A Program Library for Computing Pure Spin–Angular Coefficients for One- and Two-Particle Operators in Relativistic Atomic Theory
Abstract
:1. Introduction
2. Theory
2.1. Recoupling Coefficients and Second Quantization
2.2. Quasi-Spin Notation for Antisymmetric Subshell States
2.3. Quasi-Spin Formalism and Second Quantization
2.4. Spin-Angular Integrals
2.4.1. Matrix Elements for One-Particle Operator between Complex Configurations
2.4.2. Matrix Elements for Scalar Two-Particle Operator between Complex Configurations
- Recoupling matrix .
- Submatrix elements .
- Phase factor .
- .
2.4.3. The Method Implementation in Software Packages
- A general-purpose relativistic atomic structure program (GRASP) [22,35]. The current library (the library presented in this paper) is implemented in GRASP packages. The library version written in Fortran 77 programming language is installed in [35] and written in Fortran 95 and is installed in [22].
- The Racah program presents Maple procedures for the coupling of angular momenta [48,49]. The program is written in the Maple programming language. This implementation is not suitable for large-scale calculations. It serves for the manipulation of reduced matrix elements and some simple expressions of spin–angular integrations from this theory in both - and -couplings.
- The Hfs program presents Maple procedures as an environment for hyperfine structure parametrization [50]. The program is written in the Maple programming language. This implementation is realized in coupling and is not suitable for ab initio large-scale calculations.
2.4.4. The Spin–Angular Coefficients for Some Simple Cases and Average Energy of a Configuration
3. Structure of the Library
3.1. The METWO Routines Group
3.1.1. The Subroutine ONEPARTICLEJJ1
- NS is a number of peel subshells from the module m_C.
- KA is a rank k of the operator (see (21)).
- JJA and JJB are the numbers of configuration state functions for the matrix element to be evaluated.
- JA is the index in the array JLIST of the orbital on which the creation operator acts.
- JB is the index in the array JLIST of the orbital on which the annihilation operator acts.
- COEFF is the value of the spin–angular part of the matrix element.
3.1.2. The Subroutine ONEPARTICLEJJ2
- NS is a number of peel subshells from the module m_C.
- KA is a rank k of the operator (see (21)).
- JA is the index in the array JLIST of the orbital on which the creation operator acts.
- JB is the index in the array JLIST of the orbital on which the annihilation operator acts.
- COEFF is the value of the spin–angular part of the matrix element.
3.1.3. The Subroutines ONESCALAR1 and ONESCALAR2
3.1.4. The Subroutine EL1
- JJA and JJB are the numbers of configuration state functions for the matrix element to be evaluated.
- IIRE must be set equal to with respect to configuration state functions.
- ICOLBREI determines the calculation of spin–angular coefficients
3.1.5. The Subroutine EL2
- JJA and JJB are the numbers of configuration state functions for the matrix element to be evaluated.
- JA is the index in the array JLIST of the orbital on which the two creation operators act.
- JB is the index in the array JLIST of the orbital on which the two annihilation operators act.
- ICOLBREI determines the calculation of spin–angular coefficients
3.1.6. The Subroutine EL3
3.1.7. The Subroutine EL4
3.1.8. The Subroutine EL5
3.2. The REC Routines Group
3.2.1. The Subroutine RECOONESCALAR
- NS is the number of peel subshells from the module m_C for the diagonal case and NS = -1 for the off-diagonal matrix element.
- JA1 identifies the first subshell, on which the creation operator or annihilation tensor acts in the array JLIST.
- JA2 identifies the second subshell, on which the creation operator or annihilation tensor acts in the array JLIST. The subshells must be numbered so that the arguments JA1 and JA2 are in an increasing order.
- KA is the parameter which determines the number of subshells coupled by the interaction, taking the values
- The subroutine returns the value of IAT, which is
3.2.2. The Subroutine RECOP00
- NS is the number of peel subshells from the module m_C for the diagonal case and NS = -1 for the off-diagonal matrix element.
