Analytical and numerical expressions for the number of atomic configurations contained in a supershell
AAnalytical and numerical expressions for the numberof atomic configurations contained in a supershell
Jean-Christophe Pain a, and Michel Poirier b a CEA, DAM, DIF, F-91297 Arpajon, France b Universit´e Paris-Saclay, CEA, CNRS, LIDYL, F-91191 Gif-sur-Yvette, France
Abstract
We present three explicit formulas for the number of electronic configurations in an atom, i.e. thenumber of ways to distribute Q electrons in N subshells of respective degeneracies g , g , ..., g N . Thenew expressions are obtained using the generating-function formalism. The first one contains sumsinvolving multinomial coefficients. The second one relies on the idea of gathering subshells havingthe same degeneracy. A third one also collects subshells with the same degeneracy and leads to thedefinition of a two-variable generating function, allowing the derivation of recursion relations. Allthese formulas can be expressed as summations of products of binomial coefficients. Concerning thedistribution of population on N distinct subshells of a given degeneracy g , analytical expressions forthe first moments of this distribution are given. The general case of subshells with any degeneracyis analyzed through the computation of cumulants. A fairly simple expression for the cumulants atany order is provided, as well as the cumulant generating function. Using Gram-Charlier expansion,simple approximations of the analyzed distribution in terms of a normal distribution multiplied by asum of Hermite polynomials are given. These Gram-Charlier expansions are tested at various ordersand for various examples of supershells. When few terms are kept they are shown to provide simpleand efficient approximations of the distribution, even for moderate values of the number of subshells,though such expansions diverge when higher order terms are accounted for. The Edgeworth expansionhas also been tested. Its accuracy is equivalent to the Gram-Charlier accuracy when few terms arekept, but it is much more rapidly divergent when the truncation order increases. While this analysisis illustrated by examples in atomic supershells it also applies to more general combinatorial problemssuch as fermion distributions. The knowledge of the number of atomic configurations (i.e. the number of possible ways to distribute Q electrons in N subshells of respective degeneracies g , g , ..., g N ) is important for the computation ofatomic structure and spectra [4–6, 12, 22, 30] and is a fundamental problem of statistical physics [8, 17,25, 26]. However, it is a difficult combinatorial problem (belonging to the class of the so-called “boundedpartitions” [2, 14, 29]) and the number of electronic configurations is usually evaluated numerically bydirect multiple summations requiring the computation of nested-loops. A few years ago, efficient doublerecursion relations, on the number of electrons and the number of orbitals, were published [13, 23, 28].However, we could not find in the literature an analytical expression valid in any case. For this reason, inthis paper we develop various analytical and numerical methods providing this number of configurations.As part of the above quoted bibliography suggests, the present analysis is not limited to the number ofconfigurations obtained by distributing Q electrons in a list of subshells, but deals with more generalcombinatorial questions related, e.g., to fermion statistics.The generating function for the number of configurations is introduced in section 2, along with some ofits interesting properties. The first expressions involving multinomial coefficients is presented in section 3, [email protected] a r X i v : . [ phy s i c s . a t o m - ph ] A ug nd the second expression, obtained by partitioning the subshells into iso-degeneracy groups, is derivedin section 4. Focusing on the case of susbshells with the same degeneracy, a two-variable generatingfunction allows us to obtain several recurrence relations (section 5) and to compute moments at anyorder (section 6). Furthermore, the cumulants of this distribution as well as the cumulant generatingfunction are obtained analytically in section 7. The availability of these cumulants allows us to derivesimple approximations for this number of configurations using a Gram-Charlier and Edgeworth expansionin sections 8 and 8 respectively. Concluding remarks are finally given. We have to find the number of integer solutions of Q = q + q + q + ... with the restrictions : 0 ≤ q ≤ g ,..., 0 ≤ q N ≤ g N . Such constraints can be efficiently accounted for using generating functions [15, 16].This number of solutions being denoted C ( Q, N ), we define the generating function with G ( x, N ) = ∞ (cid:88) Q =0 x Q C ( Q, N ) (1a)= ∞ (cid:88) Q =0 x Q (cid:88) { q ,q , ··· ,q N } δ Q,q + q + ··· + q N θ ( g − q ) · · · θ ( g N − q N ) , (1b)where δ represents the Kronecker symbol and θ the Heaviside function. One gets G ( x, Q ) = (cid:88) { q i } x q + q + ··· + q N θ ( g − q ) · · · θ ( g N − q N ) . (2)Since the quantities q i are independent, one has G ( x, Q ) = g (cid:88) q =0 x q · · · g N (cid:88) q N =0 x q N , (3)i.e., G ( x, N ) = N (cid:89) i =1 (cid:20) − x g i +1 − x (cid:21) = 1(1 − x ) N N (cid:89) i =1 (cid:0) − x g i +1 (cid:1) , (4)with 1(1 − x ) N = ∞ (cid:88) i =0 (cid:18) N − iN − (cid:19) x i . (5)If all the orbitals had the same degeneracy, we would have N (cid:89) i =1 (cid:0) − x g +1 (cid:1) = (cid:0) − x g +1 (cid:1) N = N (cid:88) k =0 ( − k (cid:18) Nk (cid:19) x k ( g +1) (6)and, combining Eqs. (5) and (6) C ( Q, N ) = (cid:98) N/ ( g +1) (cid:99) (cid:88) k =0 (cid:18) Nk (cid:19) ( − k (cid:18) N − Q − k ( g + 1) N − (cid:19) , (7)where (cid:98) x (cid:99) denotes the integer part of x . However, since all the orbitals do not in general have the samedegeneracy, the problem is more complicated. Let us take the example of four orbitals with degeneracy g , g , g , g . In the present case, this generating function involves the product2 (cid:89) i =1 (cid:0) − x g i +1 (cid:1) =1 − x g +1 − x g +1 − x g +1 − x g +1 + x g + g +2 + x g + g +2 + x g + g +2 + x g + g +2 + x g + g +2 + x g + g +2 − x g + g + g +3 − x g + g + g +3 − x g + g + g +3 − x g + g + g +3 + x g + g + g + g +4 , (8)which can be expressed in terms of the so-called symmetric functions [3, 24]. Knowing the generatingfunction, one can now write C ( Q, N ) as a contour integral C ( Q, N ) = 12 iπ (cid:73) dzz Q +1 G ( z, N ) (9a)= 12 iπ (cid:73) dzz Q +1 N (cid:89) i =1 (cid:20) − z g i +1 − z (cid:21) . (9b)Assuming that the number of electrons Q and the number of orbitals N are large, one finds (followingthe asymptotics of partitions of Hardy-Ramanujan [2]) C ( Q, N ) = 12 iπ (cid:73) dzz e S N,Q ( z ) , (10)with S N,Q ( z ) = N (cid:88) i =1 ln (cid:18) − z g i +1 − z (cid:19) − Q ln z, (11)and one has to find z such that dS N,Q dz (cid:12)(cid:12)(cid:12) z = 0. However, it is difficult to find some large quantities inthe present case. Therefore, we usually make the calculation using a recursion relation [19] C ( Q, N ) = Q (cid:88) i =0 C ( Q − i, N − θ ( g N − i )= min( Q,g N ) (cid:88) i =0 C ( Q − i, N − , (12)where g N is the last-orbital degeneracy. The recurrence is initialized by C ( Q,
