ( (cid:37) ) + ∞ (cid:88) m =1 ( − m (2 m )! σ (1) c ( m, (cid:37) ) cos m θ (cid:35) − J ( (cid:37) ) (cid:34) (cid:98) σ (2) c ( (cid:37) ) + ∞ (cid:88) m =1 ( − m (2 m )! σ (2) c ( m, (cid:37) ) cos m θ (cid:35) , and sin( (cid:37) cos θ ) = 2 J ( (cid:37) ) (cid:34) ∞ (cid:88) m =0 ( − m (2 m + 1)! σ (1) s ( m, (cid:37) ) cos m +2 θ (cid:35) − J ( (cid:37) ) (cid:34) ∞ (cid:88) m =0 ( − m (2 m + 1)! σ (2) s ( m, (cid:37) ) cos m +2 θ (cid:35) . The integrals in Eq. (28) in the main text can then be rewritten in terms of the J and H integralsas (The sums over m and κ converge rapidly and the upper limits can be replaced with the first49our terms i.e. 0 ≤ m ≤ m ≤ κ ≤ C cc ( κ, z, (cid:37) ) = J ( (cid:37) ) (cid:34) (cid:98) σ (1) c ( (cid:37) ) H ( κ, , , z ) + ∞ (cid:88) m =1 ( − m (2 m )! σ (1) c ( m, (cid:37) ) H ( κ, m, , z ) (cid:35) − J ( (cid:37) ) (cid:34) (cid:98) σ (2) c ( (cid:37) ) H ( κ, , , z ) + ∞ (cid:88) m =1 ( − m (2 m )! σ (2) c ( m, (cid:37) ) H ( κ, m, , z ) (cid:35) , S ss ( κ, z, (cid:37) ) = 2 J ( (cid:37) ) ∞ (cid:88) m =0 ( − m (2 m +1)! σ (1) s ( m, (cid:37) ) J ( κ, m, , z ) − J ( (cid:37) ) ∞ (cid:88) m =0 ( − m (2 m +1)! σ (2) s ( m, (cid:37) ) J ( κ, m, , z ) , C cs ( κ, z, (cid:37) ) = 2 J ( (cid:37) ) ∞ (cid:88) m =0 ( − m (2 m +1)! σ (1) s ( m, (cid:37) ) H ( κ, m, , z ) − J ( (cid:37) ) ∞ (cid:88) m =0 ( − m (2 m +1)! σ (2) s ( m, (cid:37) ) H ( κ, m, , z ) , S sc ( κ, z, (cid:37) ) = J ( (cid:37) ) (cid:34) (cid:98) σ (1) c ( (cid:37) ) J ( κ, , , z ) + ∞ (cid:88) m =1 ( − m (2 m )! σ (1) c ( m, (cid:37) ) J ( κ, m, , z ) (cid:35) − J ( (cid:37) ) (cid:34) (cid:98) σ (2) c ( (cid:37) ) J ( κ, , , z ) − ∞ (cid:88) m =1 ( − m (2 m )! σ (2) c ( m, (cid:37) ) J ( κ, m, , z ) (cid:35) . where the integrals J ( κ, j, a, z ) and H ( κ, j, a, z ) are given by Eq. (23)The C cc ( κ, z, (cid:37) ) , S ss ( κ, z, (cid:37) ) , C cs ( κ, z, (cid:37) ) , S sc ( κ, z, (cid:37) ) integrals can also be written in terms of the50essel and Struve functions using Eqs (23). We have, C cc ( κ, z, (cid:37) ) = πJ ( (cid:37) )[ H ( z ) (cid:98) P (1) c ( κ, , z, (cid:37) ) + H ( z ) (cid:98) Q (1) c ( κ, , z, (cid:37) )+ 2 π (cid:98) R (1) c ( κ, , z, (cid:37) )] (C15) − πJ ( (cid:37) )[ H ( z ) (cid:98) P (2) c ( κ, , z, (cid:37) )+ H ( z ) (cid:98) Q (2) c ( κ, , z, (cid:37) )+ 2 π (cid:98) R (2) c ( κ, , z, (cid:37) )] , S ss ( κ, z, (cid:37) ) = πJ ( (cid:37) )[ J ( z ) P (1) s ( κ, , z, (cid:37) )+ J ( z ) Q (1) s ( κ, , z, (cid:37) )] − πJ ( (cid:37) ) [ J ( z ) P (2) s ( κ, , z, (cid:37) ) + J ( z ) Q (2) s ( κ, , z, (cid:37) )] , (C16) C cs ( κ, z, (cid:37) ) = − πJ ( (cid:37) )[ H ( z ) P (1) s ( κ, , z, (cid:37) ) + H ( z ) Q (1) s ( κ, , z, (cid:37) )+ 2 π R (1) s ( κ, , z, (cid:37) )]+ πJ ( (cid:37) )[ H ( z ) P (2) s ( κ, , z, (cid:37) ) + H ( z ) Q (2) s ( κ, , z, (cid:37) )+ 2 π R (2) s ( κ, , z, (cid:37) )] , (C17) S sc ( κ, z, (cid:37) ) = − πJ ( (cid:37) )[ J ( z ) (cid:98) P (1) c ( κ, , z, (cid:37) )+ J ( z ) (cid:98) Q (1) c ( κ, ,z, (cid:37) )]+ πJ ( (cid:37) )[ J ( z ) (cid:98) P (2) c ( κ, , z, (cid:37) )+ J ( z ) (cid:98) Q (2) c ( κ, ,z, (cid:37) )] . (C18)where P ( η ) s ( κ, a, z, (cid:37) )= ∞ (cid:88) i =0 ( − i i (2 i + 1)! σ ( η ) s ( i, (cid:37) ) ℘ ( κ, i + 2 a + 1 , z ) , Q ( η ) s ( κ,a,z, (cid:37) )= ∞ (cid:88) i =0 ( − i i (2 i + 1)! σ ( η ) s ( i, (cid:37) ) Q ( κ, i + 2 a + 1 , z ) , R ( η ) s ( κ, a, z, (cid:37) )= ∞ (cid:88) i =0 ( − i i (2 i + 1)! σ ( η ) s ( i, (cid:37) ) R ( κ, i + 2 a + 1 , z ) , and P ( η ) c ( κ, a, z, (cid:37) )= ∞ (cid:88) i =1 ( − i i (2 i )! σ ( η ) c ( i, (cid:37) ) ℘ ( κ, i + 2 a + 1 , z ) , Q ( η ) c ( κ,a,z, (cid:37) )= ∞ (cid:88) i =1 ( − i i (2 i )! σ ( η ) c ( i, (cid:37) ) Q ( κ, i + 2 a + 1 , z ) , R ( η ) c ( κ, a, z, (cid:37) )= ∞ (cid:88) i =1 ( − i i (2 i )! σ ( η ) c ( i, (cid:37) ) R ( κ, i + 2 a + 1 , z ) , (cid:98) P ( η ) c ( κ, , z, (cid:37) )= P ( η ) c ( κ, , z, (cid:37) )+ (cid:98) σ ( η ) c ( (cid:37) ) ℘ ( κ, , z ) , (cid:98) Q ( η ) c ( κ, , z, (cid:37) ) = Q ( η ) c ( κ, , z, (cid:37) )+ (cid:98) σ ( η ) c ( (cid:37) ) Q ( κ, , z ) , (cid:98) R ( η ) c ( κ, , z, (cid:37) )= R ( η ) c ( κ, , z, (cid:37) )+ (cid:98) σ ( η ) c ( (cid:37) ) R ( κ, , z ) , and where η is 1 or 2. 51he leading terms for P ( η ) s ( κ, a, z, (cid:37) ) , P ( η ) c ( κ, a, z, (cid:37) ) and Q ( η ) s ( κ,a,z, (cid:37) ) , Q ( η ) c ( κ,a,z, (cid:37) ) are P (1) s (2 κ, a, z, (cid:37) ) ∼ ( − κ + a (4 κ +1)!! z κ +1 , P (1) s (2 κ + 1 , a, z, (cid:37) ) ∼ ( − κ + a (4 κ +3)!! z κ +3 [10 κ + κ (20 a + 670) + 25 a + 820] , P (2) s (2 κ, a, z, (cid:37) ) ∼ ( − κ + a +1 (4 κ +1)!! (cid:37) z κ +1 , P (2) s (2 κ + 1 , a, z, (cid:37) ) ∼ ( − κ + a +1 (4 κ +3)!! (cid:37) z κ +3 [10 κ + κ (20 a + 670) + 25 a + 820] , P (1) c (2 κ, a, z, (cid:37) ) ∼ ( − κ + a +1 (4 κ +1)!! (cid:37) z κ +1 , P (1) c (2 κ + 1 , a, z, (cid:37) ) ∼ ( − κ + a +1 (4 κ +3)!! (cid:37) z κ +3 [80 κ + κ (160 a + 11568) + 200 a + 14320] , P (2) c (2 κ, a, z, (cid:37) ) ∼ ( − κ + a +1 (4 κ +1)!! z κ +1 , P (2) c (2 κ + 1 , a, z, (cid:37) ) ∼ ( − κ + a +1 (4 κ +3)!! z κ +3 [16 κ + κ (32 a + 1008) + 40 a + 1232] , (cid:98) P (1) c (2 κ, , z, (cid:37) ) ∼ ( − κ +1 (4 κ +1)!! (cid:37) z κ +1 , (cid:98) P (1) c (2 κ + 1 , , z, (cid:37) ) ∼ ( − κ +1 (4 κ +3)!! (cid:37) z κ +3 [120 κ + 11574 κ + 14324] , (cid:98) P (2) c (2 κ, , z, (cid:37) ) ∼ ( − κ +1 (4 κ +1)!! z κ +1 , (cid:98) P (2) c (2 κ + 1 , , z, (cid:37) ) ∼ ( − κ +1 (4 κ +3)!! z κ +3 [9 κ + 511 κ + 620] , Q (1) s (2 κ,a,z, (cid:37) ) ∼ ( − κ + a +1 (4 κ +1)!! z κ +2 [10 κ + κ (20 a + 660) + 15 a + 490] , Q (1) s (2 κ + 1 ,a,z, (cid:37) ) ∼ ( − κ + a (4 κ +3)!! z κ +2 , Q (2) s (2 κ,a,z, (cid:37) ) ∼ ( − κ + a (4 κ +1)!! (cid:37) z κ +2 [80 κ + κ (160 a + 9504) + 120 a + 7088] , Q (2) s (2 κ + 1 ,a,z, (cid:37) ) ∼ ( − κ + a +1 (4 κ +3)!! (cid:37) z κ +2 , Q (1) c (2 κ,a,z, (cid:37) ) ∼ ( − κ + a (4 κ +1)!! (cid:37) z κ +2 [80 κ + κ (160 a + 9464) + 120 a + 8576] , Q (1) c (2 κ + 1 ,a,z, (cid:37) ) ∼ ( − κ + a +1 (4 κ +3)!! (cid:37) z κ +2 , Q (2) c (2 κ,a,z, (cid:37) ) ∼ ( − κ + a (4 κ +1)!! z κ +2 [16 κ + κ (32 a + 992) + 24 a + 736] , Q (2) c (2 κ + 1 ,a,z, (cid:37) ) ∼ ( − κ + a +1 (4 κ +3)!! z κ +2 , (cid:98) Q (1) c (2 κ, ,z, (cid:37) ) ∼ ( − κ (4 κ +1)!! (cid:37) z κ +2 [120 κ + 4 κ + 2] , (cid:98) Q (1) c (2 κ + 1 , ,z, (cid:37) ) ∼ ( − κ +1 (4 κ +3)!! (cid:37) z κ +2 , (cid:98) Q (2) c (2 κ, ,z, (cid:37) ) ∼ ( − κ (4 κ +1)!! z κ +2 [9 κ + 498 κ + 369] , (cid:98) Q (2) c (2 κ + 1 , ,z, (cid:37) ) ∼ ( − κ +1 (4 κ +3)!! z κ +2 . R ( η ) s ( κ, a, z, (cid:37) ) and R ( η ) c ( κ, a, z, (cid:37) ) for a equal to 0 or 1 being R (1) s (2 κ, , z, (cid:37) ) R (1) s (2 κ + 1 , , z, (cid:37) ) (cid:41) ∼ − z + O ( 1 z κ +2 ) , R (1) s (2 κ, , z, (cid:37) ) R (1) s (2 κ + 1 , , z, (cid:37) ) (cid:41) ∼ O ( 1 z κ +2 ) , R (2) s (2 κ, , z, (cid:37) ) R (2) s (2 κ + 1 , , z, (cid:37) ) (cid:41) ∼ (cid:37) (cid:26) z + O ( 1 z κ +2 ) (cid:27) , R (2) s (2 κ, , z, (cid:37) ) R (2) s (2 κ + 1 , , z, (cid:37) ) (cid:41) ∼ (cid:37) O ( 1 z κ +2 ) , R (1) c (2 κ, , z, (cid:37) ) R (1) c (2 κ + 1 , , z, (cid:37) ) R (1) c (2 κ, , z, (cid:37) ) R (1) c (2 κ + 1 , , z, (cid:37) ) ∼ (cid:37) O ( 1 z κ +2 ) , R (2) c (2 κ, , z, (cid:37) ) R (2) c (2 κ + 1 , , z, (cid:37) ) R (2) c (2 κ, , z, (cid:37) ) R (2) c (2 κ + 1 , , z, (cid:37) ) ∼ O ( 1 z κ +2 ) , (cid:98) R (1) c (2 κ, , z, (cid:37) ) (cid:98) R (1) c (2 κ + 1 , , z, (cid:37) ) (cid:41) ∼ (cid:37)z + 1 (cid:37) O ( 1 z k +2 ) , (cid:98) R (2) c (2 κ, , z, (cid:37) ) (cid:98) R (2) c (2 κ + 1 , , z, (cid:37) ) (cid:41) ∼ O ( 1 z κ +2 ) , (cid:98) R (2) c (2 κ, , z, (cid:37) ) (cid:98) R (2) c (2 κ + 1 , , z, (cid:37) ) (cid:41) ∼ z + O ( 1 z k +2 ) . Lastly the integrals T cc , T ss , T sc , T cs defined by T cc ( K ε , z, (cid:37) ) = (cid:90) π/ I ( K ε sin θ ) sin θ cos θ cos( z cos θ ) cos( (cid:37) cos θ ) dθ,T ss ( K ε , z, (cid:37) ) = (cid:90) π/ I ( K ε sin θ ) sin θ cos θ sin( z cos θ ) sin( (cid:37) cos θ ) dθ,T cs ( K ε , z, (cid:37) ) = (cid:90) π/ I ( K ε sin θ ) sin θ cos θ cos( z cos θ ) sin( (cid:37) cos θ ) dθ,T sc ( K ε , z, (cid:37) ) = (cid:90) π/ I ( K ε sin θ ) sin θ cos θ sin( z cos θ ) cos( (cid:37) cos θ ) dθ, I ( z ) = ∞ (cid:88) κ =0 ( − k z κ (2 κ + 3) , can be rewritten T cc ( K ε , z, (cid:37) ) = ∞ (cid:88) κ =0 ( − κ K κε (2 κ + 3) C cc ( κ + 1 , z, (cid:37) ) ,T ss ( K ε , z, (cid:37) ) = ∞ (cid:88) κ =0 ( − κ K κε (2 κ + 3) S ss ( κ + 1 , z, (cid:37) ) ,T cs ( K ε , z, (cid:37) ) = ∞ (cid:88) κ =0 ( − κ K κε (2 κ + 3) C cs ( κ + 1 , z, (cid:37) ) ,T sc ( K ε , z, (cid:37) ) = ∞ (cid:88) κ =0 ( − κ K κε (2 κ + 3) S sc ( κ + 1 , z, (cid:37) ) , able 3The ℘ ( k, , z ) polynomials k ℘ ( k, , z )1 − /z − /z /z − /z /z − /z − /z + 1496880 /z − /z − /z + 259459200 /z − /z /z − /z + 58378320000 /z − /z Table 4The Q ( k, , z ) polynomials k Q ( k, , z )1 6 /z − /z + 120 /z − /z + 5040 /z /z − /z + 362880 /z /z − /z + 39916800 /z − /z + 38918880 /z − /z + 6227020800 /z − /z + 9729720000 /z − /z + 1307674368000 /z Table 5The R ( k, , z ) polynomials k R ( k, , z )1 1 /z /z + 20 /z /z + 7 /z + 840 /z /z + 9 /z − /z + 60480 /z /z + 11 /z + 297 /z − /z + 6652800 /z /z + 13 /z + 429 /z + 154440 /z − /z + 1037836800 /z /z + 15 /z + 585 /z + 32175 /z + 64864800 /z − /z + 217945728000 /z Table 6The ℘ (1 , ν, z ) polynomials ν ℘ (1 , ν, z )0 − /z /z /z − /z − /z + 360 /z − /z + 225 /z − /z Table 7The Q (1 , ν, z ) polynomials Q (1 , ν, z )0 6 /z /z − /z − /z + 120 /z − /z + 144 /z − /z /z − /z + 5040 /z Table 8The R (1 , ν, z ) polynomials ν R (1 , ν, z )0 1 /z − /z /z /z − /z − /z + 840 /z eferences [1] Parr R G and Yang W., 1982, Density functional theory for atoms and molecules (New York:Oxford University Press), Pearson, R.G.
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