- JA1 identifies the first subshell, on which the creation operator or annihilation tensor acts in the array JLIST.
- JA2 identifies the second subshell, on which the creation operator or annihilation tensor acts in the array JLIST. The subshells must be numbered so that the arguments JA1 and JA2 are in an increasing order.
- KA is the rank k of the operator.
- The subroutine returns the value of IAT, which is
3.2.3. The Subroutine RECO
- KA = 0, in one subshell or in two subshells when the intermediate rank of the operator is equal to zero.
- KA = 1, in two subshells when the intermediate rank of the operator is nonzero.
- KA = 2, three subshells.
- KA = 3, four subshells.
- IAT = 0 if the recoupling coefficient is zero.
- IAT = 1 otherwise.
3.2.4. The Subroutine RECO2
- JA1 identifies the first subshell on which the operator acts in the array JLIST.
- JA2 identifies the second subshell on which the operator acts in the array JLIST.
- KA is the intermediate rank k.
- IRE takes the input value
- When IRE = 0, the subroutine returns the value of IAT, which is
- REC is the value of the recoupling coefficient computed when IRE = 1.
3.2.5. The Subroutine REC3
- JA1, JA2, and JA3, which point to the orbitals i, j, and m in the array JLIST.
- K1, K2, and KA, which are the ranks , , and k of the component tensor operators (86).
- IRE takes the input value
- When IRE = 0, the subroutine returns the value of IAT, which is
- REC is the value of the recoupling coefficient computed when IRE = 1.
3.2.6. The Subroutine RECO4
- JA1, JA2, JA3, and JA4, which point to the orbitals 1, 2, 3, and 4 in the array JLIST.
- K1, K2, K3, K4, and KA, which are the ranks , , , , and k of the component tensor operators (87).
- IRE takes the input value
- When IRE = 0, the subroutine returns the value of IAT, which is
- REC is the value of the recoupling coefficient computed when IRE = 1.
3.2.7. The Subroutine RECOP1
- NS is the number of peel subshells from the module m_C.
- JA1 identifies the subshell on which the operator acts in the array JLIST.
- KA is the intermediate rank k.
- IRE takes the input value
- When IRE = 0, the subroutine returns the value of IAT, which is
- RECC is the value of the recoupling coefficient computed when IRE = 1.
3.2.8. The Subroutine RECOP2
- NS is the number of peel subshells from the module m_C.
- JA1 and JA2, which point to the orbitals 1 and 2 in the array JLIST.
- K1, K2, and KA, which are the ranks , , and k of the component tensor operators (88).
- IRE takes the input value
- When IRE = 0, the subroutine returns the value of IAT, which is
- RECC is the value of the recoupling coefficient computed when IRE = 1.
3.3. The SQ Routines Group
- I(1), which is the state number of the subshell (see column in Table 4).
- I(2), which is the principal quantum number n.
- I(3), which is the quantum number j multiplied by two.
- I(4), which is the number of electrons in the subshell.
- I(5), which is the orbital quantum number l.
- I(6), which is the subshell total angular momentum J multiplied by two.
- I(7), which is the subshell total quasi-spin Q multiplied by two.
- B(1), which is the subshell quasi-spin Q.
- B(2), which is the subshell total angular momentum J.
- B(3), which is the subshell quasi-spin projection .
3.3.1. The Subroutine RMEAJJ
- LL is the quantum number j multiplied by two.
- IT is the state number of the bra function (see column in Table 4).
- LQ is the quasi-spin Q for the bra function multiplied by two.
- J is the total angular momentum J for the bra function multiplied by two.
- ITS is the state number of the ket function.
- LQS is the quasi-spin Q for the ket function multiplied by two.
- J1S is the total angular momentum J for the ket function multiplied by two.