0) = δ Q, .One may note that, in a different context, formula (7) has been used by Crance (see Appendix inRef. [10]) to calculate the proportion of neutral atoms in a statistical description of multiple ionization. The number of atomic configurations of Q electrons in N subshells is related to the generating function G ( x, N ) by C ( Q, N ) = 1 Q ! ∂ Q ∂x Q G ( x, N ) (cid:12)(cid:12)(cid:12)(cid:12) x =0 . (13)The recursion relation (12) can be obtained from this relation. Using the Leibniz rule for the derivativeof a product of two functions, we obtain 3 ( Q, N ) = 1 Q ! Q (cid:88) i =0 (cid:18) Qi (cid:19) ∂ i ∂x i − x ) N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x =0 ∂ Q − i ∂x Q − i N (cid:89) i =1 (cid:0) − x g i +1 (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x =0 . (14)We have ∂ i ∂x i − x ) N (cid:12)(cid:12)(cid:12)(cid:12) x =0 = i ! (cid:18) i + N − i (cid:19) (15)and ∂ Q − i ∂x Q − i N (cid:89) i =1 (cid:0) − x g i +1 (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x =0 = (cid:88) (cid:126)α/ (cid:80) Nj =1 α j = Q − i ( Q − i )! α ! α ! α ! ...α N ! N (cid:89) j =1 ∂ α j ∂x α j (cid:0) − x g j +1 (cid:1)(cid:12)(cid:12) x =0 , (16)where (cid:126)α = ( α , α , · · · , α N ). The quantity (cid:18) Q − iα , α · · · , α N (cid:19) = ( Q − i )! α ! α ! · · · α N ! (17)is the multinomial coefficient. It can be expressed in numerous ways, including a product of binomialcoefficients (cid:18) Q − iα , α · · · , α N (cid:19) = δ Q − i,α + ··· α N (cid:18) α α (cid:19)(cid:18) α + α α (cid:19) · · · (cid:18) Q − iα N (cid:19) . (18)We have also, if α j (cid:54) = 0 ∂ α j ∂x α j (cid:0) − x g j +1 (cid:1)(cid:12)(cid:12) x =0 = − ( g j + 1)! × δ α j ,g j +1 (19)and we get finally C ( Q, N ) = 1 Q ! Q (cid:88) i =0 i ! (cid:18) Qi (cid:19)(cid:18) i + N − i (cid:19) (cid:88) (cid:126)α/ (cid:80) Nj =1 α j = Q − i ( Q − i )! α ! α ! ...α N ! N (cid:89) j =1 (cid:0) δ α j , − ( g j + 1)! δ α j ,g j +1 (cid:1) , (20)which can also be put in the form C ( Q, N ) = Q (cid:88) i =0 (cid:18) i + N − i (cid:19) (cid:88) (cid:126)α/ (cid:80) Nj =1 α j = Q − i α ! α ! ...α N ! N (cid:89) j =1 (cid:0) δ α j , − ( g j + 1)! δ α j ,g j +1 (cid:1) , (21)which is the first main result of the present work. Let us consider the case where n orbitals have the same degeneracy g and n orbitals have the samedegeneracy g , with N = n + n . For instance (2 p p p ) and (3 d d ) correspond to g =6, g =10, n =4and n =6, i.e. N =10. The generating function can be put in the form: G ( x, N ) = (cid:18) − x g +1 − x (cid:19) n (cid:18) − x g +1 − x (cid:19) n . (22)Using the Leibniz formula for the derivative of a product of two functions, we get4 ( Q, N ) = 1 Q ! Q (cid:88) i =0 (cid:18) Qi (cid:19) ∂ i ∂x i − x ) n + n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x =0 × ∂ Q − i ∂x Q − i (cid:104)(cid:0) − x g +1 (cid:1) n (cid:0) − x g +1 (cid:1) n (cid:105)(cid:12)(cid:12)(cid:12) x =0 . (23)We still have ∂ i ∂x i − x ) n + n (cid:12)(cid:12)(cid:12)(cid:12) x =0 = i ! (cid:18) i + n + n − i (cid:19) (24)and since (cid:0) − x g +1 (cid:1) n = n (cid:88) i =0 ( − i (cid:18) n i (cid:19) x i ( g +1) , (25)one can write ∂ Q − i ∂x Q − i (cid:104)(cid:0) − x g +1 (cid:1) n (cid:0) − x g +1 (cid:1) n (cid:105)(cid:12)(cid:12)(cid:12)(cid:12) x =0 = n (cid:88) i =0 n (cid:88) i =0 ( − i + i (cid:18) n i (cid:19)(cid:18) n i (cid:19) × [ i ( g + 1) + i ( g + 1) − Q + i + 1] [ i ( g + 1) + i ( g + 1) − Q + i + 2] · · ·× [ i ( g + 1) + i ( g + 1) −
2] [ i ( g + 1) + i ( g + 1) − × [ i ( g + 1) + i ( g + 1)] × x i ( g +1)+ i ( g +1) − Q + i (cid:12)(cid:12)(cid:12) x =0 . (26)The only non-zero value on the right-hand side corresponds to i = Q − i ( g + 1) − i ( g + 1) and wefinally get C ( Q, N ) = n (cid:88) i =0 n (cid:88) i =0 ( − i + i (cid:18) n i (cid:19)(cid:18) n i (cid:19)(cid:18) n + n − Q − i ( g + 1) − i ( g + 1) n + n − (cid:19) . (27)If we generalize and gather the n subshells of degeneracy g , the n subshells of degeneracy g , ..., the n s subshells of degeneracy g s (with therefore n + n + · · · n s = N ), we obtain C ( Q, N ) = n (cid:88) i =0 n (cid:88) i =0 · · · n s (cid:88) i s =0 ( − i + i + ··· n s (cid:18) n i (cid:19)(cid:18) n i (cid:19) · · · (cid:18) n s i s (cid:19) × (cid:18) n + · · · + n s − Q − i ( g + 1) − i ( g + 1) − · · · − i s ( g s + 1) n + · · · + n s − (cid:19) , (28)which is the second main result of the present work. The equation (28) is rather compact and adapted to numerical computation. However one may note thatit contains terms of alternating signs. It is possible to derive an alternate formula containing only positiveterms. Let us note N ( N , · · · N t ; g , · · · g t ; Q ) the number of configurations of Q electrons distributedwithin N distinct subshells of degeneracy g ,. . . N t subshells of degeneracy g t . For instance consideringthe non relativistic configurations constructed on the 1s 2s 2p 3s 3p 3d subshells , one has t = 3, N = 3, g = 2, N = 2, g = 6, and N = 1, g = 10. It is clear that the evaluation of this number can be5educed to the evaluation of the number of the configurations of a given degeneracy S ( g ; N ; Q ) which isthe number of configurations with Q electrons distributed on N subshells of the same degeneracy g . Thenumbers N and S are connected through the discrete convolution formula N ( N , · · · N t ; g , · · · g t ; Q ) = (cid:88) p · · · (cid:88) p t δ p + ··· + p t ,Q S ( g ; N ; p ) · · · S ( g t ; N t ; p t ) . (29)In this section we will