Publication: Calculation of coefficients of transformations between three-particle hyperspherical harmonics
Дата
2021
Авторы
Efros, V. D.
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© 2020 Elsevier B.V.New versions of the program to calculate the three-particle hyperspherical brackets 〈l1′l2′|l1l2〉KLφ are presented. Whereas the previous program, Efros (2020), computes the sets of brackets existing at given l1, l2, K, and L values, one of the present programs produces the set of all the brackets existing at L values in a range from Lmin up to Lmax and all K values of a given parity up to some Kmax. The other of the present programs provides the sets of all the brackets existing at given l1, l2, L values, and all K values of a given parity up to some Kmax. Use of such sets of brackets is more convenient. The present programs are considerably faster than the previous one, and they are also easier to follow. As the previous program, the present programs are easy to implement and are well applicable up to very high values of hypermomentum and orbital momenta. New version program Summary: Program Title: BRACHH CPC Library link to program files: http://dx.doi.org/10.17632/77kd74zy5k.2 Licensing provisions: GPLv3 Programming language: Fortran-90 Supplementary material: A file hhbrackets.pdf that describes the new version in more detail. Journal Reference of previous version: Comput. Phys. Comm. 255 C (2020) 107281 Does the new version supersede the previous version?: No. Nature of problem: Expansions of hyperspherical harmonics over harmonics similar in form but pertaining to different sets of Jacobi vectors are needed in the hyperspherical approach to solving three-body problems. A fast and convenient routine is to be created to construct the coefficients of such expansions, i.e., the brackets of the form 〈l1′l2′|l1l2〉KLφ. Solution method: A K→K+2 type recursion [2] is employed to compute the brackets. To start the recursion, one needs the brackets with K values that are the lowest possible ones at given l1 and l2. Such brackets are calculated with the help of the explicit expression [2] that includes only few summations. Reasons for new version: An opportunity to create versions of the program that are faster and provide outputs that are more convenient. Summary of revisions: The brackets 〈l1′l2′|l1l2〉KLφ are addressed. The notation here is as in Refs. [1, 2]. In the previous program, sets of brackets existing at given l1, l2, K, and L values are computed. In the present work, two versions of the program are given that provide different outputs and they are also faster. It is more convenient to use the present outputs. The present programs are also easier to follow. One of the new versions of the program produces the “table” of all the brackets existing at L values in a range from Lmin up to Lmax and all K values of a given parity up to some Kmax. The corresponding output array of the brackets is of the form Brac(M′,N′,int(K∕2),M,N,L). Here M and N parametrize the l1 and l2 orbital momenta as follows, M=(l1+l2−L−ϵ)∕2 and N=(l1−l2+L−ϵ)∕2, where ϵ equals zero or unity when, respectively, K−L is even or odd. The M′ and N′ variables parametrize the l1′ and l2′ orbital momenta in the same way. In difference to the orbital momenta variables, the M and N, or M′ and N′, variables densely fill intervals independent of each other, 0≤M,M′≤(K−L−ϵ)∕2, and 0≤N,N′≤L−ϵ. The size of the Brac array is normally moderate. The other new version of the program provides sets of all the brackets existing at given l1, l2, L values, and all K values of a given parity up to some Kmax. The corresponding output arrays of the brackets are of the form Brac(M′,N′,int(K∕2)). The present versions of the program are considerably faster than the previous one. The running times to produce all the brackets at a given L value and K values of a given parity up to Kmax are shorter e.g., hundreds of times at Kmax≃200 and several times at Kmax≃20. The decrease in running times is due both to the fact that now the output brackets are produced at each step of the K→K+2 recursion procedure and to that some common ingredients of the brackets are calculated in advance. The number of the existing brackets, e.g., at even K values with K≤200 and L=10 is about 3.4⋅107. The net time to compute these brackets, with the double precision being set, at use of a consumer notebook of 2009 is about one second. The structures of the present routines are the following. In the first mentioned version, the brackets are produced by the subroutine named ALLHHBRAC whose parameters are Kmax, Lmin, Lmax, cosφ, sinφ, and Brac. ALLHHBRAC calls for the subroutines that calculate in advance the coefficients of the K→K+2 recurrence formula and, up to simple factors, the brackets with K=l1+l2 that are required to start the recursion. The calculation in advance leads to the decrease of running times since the coefficients of the recurrence formula are independent of K, l1, and l2 values, and the initial brackets with K=l1+l2 depend on only three variables, l1′, l2′, and l1−l2, up to simple factors. The subroutine calculating the latter brackets calls for the function that computes 3j symbols. The subroutine ALLHHBRAC calls also for a small subroutine providing subsidiary quantities like ln(I!), ln((2I+1)!!), etc. which are used in various places. In the second mentioned version, the brackets are produced by the subroutine named HHBRAC. It calls for the routines similar to those that are called by ALLHHBRAC. The parameters of HHBRAC are Kmax, L, M, N, cosφ, sinφ, firstcall, and Brac. The M and N variables are equivalent to l1 and l2 and are defined above. The parameter firstcall is a logical variable aimed to suppress unnecessary calls, when HHBRAC is called repeatedly, of the above mentioned subroutine that produces the initial brackets with K=l1+l2. An example of use of this variable is given in the appended program TESTHHBRAC. (The unnecessary calls of two other above mentioned subroutines are then mostly removed as well. Times these subroutines run are relatively very small and this is unimportant.) It is seen that the present routines are very easy to implement. Tests of the new versions of the program have been performed. The testing programs are appended. Besides the tests, these programs provide the commented examples of use of the ALLHHBRAC and HHBRAC routines. The tests are similar to those performed in Ref. [1]. In one of the tests the quantity is computed that may be viewed as an estimate of the relative round-off error of calculated brackets. At Kmax=200 this error proves to be some units of 10−9 in case of the double precision calculation. At lower Kmax values the error is much smaller. A more detailed description of the programs and the calculations is given in the file hhbrackets.pdf appended to the programs. References [1] V.D. Efros, Comput. Phys. Comm. 255 C (2020) 107281. [2] Ja.A. Smorodinsky and V.D. Éfros, Yad. Fiz. 17, 210 (1973) [Sov. J. Nucl. Phys. 17, 107 (1973)].
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Efros, V. D. Calculation of coefficients of transformations between three-particle hyperspherical harmonics / Efros, V.D. // Computer Physics Communications. - 2021. - 261. - 10.1016/j.cpc.2020.107817
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https://www.doi.org/10.1016/j.cpc.2020.107817
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