- COEF is the value of the reduced matrix element (89) which is returned by the subroutine.
3.3.2. The Subroutine RWJJ
3.3.3. The Subroutine WJ1
- IK is the array I for the bra function.
- BK is the array B for the bra function.
- ID is the array I for the ket function.
- BD is the array B for the ket function.
- K2 is the rank k (see (91)).
- QM1, QM2 are the quasi-spin projections in (91).
- WJ is the value of the reduced matrix element (91) which is returned by the subroutine.
3.3.4. The Subroutine W1JJG
- K1 is the rank k (see (91)).
- QM1 and QM2 are the quasi-spin projections in (91).
- IK is the array I for the bra function.
- BK is the array B for the bra function.
- ID is the array I for the ket function.
- BD is the array B for the ket function.
- WW is the value of the reduced matrix element (91) which is returned by the subroutine.
3.3.5. The Subroutine AWP1
- IK is the array I for the bra function.
- BK is the array B for the bra function.
- ID is the array I for the ket function.
- BD is the array B for the ket function.
- K1 is the rank (see (92)).
- BK2 is the rank k (see (92)).
- QM1, QM2, and QM3 are the quasi-spin projections in (92).
- AW is the value of the reduced matrix element (92) which is returned by the subroutine.
3.3.6. The Subroutine AWP1JJG
- K1 is the rank (see (92)).
- BK2 is the rank k (see (92)).
- QM1, QM2, and QM3 are the quasi-spin projections in (92).
- IK is the array I for the bra function.
- BK is the array B for the bra function.
- ID is the array I for the ket function.
- BD is the array B for the ket function.
- AW is the value of the reduced matrix element (92) which is returned by the subroutine.
3.3.7. The Subroutine WAP1
- IK is the array I for the bra function.
- BK is the array B for the bra function.
- ID is the array I for the ket function.
- BD is the array B for the ket function.
- K1 is the rank (see (93)).
- BK2 is the rank k (see (93)).
- QM1, QM2, and QM3 are the quasi-spin projections in (93).
- WA is the value of the reduced matrix element (93) which is returned by the subroutine.
3.3.8. The Subroutine WAP1JJG
- K1 is the rank (see (93)).
- BK2 is the rank k (see (93)).
- QM1, QM2, and QM3 are the quasi-spin projections in (93).
- IK is the array I for the bra function.
- BK is the array B for the bra function.
- ID is the array I for the ket function.
- BD is the array B for the ket function.
- WA is the value of the reduced matrix element (93) which is returned by the subroutine.
3.3.9. The Subroutine WW1
- IK is the array I for the bra function.
- BK is the array B for the bra function.
- ID is the array I for the ket function.
- BD is the array B for the ket function.
- K2 is the rank k (see (94)).
- QM1, QM2, QM3, and QM4 are the quasi-spin projections in (94).
- WW is the value of the reduced matrix element (94) which is returned by the subroutine.
3.4. Description of the New Modules for Arrays Used in the Program Library
3.5. Interface between the Program Library and GRASP
3.6. Installation Features in the GRASP
3.7. A Speed-Up in Calculations with the Present Program Library Integrated in GRASP Package
- 3SD: Single and double excitations from to the active set containing 14 configuration state functions (CSF) for and 34 CSF (the maximum) for .
- 3SDT: Single, double, and triple excitations from to the active set . The maximum number of CSF is 145 for .
- 4SD: Single and double excitations from to the active set. The maximum number of CSF is 465 for .
- 4SDT: Single, double, and triple excitations from to the active set.
- 5SD; Single and double excitations from to the active set.