focus on the computation of the S ( g ; N ; Q ) numbers. Let us consider for in-stance the case g = 4. To each configuration corresponds a 5-uple ( n , n , n , n , n ) of numbers ofsubshells with population from 0 to 4 respectively. Obviously two configurations with distinct 5-uplesare different. Conversely, there are several distinct configurations for a given set ( n , n , n , n , n ), thatcan be straightforwardly numbered. One has (cid:0) Nn (cid:1) ways to choose the subshell(s) with 4 electrons, then (cid:0) N − n n (cid:1) ways to choose the remaining subshell(s) with 3 electrons, etc. Therefore the total number ofconfigurations writes S ( g ; N ; Q ) | g =4 = (cid:88) (cid:18) Nn (cid:19)(cid:18) N − n n (cid:19)(cid:18) N − n − n n (cid:19)(cid:18) N − n − n − n n (cid:19)(cid:18) N − n − n − n − n n (cid:19) (30a)where the summation is performed on all ( n , n , n , n , n ) verifying N = n + n + n + n + n (30b) Q = n + 2 n + 3 n + 4 n . (30c)The product of binomial coefficients in the above sum simplifies, and one gets in the general case, S ( g ; N ; Q ) = (cid:88) n ,n ··· n g δ n + ··· + n g ,N δ n + ··· + gn g ,Q N ! n ! n ! · · · n g ! (31a)which, introducing the multinomial coefficient (17), writes S ( g ; N ; Q ) = (cid:88) n ,n ··· ng C (cid:18) Nn , n , · · · n g (cid:19) (31b)where the multiple sum is constrained by the double condition C N = n + n + · · · + n g (31c) Q = n + 2 n + · · · + gn g . (31d)This equation, in conjunction with (29), provides a third expression for the total number of configurations.Let us now consider the generating function G ( g ; z, X ) = ∞ (cid:88) n =0 z n n ! ∞ (cid:88) n =0 z n X n n ! ∞ (cid:88) n =0 z n X n n ! · · · ∞ (cid:88) n g =0 z n g X gn g n g ! (32a)= exp( z + zX + zX · · · + zX g ) (32b)= exp (cid:18) z − X g +1 − X (cid:19) . (32c)Comparing the above expansion with the value (31a) one checks that G ( g ; z, X ) = ∞ (cid:88) Q =0 ∞ (cid:88) N =0 S ( g ; N ; Q ) z N N ! X Q . (33)6herefore one may express the number of configurations as the partial derivative S ( g ; N ; Q ) = 1 Q ! ∂ N + Q ∂z N ∂X Q G ( g ; z, X ) (cid:12)(cid:12)(cid:12)(cid:12) z =0 ,X =0 . (34)The above expansion allows us to derive various properties. Using the form (32c) one easily verifies that G ( g ; z, X ) = G ( g ; zX g , /X ) (35)which implies S ( g ; N ; Q ) = S ( g ; N ; gN − Q ) . (36)Recursion relations can be obtained by deriving the generating function (32c) with respect to z or X .Writing the ratio (1 − X g +1 ) / (1 − X ) (resp. its derivative) as the polynomial 1 + X + · · · + X g (resp.1 + 2 X + · · · + gX g − ), one gets two identities. First, using derivation versus z and identifying terms in z N X Q one has S ( g ; N + 1; Q ) = min( g,Q ) (cid:88) j =0 S ( g ; N ; Q − j ) . (37)Then, using derivation versus X , assuming Q >
0, one obtains S ( g ; N + 1; Q ) = N + 1 Q min( g,Q ) (cid:88) j =1 j S ( g ; N ; Q − j ) . (38)In a similar way, dealing with (1 − X g +1 ) / (1 − X ) or its derivative as a rational fraction one first gets byderiving with respect to z − X g +1 − X exp (cid:18) z − X g +1 − X (cid:19) = 1 − X g +1 − X (cid:88) N,Q S ( g ; N ; Q ) z N N ! X Q (39a)= (cid:88) N,Q S ( g ; N ; Q ) z N − ( N − X Q (39b)and after multiplying the right-hand sides of these subequations by (1 − X ) and identifying the factor of z N X Q , one has S ( g ; N + 1; Q ) − S ( g ; N + 1; Q −
1) = S ( g ; N ; Q ) − S ( g ; N ; Q − g − . (39c)Then after deriving the generating function G with respect to X and multiplying both sides by (1 − X ) , z (cid:2) − ( g + 1) X g + gX g +1 (cid:3) (cid:88) NQ S ( g ; N ; Q ) z N N ! X Q = (1 − X ) (cid:88) NQ Q S ( g ; N ; Q ) z N N ! X Q − (40a)and term-by-term identification leads to the recurrence relation( Q + 1) S ( g ; N + 1; Q + 1) − Q S ( g ; N + 1; Q ) + ( Q − S ( g ; N + 1; Q − N + 1) (cid:16) S ( g ; N ; Q ) − ( g + 1) S ( g ; N ; Q − g ) + g S ( g ; N ; Q − g − (cid:17) . (40b)The first recurrence (37) has been mentioned previously (12). If the S ( g ; N ; Q ) numbers are written ina Pascal-like triangle where lines are indexed by N and columns by Q , this equation implies that anynumber in the array is equal to the sum of the numbers located on the row above at the g + 1 positionsending at the current column — ignoring elements with negative column indices. In the special case g = 1 this rule reverts to the usual triangle rule so that S (1; N ; Q ) = (cid:18) NQ (cid:19) . (41)7f course this relation could also have been obtained by a direct argument. Noting that the generatingfunction (33) verifies G ( g ; z, X ) = G ( g − z, X ) exp( zX g ) (42)one obtains an additional recurrence relation on the degeneracy g . This equation may be written, withthe above definitions (cid:88) NQ S ( g ; N ; Q ) z N N ! X Q = S ( g − N ; Q ) (cid:88) j z j X jg j ! (43)and identifying the terms in z N X Q on both sides one gets S ( g ; N ; Q ) = (cid:88) j (cid:18) Nj (cid:19) S ( g − N − j ; Q − jg ) (44)the minimum index j being max(0 , Q − ( g − N ) so that one has Q − jg ≤ ( g − N − j ), and themaximum index j being min( N, (cid:98) Q/g (cid:99) ). With the initial value (41), this relation may be used to get all S ( g ; N ; Q ). Because of the symmetry property (36), for a given number of subshells N the evaluationneeds only to be done for 0 ≤ Q ≤ p max = (cid:98) ( gN + 1) / (cid:99) . For low Q values, the sum (44) contains veryfew terms since one must have Q − jg ≥
0. For Q = p max , the maximum index j is only (cid:98) ( N + 1) / (cid:99) .Up to our knowledge, the recurrence relations (38, 39c,40b,44) have not been published previously.Using a batch of test values (mostly in the g = 6 case) we have checked that the various recurrencesobtained here are numerically correct. Moreover, at variance with the relations derived in the previoussections, the sums in the right-hand side of (37,38) involve only positive terms and therefore cannot giverise to a loss of accuracy or instability after repeated use of the recurrence. N distinctsubshells with the same degeneracy using moments calcula-tion The formulas given in the preceding sections, and mostly those involving recurrence relations, provide avery fast method to get a large set of S ( g ; N ; Q ) values. As mentioned before, if g = 1 the distributionof S as a function of Q is binomial. A very efficient characterization of such distributions lies in theanalysis of moments defined, for a given degeneracy g and subshell number N , as M ( g ; N ; k ) = gN (cid:88) Q =0 Q k S ( g ; N ; Q ) . (45)The moment analysis is, in particular, crucial in the study of unresolved transition arrays as proven byBauche et al. [6]. It allows to give a simple and often accurate description of such arrays through thedefinition of a small number of such moments.We have been able to derive analytically or numerically the corresponding formulas for the moments.Indeed, it has been mentioned in several works [18, 20, 21] that, in some cases, the knowledge of themoments up to the second (variance) is far from sufficient to describe distributions significantly differentfrom the normal distribution. This is why a certain effort is devoted here to moments up to a quite largeorder.First one easily finds that M ( g ; N ; 0) = ( g + 1) N (46)since this is the total number of configurations with any number of electrons distributed over N subshellsof degeneracy g . As mentioned in Eq. (36) the S ( g ; N ; Q ) distribution is symmetric with respect to itsmedian value gN/
2, and this provides immediately the next moment M ( g ; N ; 1) = 12 gN ( g + 1) N . (47)8he generating function (33) also allows us to derive expressions for moments at any order in a closedform. Explicitly, one has for the k -th order derivative with respect to X∂ k G ( g ; z ; X ) ∂X k = ∞ (cid:88) Q =0 ∞ (cid:88) n =0 ( Q ) k S ( g ; n ; Q ) z n n ! X Q − k . (48)where, for integer n , ( A ) n = A ( A − ... ( A − n + 1) (49)is the so-called descending factorial. Evaluating this quantity for X = 1 provides the successive momentsof the S distribution. Indeed, one easily checks using the analytical form (32b) ∂ k ∂X k exp ( z (1 + X + · · · + X g )) (cid:12)(cid:12)(cid:12)(cid:12) X =1 = ∞ (cid:88) z =0 z N N ! M ( g ; N ; k ) (50a)with M ( g ; N ; k ) = gN (cid:88) Q =0 ( Q ) k S ( g ; N ; Q ) . (50b)Therefore the modified moments M appear as the ( N + k )-th partial derivative M ( g ; N ; k ) = ∂ N + k ∂X k ∂z N exp ( z (1 + X + · · · + X g )) (cid:12)(cid:12)(cid:12)(cid:12) X =1 ,z =0 (50c)= ∂ k ∂X k (cid:104) (1 + X + · · · + X g ) N exp (cid:16) z (1 + X + · · · + X g ) (cid:17)(cid:105)(cid:12)(cid:12)(cid:12)(cid:12) X =1 ,z =0 . (50d)For instance, in the case k = 0, one gets immediately ( g + 1) N as mentioned above (46). The variousmoments (45) can be easily related to the sums obtained above (50d) since one has x n = n (cid:88) j =0 (cid:26) nj (cid:27) ( x ) j (51)the coefficients on the right-hand side being the Stirling numbers of the second kind [9]. These numberscan be easily generated from the recurrence [1] (cid:26) n + 1 m (cid:27) = m (cid:26) nm (cid:27) + (cid:26) nm − (cid:27) with, by convention, (cid:26) (cid:27) = 1 , (cid:26) n (cid:27) = 0 if n > . (52)Furthermore, the Arbogast-Fa di Bruno’s formula allows us to write [1, 9] ∂ k S ( X ) N ∂X k = (cid:88) n ,n , ··· ,n k δ k,n +2 n ··· + kn k P ( k ; n , n · · · , n k ) S (1) ( X ) n S (2) ( X ) n · · · S ( k ) ( X ) n k ( N ) d S ( X ) N − d (53)where d = n + n + · · · + n k (54)and where P ( k ; n , n · · · , n k ) is the number of partitions of k distinct objects with n groups containing1 element, n groups containing 2 elements,. . . n k groups containing k elements. The number P is givenby Eq. (97) of Appendix A. In order to close the computation, one needs to substitute 1 + X + · · · + X g to S ( X ) in the derivative formula (53) and therefore to compute the partial derivative T j = ∂ j ∂X j (1 + X + · · · X g ) (cid:12)(cid:12)(cid:12)(cid:12) X =1 . (55)9his can be easily performed by explicitly deriving the first values T = g + 1 , T = 12 g ( g + 1) , T = 13 ( g − g ( g + 1) , (56)from which one infers the general form T r = 1 r + 1 ( g + 1)!( g − r )! = r ! (cid:18) g + 1 r + 1 (cid:19) . (57)The proof of the above alleged expression can be established by a simple recurrence on the index g . Theaverage over the distribution S ( g ; N ; Q ) of any function of Q X ( Q ) is defined as (cid:104) X ( Q ) (cid:105) = (cid:88) Q X ( Q ) S ( g ; N ; Q ) / (cid:88) Q S ( g ; N ; Q ) = (cid:88) Q X ( Q ) S ( g ; N ; Q ) / ( g + 1) N . (58)Collecting formulas (50c,53,57, 58,97), and noting that the factor S ( X ) N − d in Eq. (53) may be writtenas S ( X ) N − d = ( g + 1) N − n − n ···− n k , (59)one gets finally the average value of the descending factorials ( Q ) k , (cid:104) ( Q ) k (cid:105) = M ( g ; N ; k )( g + 1) N (cid:88) n ··· n k δ j,n +2 n + ··· kn k k ! (cid:81) kq =1 n q !( q !) n q ( N ) n + n ··· + n k k (cid:89) r =1 (cid:20) r + 1 g !( g − r )! (cid:21) n r (60)and the normalized moments, using the sum (51), M ( g ; N ; k ) / ( g + 1) N = k (cid:88) j =0 (cid:26) kj (cid:27) (cid:88) n ··· n j δ j,n +2 n + ··· jn j j ! (cid:81) jq =1 n q !( q !) n q ( N ) n + n ··· + n j j (cid:89) r =1 (cid:20) r + 1 g !( g − r )! (cid:21) n r . (61)Using the second form for T as written in Eq. (57) one may also write the somewhat simpler result (cid:104) ( Q ) k (cid:105) / ( g + 1) N = (cid:88) n ··· n k δ k,n +2 n + ··· kn k k ! (cid:81) kq =1 n q ! ( N ) d ( g + 1) d k (cid:89) r =1 (cid:18) g + 1 r + 1 (cid:19) n r (62)with d is the sum of the n j indices (54).The moments with k ≤ n j in the expression (60), the analytical expressions for moments up to k = 10 can be obtained at a verylow computational cost. Indeed, considering for instance the 4-th order moment, the nested loop on n j indices only contains four terms, namely ( n = 4) , ( n = 2 , n = 1) , ( n = 1 , n = 1) , ( n = 1), where allthe unmentioned n j are 0. From the above expression one may also notice that each of these normalizedmoments is given by a polynomial form M ( g ; N ; k ) / ( g + 1) N = k (cid:88) p =1 k (cid:88) q =1 c pq ( g ; N ; k ) g p N q . (63)To get moments for large k values, it may be easier to use such formula instead of (61). One first computesnumerically a series of moments for various g and N values using the previously mentioned recurrencerelation, and one then solves the linear system (63) to obtain the c pq .The normalized centred moments are defined as M c ( g ; N ; k ) = gN (cid:88) Q =0 ( Q − gN/ k S ( g ; N ; Q ) / ( g + 1) N . (64)10 Non-centred moment2 14 g N + 112 g ( g + 2) N g N + 18 g ( g + 2) N g N + 18 g ( g + 2) N + 148 g ( g + 2) N − g ( g + 2)( g + 2 g + 2) N g N + 548 g ( g + 2) N + 596 g ( g + 2) N − g ( g + 2)( g + 2 g + 2) N g N + 564 g ( g + 2) N + 564 g ( g + 2) N − g ( g + 2)(13 g + 16 g + 16) N − g ( g + 2) ( g + 2 g + 2) N + 1252 g ( g + 2)( g + g + 1)( g + 3 g + 3) N g N + 7128 g ( g + 2) N + 35384 g ( g + 2) N − g ( g + 2)( g − g − N − g ( g + 2) ( g + 2 g + 2) N + 172 g ( g + 2)( g + g + 1)( g + 3 g + 3) N g N + 7192 g ( g + 2) N + 35384 g ( g + 2) N + 7288 g ( g + 2)( g + 7 g + 7) N − g ( g + 2) (67 g + 124 g + 124) N + 1576 g ( g + 2)(9 g + 22 g + 14 g − g − N + 18640 g ( g + 2) (101 g + 404 g + 728 g + 648 g + 324) N − g ( g + 2)( g + 2 g + 2)( g + 4 g + 6 g + 4 g + 2) N Table 1: Normalized non-centered moments M ( g ; N ; k ) / ( g + 1) N of the distribution S ( g ; N ; Q ).11ecause of the symmetry property (36), these moments cancel if k is odd. Using the above equation andthe general expression (61) one obtains moments with a somewhat simpler form than the non-centredmoments. Namely one has M c ( g ; N ; 2) = 112 g ( g + 2) N = D N (65a) M c ( g ; N ; 4) = 148 g ( g + 2) N − g ( g + 2)( g + 2 g + 2) N = D N − D N (65b) M c ( g ; N ; 6) = 5576 g ( g + 2) N − g ( g + 2) ( g + 2 g + 2) N + 1252 g ( g + 2)( g + g + 1)( g + 3 g + 3) N = 5 D N − D D N + D N (65c) M c ( g ; N ; 8) = 356912 g ( g + 2) N − g ( g + 2) ( g + 2 g + 2) N + 18640 g ( g + 2) (101 g + 404 g + 728 g + 648 g + 324) N − g ( g + 2)( g + 2 g + 2)( g + 4 g + 6 g + 4 g + 2) N = 35 D N − D D N + D D N + 7 D N − D N. (65d)where we have defined, for the sake of simplification, D k = ( g + 1) k − . (65e)The first centred moment of this list is the variance σ = 112 g ( g + 2) N (66)and the second one is related to the excess kurtosis, given by [27] κ /σ = M c ( g ; N ; 4) /σ − − g + 2 g + 2)5 g ( g + 2) N (67)which would be zero for a normal distribution. As one will verify below, the excess kurtosis can besignificantly different from 0, especially for large g and moderate N . This negative value means that suchdistributions, named platykurtic, are flatter than the normal distribution. Conversely, for a given g , onehas lim N →∞ κ /σ = 0. The previous considerations are useful to characterize the S population distribution, e.g., by comparing itto a normal distribution. They can be used to compute Gram-Charlier approximations (by truncating thisseries at various orders). However they suffer from two limitations. The first one is that the expressionsfor the moments increase in complexity with the order k . The second one is that they do not apply whenseveral subshells with different degeneracies g are present in the supershell.To circumvent these limitations, one must resort to the cumulant formalism. The global distributionis given by the discrete convolution formula (29). While the normalized centred moments cannot in thegeneral case be expressed as the sum of the ( g j , n j ) moments (65), the additivity holds for the cumulants .The generating function for the cumulants is defined as [27] K ( t ) = ∞ (cid:88) n =1 κ n t k n ! (68a)= log ( (cid:104) exp( tQ ) (cid:105) ) . (68b)12onsidering first the case of N distinct subshells with the same degeneracy g , the above average value (cid:104) exp( tQ ) (cid:105) can be easily computed. Using the well-known property arising from the convolution relation(12) S ( g ; N + N ; Q ) = (cid:88) j S ( g ; N ; j ) S ( g ; N ; Q − j ) (69)one has for the Laplace-transformed expression for any natural integers N , N (cid:88) Q S ( g ; N + N ; Q ) e Qt = (cid:88) Q,j S ( g ; N ; j ) e jt S ( g ; N ; Q − j ) e ( Q − j ) t (70a)= (cid:88) j S ( g ; N ; j ) e jt (cid:32)(cid:88) k S ( g ; N ; k ) e kt (cid:33) (70b)and by repeated application of the convolution formula (cid:88) Q S ( g ; N ; Q ) e Qt = (cid:88) j S ( g ; 1; j ) e jt N . (71)The sum raised to the N -th power is evaluated straightforwardly. Using the N = 1 value S ( g ; 1; Q ) = θ ( g − Q ) (72)which comes directly from the definition of S , one gets (cid:88) j S ( g ; 1; j ) e jt = 1 + e t + · · · + e gt = e ht − e t − (cid:18) gt (cid:19) sinh( ht/ t/
2) (73)where h = g + 1. Using the normalization relation (46), one obtains the average value for the case with N distinct subshells with the same degeneracy (cid:10) e Qt (cid:11) = (cid:88) Q S ( g ; N ; Q ) e Qt (cid:44)(cid:88) Q S ( g ; N ; Q ) = e Ngt/ (cid:18) sinh( ht/ h sinh( t/ (cid:19) N (74)and considering the centred variable Q − (cid:104) Q (cid:105) one has, since (cid:104) Q (cid:105) = gN/ (cid:68) e ( Q −(cid:104) Q (cid:105) ) t (cid:69) = (cid:18) sinh( ht/ h sinh( t/ (cid:19) N . (75)Let us note that the above relations are formally equivalent to the ones providing the partition functionof a quantum magnetic momentum interacting with a magnetic field in the theory of paramagnetism. Inorder to get the cumulants one must according to the definition (68b), compute the k -th derivative of thegenerating function K ( t ) = N log (cid:18) sinh( ht/ h sinh( t/ (cid:19) . (76)These derivatives may be obtained by various methods. Let us now consider the Taylor series K ( t ) = N ∞ (cid:88) k =1 B k k h k − k )! t k (77a)= N ( G ( ht ) − G ( t )) (77b)13here B j are the Bernoulli numbers and where G ( X ) = ∞ (cid:88) k =1 B k k X k (2 k )! . (77c)One gets, using a well known property of the Bernoulli numbers, G ( X ) = (cid:90) X duu ∞ (cid:88) k =1 B k u k (2 k )! = (cid:90) X duu (cid:18)
12 coth (cid:16) u (cid:17) − u (cid:19) = log (cid:18) X sinh (cid:18) X (cid:19)(cid:19) . (77d)Inserting formula (77d) in the above expression (77b) for the cumulant generating function, and comparingthe analytical expressions (76,77b), one readily obtains K ( t ) = K ( t ) . (78)From the expansion (68a) one obtains directly the even-order cumulants κ k = N B k k (cid:0) ( g + 1) k − (cid:1) (79)in the case of a unique g value. Because of the definition (68b), when subshells of various g are involved,the average value (cid:10) e Qt (cid:11) is simply the product of the average on each subshell, the global K ( t ) is the sumof the individual generating functions, and the 2 k -th derivative provides the cumulant κ k = B k k (cid:88) j N j (cid:0) ( g j + 1) k − (cid:1) (80)for the most general supershell.Assuming µ j are centred moments, then κ cancels, and the general relation giving moments asfunction of cumulants is [27] µ n = (cid:88) a ··· ,a n a ··· + na n = n P ( n ; a · · · , a n ) κ a · · · κ a n n (81)where the coefficient P is defined in Appendix A. Since in the present case, all odd-order moments (orcumulants) cancel, one may limit the index sets to even-order sets a , a · · · a k with n = 2 k . As anexample, defining C k = t (cid:88) j =1 ( h kj − N j with h j = g j + 1 , (82)one gets new expressions for the first centred moments µ = C
12 (83a) µ = C − C
120 (83b) µ = 5 C − C C
96 + C
252 (83c) µ = 35 C − C C
576 + C C
108 + 7 C − C
240 (83d) µ = 35 C − C C C C
288 + 7 C C − C C − C C
144 + C
132 (83e) µ = 385 C − C C C C C C − C C − C C C C C − C C C C − C g j are present. According to statistical treaties, any distribution such as (29) may be approximated by a Gram-Charlierexpansion, which is defined as (see Sec. 6.17 in Ref. [27]) F GC ( Q ) = G (2 π ) / σ exp (cid:20) − ( Q − (cid:104) Q (cid:105) ) σ (cid:21) (cid:88) k ≥ c k He k (cid:18) Q − (cid:104) Q (cid:105) σ (cid:19) (84)where the He n is the Chebyshev-Hermite polynomial [27] He k ( X ) = k ! (cid:98) k/ (cid:99) (cid:88) m =0 ( − m X k − m m m !( k − m )! (85)and (cid:98) x (cid:99) is the integer part of x . The Gram-Charlier coefficients c k are related to the centred moments µ k through the relation c k = (cid:98) k/ (cid:99) (cid:88) j =0 ( − j µ k − j /σ k − j j j !( k − j )! (86)and from this definition the coefficients c and c cancel. For a symmetric distribution as the oneconsidered here, all the odd-order terms c k cancel too. In the present case, the coefficient G in Eq.(84)is given by the normalization condition G = (cid:90) ∞−∞ dQ F GC ( Q ) = (cid:89) j N j g j (cid:88) Q =0 S ( g j ; N j ; Q ) = (cid:89) j ( g j + 1) N j , (87)the average value is (cid:104) Q (cid:105) = (cid:80) j ( g j N j ) / σ = (cid:80) j g j ( g j + 2) N j . As shown by Eq.(108a) of Appendix B, one may also express the Gram-Charlier coefficients as a function of the cumulants. We first consider here the case where only one degeneracy g is present. In Eq. (84), one chooses (cid:104) Q (cid:105) = gN/ σ given by (66). Using the general relation between c k coefficients and cumulants (108a)and the cumulant value (79) one gets c = − h + 120( h − N (88a) c = h − h − N (88b) c = − ( h + 1) (cid:2) h + 1) − h − N (cid:3) h − N (88c) c = 12( h − − h − h − N h − N (88d)where we have again introduced h = g + 1. It is remarkable that c k coefficients with k as high as 10 keepa quite tractable formulation. These formulas allow us to build a fast analytical approximation for S ,either as a normal distribution, or as a Gram-Charlier series.15sing the above relations (84,88) we have compared the exact distribution S ( g ; N ; Q ) with Gram-Charlier expansions for several ( g, N ) pairs on the whole Q = 0 − g.N range of populations. Examplesare given in Figs. 1 and 2 for g = 2 and g = 10 respectively. In each figure, cases N = 2 ,
5, and 10 havebeen studied. One observes that even the normal distribution, i.e., formula (84) with all c k canceled,provides a reasonable approximation of the S ( g ; N ; Q ) value. Looking in more detail, in the wings of thedistribution, the inclusion of at least the 2nd-order correction c He ( X ) in the Gram-Charlier expansionsignificantly improves the quality of the approximation. As mentioned above, the evaluation of suchcorrection using the expression (88a) is straightforward. One may notice a visible, though moderate,discrepancy in the case N = 2, whatever the g value. This may be easily understood by computingdirectly the S ( g ; N = 2; Q ) value. Using the recursion relations (12) and the initial value (72) one maycheck that S ( g ; N ; Q ) expressed versus Q are piecewise polynomials of degree N −
1, with a uniquedefinition on intervals of length g . Namely, one obtains S ( g ; 2; Q ) = g + 1 − | Q − g | (89) S ( g ; 3; Q ) = ( Q + 1)( Q + 2) if 0 ≤ Q ≤ g ( g + 1)( g + 2) − ( Q − g )( Q − g ) if g ≤ Q ≤ g ( Q − g − Q − g −
2) if 2 g ≤ Q ≤ g . (90)Obviously, it quite difficult to approximate the triangle-shaped function (89) with a normal distribution.The approximations at the various orders Gram-Charlier of S (2; 2; Q ) are given in table 2. It turns outthat the maximum discrepancy is about 10 %. For Q = 0, the discrepancy decreases with the expansionorder, while for Q = 1 , N values, as seen in the above mentioned examples, simple piecewisepolynomial expressions are available. Q Exact Order 1 Order 2 Order 3 Order 4 Order 5 Order 6 Order 7 Order 80 1 0.694 0.824 0.852 0.855 0.904 0.991 1.070 1.1141 2 2.137 2.200 2.277 2.264 2.155 2.002 1.861 1.7692 3 3.109 2.818 2.660 2.679 2.818 3.003 3.174 3.294Table 2: Number of configurations as a function of the population Q for N = 2 subshells of degeneracy g = 2: exact values and Gram-Charlier approximations. Order one is the normal distribution, order 2includes the kurtosis contribution, etc.As seen in figure 2 dealing with a greater g value, while the Gram-Charlier expansion at 2nd order (withthe excess kurtosis accounted for) is quite acceptable in most of the Q = 0 to gN range, discrepanciesare clearly visible for Q (cid:46) ( gN ) / , Q (cid:38) gN − ( gN ) / . For such population values, the number ofconfigurations S is usually orders of magnitude below its peak value ( g + 1) N / (2 πσ ) / , however onemay be interested in approximations uniformly valid whatever Q . In this case it appears that the inclusionof more terms in the Gram-Charlier expansion improves its accuracy in the wings. Though this behavioris clear on subfigure 2(c), we did not try to get a quantitative estimate of the Gram-Charlier order whichprovides a uniform approximation for the S ( g = 10; N = 10; Q ) values.16 Total population N u m be r o f c on f i gu r a t i on s ExactGC order 1GC order 2GC order 3GC order 4GC order 5 g = 2 N = 2 (a) N = 2 Total population N u m be r o f c on f i gu r a t i on s ExactGC order 1GC order 2GC order 3GC order 4GC order 5 g = 2 N = 5 (b) N = 5 Total population N u m be r o f c on f i gu r a t i on s ExactGC order 1GC order 2GC order 3GC order 4GC order 5 g = 2 N = 10 (c) N = 10 Figure 1: Comparison of the exact population distribution in N subshells of degeneracy g = 2 withGram-Charlier expansions at various orders. The Gram-Charlier expansion is plotted as a continuousfunction of the total population Q . In this figure, “order p ” means that moments up to k = 2 p have beenincluded in the expansion. 