3.8. Limitations of the Program Library
4. Conclusions
- A number of theoretical methods known in atomic physics facilitate the treatment of spin–angular parts of matrix elements, among them are the theory of angular momentum, its graphical representation, the quasi-spin, and the second quantization in its coupled tensorial form. However, while treating the matrix elements of physical operators in general, including the off-diagonal ones, with respect to configurations, the above-mentioned methods are usually applied either only partly and/or inefficiently.An idea to combine all of these methods in order to optimize the treatment of the general form of matrix elements of physical operators in atomic spectroscopy is presented and carried out in this work. It allows us to investigate even the most complex cases of atoms and ions efficiently in relativistic approaches.
- The general tensorial expressions of one- and two-particle operators, presented in this work, allow one to exploit all the advantages of tensorial algebra. In particular, this is not only a reformulation of spin–angular calculations in terms of standard quantities but also the prior determination from symmetry properties in which matrix elements are equal to zero without performing further explicit calculations (the first and third grout of selection rules (see Table 1, Table 2 and Table 3)).
- The tensorial forms of a one- and two-particle operator (see (8) and (24)), allow one to obtain simple expressions for the recoupling matrices. Hence, the computer code based on this approach would immediately use the analytical formulas for the recoupling matrices . Among the rest, this feature saves computing time because (i) complex calculations lead finally to simple analytical expressions [11,12], and (ii) a number of momenta triads (triangular conditions (the second group of selection rules (see Table 1, Table 2 and Table 3)) can be checked before the explicit calculation of a recoupling matrix leading to a zero value. These triangular conditions may be determined not only for the terms of subshells that the operators of the second quantization act upon but also for the rest of the subshells and resulting terms.
- The tensorial form of any operator presented as products of tensors of the second quantizaqtion , , , , and allows one to exploit all the advantages of a new version of Racah algebra based on quasi-spin formalism. So, the application of the Wigner–Eckart theorem in quasi-spin space for the submatrix element of the operator of the second quantization or its combinations provides an opportunity to use the tables of reduced coefficients of fractional parentage and tables of the other standard quantities, which do not depend on the occupation number of a subshell of equivalent electrons. Thus, the volume of tables of standard quantities is reduced considerably in comparison with the analogous tables of submatrix elements of (for -coupling) [16,56] and the tables of coefficients of fractional parentage. These tables cover all the electronic configurations needed in practice. Therefore, the process of selecting the standard quantities from the tables becomes simpler. It also allows to determine the third group of selection rules (see Table 1, Table 2 and Table 3).
- In this approach, which is both diagonal and off-diagonal with respect to configurations, matrix elements are considered in a uniform way and are expressed in terms of the same quantities, namely, reduced coefficients of fractional parentage or reduced submatrix elements of standard tensors, which are independent of the number of electrons in a subshell. The difference is only in the values of the projections of the quasi-spin momenta of separate subshells. The complete numerical tables of these quantities allow practical studies of any atom or ion in the periodical table.
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