17 Total population N u m be r o f c on f i gu r a t i on s ExactGC order 1GC order 2GC order 3GC order 4GC order 5 g = 10 N = 2 (a) N = 2 Total population N u m be r o f c on f i gu r a t i on s ExactGC order 1GC order 2GC order 3GC order 4GC order 5 g = 10 N = 5 (b) N = 5 Total population N u m be r o f c on f i gu r a t i on s ExactGC order 1GC order 2GC order 3GC order 4GC order 5 g = 10 N = 10 (c) N = 10 Figure 2: Comparison of the exact population distribution in N subshells of degeneracy g = 10 withGram-Charlier expansions at various orders. 18 .2 Multiple-degeneracy case Using the general expression (108a) of the Gram-Charlier coefficients, and the cumulant value (80), oneeasily gets the first terms of the expansion c = − C C (91a) c = C C (91b) c = 7 C − C C (91c) c = 12 C − C C C (91d)which generalize the Eqs. (88) in the multi-degeneracy case. Such a procedure has been used first toanalyze the population distribution in the case t = 2 , g = 2 , N = 2 , g = 6 , N = 2, labeled s[2]p[2] forshort. The Gram-Charlier analysis is presented in figure 3(a). We note that, even though the numberof subshells is small (4), the Gram-Charlier expansion with the first correction c (orange curve andtriangles) provides a fair approximation of the exact number. Moreover the Gram-Charlier formula, ofstatistical nature, would perform even better for more complex configurations with a greater number ofsubshells.As a second example the Gram-Charlier approximation for the more complex supershell s[3]p[2]d[1](for instance 1s2s2p3s3p3d) is analyzed on figure 3(b). One checks that Gram-Charlier at second order( k = 4) is in fair agreement with the exact data. The 3rd order ( k = 6) improves again the agreement,with no significant gain at 4th order ( k = 8). The higher-order expansions k = 12 ,
16 bring an improvedagreement with the exact value, especially for the smallest and largest Q values.As a rule one may check that the accuracy of the Gram-Charlier expansion globally increases withthe order, though some oscillations are noticed. As an example, in figure 4 we have plotted the differencebetween the Gram-Charlier approximation (84) truncated at various orders and the exact number ofconfigurations. In this particular case, a good compromise between the quality of the expansion andthe computational cost is reached for k = 10, i.e., with five terms in the sum. As shown below, a morecomplete numerical analysis involving higher orders demonstrates that the Gram-Charlier series is indeeddivergent. It has been mentioned that some distributions get a better representation in terms of Edgeworth seriesrather than of Gram-Charlier series [7]. Another interest of the Edgeworth expansion is that it is directlyexpressed in terms of cumulants rather than of centred moments. The Edgeworth series is an expansionversus powers of the standard deviation σ , defined as E ( Q ) = G exp( − x / √ πσ ∞ (cid:88) s =1 σ s (cid:88) { k m } He s +2 r ( x ) s (cid:89) m =1 k m ! (cid:18) S m +2 ( m + 2)! (cid:19) k m (92a)with S n = κ n /σ n − , r = k + k + · · · k s (92b) x being the reduced variable x = ( Q − (cid:104) Q (cid:105) ) /σ (92c)and where the index { k m } refer to all s -uple indices verifying k + 2 k + · · · + sk s = s. (92d)19 Total population N u m be r o f c on f i gu r a t i on s ExactGC k = 2GC k = 4GC k = 6GC k = 8 g = 2, N = 2 g = 6, N = 2 (a) g = 2 , N = 2, g = 6 , N = 2 Total population N u m be r o f c on f i gu r a t i on s Exact k = 2 k = 4 k = 6 k = 8 k = 12 k = 16 g = 2, N = 3 g = 6, N = 2 g = 10, N = 1 (b) g = 2 , N = 3, g = 6 , N = 2, g = 10 , N = 1 Figure 3: Comparison of the exact population distribution with Gram-Charlier expansions at variousorders for the supershell s[2]p[2] (two subshells s and two subshells p, for instance 2s2p3s3p) (a), and forthe supershell s[3]p[2]d[1] (for instance 1s2s2p3s3p3d) (b).20
Total population -50050100 N u m be r o f c on f i gu r a t i on s : G r a m - C ha r li e r − e x a c t k = 2 k = 4 k = 6 k = 8 k = 10 k = 12 k = 14 k = 16 g = 2, N = 3 g = 6, N = 2 g = 10, N = 1 Figure 4: Difference between the number of configurations obtained with the Gram-Charlier expansiontruncated at various orders and the exact value for the supershell 1s2s2p3s3p3d. The k = 2 curve corre-sponds to the normal distribution, k = 4 is the Gram-Charlier series involving up to the He polynomialor second-order approximation, etc. Though exact values are only defined for integer populations Q , linesare drawn as a visual guide. 21s for Gram-Charlier expansion, this series involves only even s orders. The sum over s is replaced by afinite sum up to some s trunc , which is chosen as discussed below.In order to compare Edgeworth and Gram-Charlier expansions we have plotted in figure 5 the averagedeviation ∆ app ( s trunc ) = Q max (cid:88) Q =0 ( N app ( Q ; s trunc ) − N exact ( Q )) / ( Q max + 1) / (93)for the 1 s s p s p d supershell as a function of s trunc . In the above formula Q max is the maximumoccupation number of the supershell (cid:80) i g i N i , 28 in the present case, N app ( Q ; s trunc ) is the approximatenumber of configurations with occupation Q computed with Gram-Charlier (84) or Edgeworth (92a)truncated series. A truncation order s trunc = 2 corresponds to the normal distribution, the truncation s trunc = 4 corresponds to terms involving the Chebyshev-Hermite polynomial He ( X ), etc. On this graph,it appears that both expansions provide an acceptable representation of the number of configurations forthe low values of s trunc . Truncating the expansion at s trunc = 10, i.e., keeping four correction terms tothe normal distribution, provides the best approximation in case of Edgeworth series. In this case, therelative error (cid:104)(cid:80) Q max Q =0 ( N app ( Q ; s trunc ) /N exact ( Q ) − / ( Q max + 1) (cid:105) / (cid:39) . Q = 0 approximate value : N Edgeworth ( Q = 0; s trunc = 10) (cid:39) − .
519 while N exact ( Q = 0) = 1. However large values around Q max / N Edgeworth ( Q = 14; s trunc = 10) (cid:39) . , N exact ( Q = 14) = 1217. The general behavior is quitedifferent for s trunc above 10: while Gram-Charlier accuracy still improves with s trunc , the Edgeworth-expansion accuracy deteriorates rapidly. As seen on the graph, for very large values ( s trunc > et al. [11] who claim thatEdgeworth series strongly outperforms Gram-Charlier series. In our opinion this difference comes fromthe fact that we are dealing here with a discrete distribution, defined only for integer values, and thatthis distribution is not an analytical function of Q but a piecewise polynomial.
10 Conclusion
We found three explicit formulas for the number of atomic configurations. Although the best way to com-pute such a quantity remains probably the double recurrence on the numbers of electrons and orbitals, thenew expressions may be of interest in order to get new relations for the number of atomic configurations,using the numerous properties, identities and sum rules for binomial and multinomial coefficients. Usinga two-variable generating function, we have derived several recurrence relations, not published before upto our knowledge. Using the same generating function, the moments of the distribution have received ananalytical expression. It allowed us to provide explicit expressions for moments up to the twelfth, thoughhigher-order moments could be obtained too. The case of multiple value for the subshell degeneracyhas been addressed using the cumulant formalism. We have shown that the cumulants receive a verysimple expression whatever the order. This allowed us to obtain centred moments explicitly for k up to12. A Gram-Charlier analysis has shown that an expansion with two terms is in acceptable if not fairagreement with the exact number of configurations, though the series is not convergent. We have foundthat the Edgeworth expansion provides an equivalent accuracy if few terms are kept, though it divergesmuch more rapidly than the Gram-Charlier series. A Numbering the partitions defined by subset populations
The purpose of this appendix is to enumerate the partitions of n distinct objects knowing that there are n subsets of population 1, n subsets of population 2, . . . n k subsets of population k . In the main text22
10 20 30 40 50 60
Expansion order S t a nd a r d d e v i a ti on Gram-CharlierEdgeworth
Figure 5: Standard mean deviation (cid:104)(cid:80) Q max Q =0 ( N app ( Q ; s trunc ) − N exact ( Q )) / ( Q max + 1) (cid:105) / for the num-ber of configurations Q computed exactly or using expansions truncated at various orders s trunc . Con-figurations are generated from the 1 s s p s p d supershell and approximations are those obtained fromGram-Charlier and Edgeworth series. Only even s trunc values are plotted since odd-order terms in theexpansions vanish. 23ne has n = k though this constraint is not required for the present derivation. Conversely one musthave n = n + 2 n + · · · kn k . (94)The generation of these partitions may be done in k +1 steps. In the first step, one selects the n elementsin single-element subsets, the 2 n elements in twofold subsets, up to the kn k elements in the subsets ofpopulation n k . The number of possibilities at this step is p = (cid:18) nn (cid:19)(cid:18) n − n n (cid:19) . . . (cid:18) n − n · · · − ( k − n k − ( k − n k − (cid:19)(cid:18) n − n · · · − ( k − n k − kn k (cid:19) = n ! (cid:81) kj =1 ( jn j )! . (95)At the next k steps one must choose, for any j from 1 to k , how to partition jn j objects in n j subsets.This operation is performed by first selecting j objects among jn j , then j more objects among j ( n j − n j − n j !identical solutions, since the order of the subsets is not significant. Therefore the number of possibilitiesat step j is p j = 1 n j ! (cid:18) jn j j (cid:19)(cid:18) ( j − n j j (cid:19) · · · (cid:18) jj (cid:19)(cid:18) jj (cid:19) = 1 n j ! ( jn j )!( j !) n j . (96)Multiplying p given by Eq. (95) by the product of p j ’s provided by Eq. (96) one gets the desired numberof partitions P ( n ; n , n · · · , n k ) = n ! (cid:81) kj =1 n j !( j !) n j . (97) B Coefficients of the Gram-Charlier expansion as a function ofthe cumulants
The generating function of the cumulants is defined as K ( t ) = ∞ (cid:88) n =1 κ n t n n ! = log ( (cid:104) exp( tQ ) (cid:105) ) . (98)In the case of the Gram-Charlier expansion the integral (cid:104) exp( tQ ) (cid:105) is easily obtained as e K ( t ) = (cid:104) exp( tQ ) (cid:105) = (cid:90) dQ exp( tQ − Q / σ ) √ πσ (cid:34) (cid:88) n> c n He n ( Q/σ ) (cid:35) . (99)Using the Rodrigues formula for He n ( X ) and repeated integration by parts one easily gets (cid:90) + ∞−∞ dQ e tQ − X / σ He n ( Q/σ ) = ( σt ) n exp( σ t /
2) (100)from which one has the average over Gram-Charlier distribution (cid:104) exp( tQ ) (cid:105) = (cid:90) + ∞−∞ dQ e tQ − X / σ (cid:34) (cid:88) n> c n He n ( Q/σ ) (cid:35) = e σ t / (cid:34) (cid:88) n> ( σt ) n c n (cid:35) . (101)The exponential of the generating function of cumulants is, for any centred distribution (i.e., such as κ = 0), e K ( t ) = exp (cid:32) ∞ (cid:88) n =1 κ n t n n ! (cid:33) = e κ t / exp (cid:32) ∞ (cid:88) n =3 κ n t n n ! (cid:33) . (102)24dentifying this expression with the average (101), one writes1 + (cid:88) n ≥ ( σt ) n c n = exp (cid:32) ∞ (cid:88) n =3 κ n t n n ! (cid:33) = exp (cid:32) ∞ (cid:88) n =1 x n t n n ! (cid:33) = ∞ (cid:88) m =0 m ! (cid:32) ∞ (cid:88) n =1 x n t n n ! (cid:33) m (103a)where we have defined x = 0, x = 0, x n = κ n if n ≥ . (103b)The m th power in the sum (103a) may be computed with the identity (see section 24.1.2 in Ref. [1]) (cid:32) ∞ (cid:88) n =1 x n t n n ! (cid:33) m = m ! ∞ (cid:88) n = m t n n ! (cid:88) a ,a , ··· a n P ( n ; a , a · · · , a n ) x a x a · · · x a n n (104a)with the above definition (97) of the partition number P , and where integer indices a , a , · · · a n areconstrained by a + a + · · · + a n = m (104b) a + 2 a + · · · + na n = n. (104c)Identifying terms in t n in Eqs. (103a, 104a), one has σ n c n = 1 n ! (cid:88) m ≤ n (cid:88) a ,a , ··· a n P ( n ; a , a · · · , a n ) x a x a · · · x a n n (105)where the sum on a i follows the constraints (104). One will note that, since the a i are nonnegative, onehas m = a + a + · · · + a n ≤ a + 2 a + · · · + na n = n, (106)therefore in the multiple sum (105) one may ignore the sum over m , since this index is only intended tocollect terms in the sum. One has then σ n c n = 1 n ! (cid:88) a ,a , ··· a n P ( n ; a , a · · · , a n ) x a x a · · · x a n n (107)where only the second constraint (104c) has been kept. Accounting for x i definitions (103b), one notesthat only terms with a = 0 , a = 0 contribute and one gets the Gram-Charlier-series coefficient c n = 1 σ n n ! (cid:88) a , ··· a n a + ··· + na n = n P ( n ; 0 , , a · · · , a n ) κ a · · · κ a n n (108a)= (cid:88) a , ··· a n a + ··· + na n = n a ! (cid:16) κ σ (cid:17) a · · · a n ! (cid:16) κ n n ! σ n (cid:17) a n . (108b) References [1] M. Abramowitz and I.A. Stegun.
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