1 | The total angular momentum of the first subshell coupled with the total angular momentum of the second subshell produces the resulting angular momentum , meanwhile the intermediate angular momentum coupled with the third subshell gives the intermediate angular momentum , etc. So, the intermediate angular momentum is obtained from the intermediate angular momentum and the u subshell angular momentum . |
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The Matrix Element | |
---|---|
Diagonal | Off-Diagonal |
The first group of selection rules (coming from (8)) | |
The second group of selection rules (coming from) | |
The first part: | |
The second part: | |
(where a = min() and b = max()) | |
The third group of selection rules (coming from (5), (6) or (12)) | |
The fourth group of selection rules (coming from effective interaction strength (16)) | |
For example, the matrix element of the Dirac operator has (17): | |
The Matrix Element | |
---|---|
Diagonal | Off-Diagonal |
The first group of selection rules (coming from (21)) | |
The second group of selection rules (coming from) | |
The first part: | |
The second part: | |
(where a=min() and b=max(); | |
additional triangular delta from | |
-coefficients; it depends on the case [11] | |
The third group of selection rules (coming from the tensorial part of matrix element) | |
The Diagonal Matrix Element | |
operator acts on one subshell | operator acts on two subshells |
for the radial integral 1 ([5], (89) and (90)) | for the radial integral 1 ([5], (89) and (90)) |
The first group of selection rules (coming from (24)) | |
The second group of selection rules (coming from) | |
The first part: | |
) | |
The second part: | |
where a = min() and b = max() | |
additional triangular delta from -coefficients; it depends on the case [11] | |
The third group of selection rules (coming from) | |
The fourth group of selection rules (coming from) | |
For example, the matrix element of Coulomb operator has: | |
Matrix element | |
Diagonal | Off-diagonal |
Operator acts on two subshells | |
for the radial integral 1 ([5], (89) and (90)) | for the radial integral 1 ([5], (89) and (90)) |
The first group of selection rules (coming from (24)) | |
The second group of selection rules (coming from) | |
The first part: | |
The second part: | |
where a = min(), b = max() and A = min | |
additional triangular delta from -coefficients; it depends on the case [11] | |
The third group of selection rules (coming from) | |
The fourth group of selection rules (coming from) | |
For example, the matrix element of Coulomb operator has: | |
j | j | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
1 | 0 | 2 | 1 | 33 | 2 | 6 | 9 | ||||
2 | 1 | 0 | 0 | 1/2 | 34 | 0 | 10 | 9 | |||
3 | 1 | 2 | 3 | 35 | 2 | 6 | 11 | ||||
4 | 2 | 0 | 0 | 3/2 | 36 | 0 | 10 | 11 | |||
5 | 0 | 4 | 4 | 37 | 2 | 6 | 13 | ||||
6 | 2 | 2 | 5 | 38 | 0 | 10 | 13 | ||||
7 | 0 | 6 | 3 | 39 | 2 | 6 | 15 | ||||
8 | 0 | 6 | 9 | 40 | 0 | 10 | 15 | ||||
9 | 3 | 0 | 0 | 5/2 | 41 | 2 | 6 | 17 | |||
10 | 1 | 4 | 4 | 42 | 0 | 10 | 17 | ||||
11 | 1 | 4 | 8 | 43 | 0 | 10 | 19 | ||||
12 | 1 | 6 | 3 | 44 | 2 | 6 | 21 | ||||
13 | 1 | 6 | 5 | 45 | 0 | 10 | 25 | ||||
14 | 3 | 2 | 7 | 46 | 5 | 0 | 0 | ||||
15 | 1 | 6 | 9 | 47 | 1 | 8 | 0 | ||||
16 | 1 | 6 | 11 | 48 | 3 | 4 | 4 | 9/2 | |||
17 | 1 | 6 | 15 | 49 | 1 | 8 | 4 | ||||
18 | 4 | 0 | 0 | 50 | 1 | 8 | 6 | ||||
19 | 2 | 4 | 4 | 7/2 | 51 | 3 | 4 | 8 | |||
20 | 0 | 8 | 4 | 52 | 1 | 1 | 8 | 8 | |||
21 | 2 | 4 | 8 | 53 | 2 | 1 | 8 | 8 | |||
22 | 0 | 8 | 8 | 54 | 1 | 8 | 10 | ||||
23 | 0 | 8 | 10 | 55 | 3 | 4 | 12 | ||||
24 | 2 | 4 | 12 | 56 | 1 | 1 | 8 | 12 | |||
25 | 0 | 8 | 16 | 57 | 2 | 1 | 8 | 12 | |||
26 | 0 | 10 | 1 | 58 | 1 | 8 | 14 | ||||
27 | 2 | 6 | 3 | 59 | 3 | 4 | 16 | ||||
28 | 2 | 6 | 5 | 60 | 1 | 8 | 16 | ||||
29 | 0 | 10 | 5 | 9/2 | 61 | 1 | 8 | 18 | |||
30 | 2 | 6 | 7 | 62 | 1 | 8 | 20 | ||||
31 | 0 | 10 | 7 | 63 | 1 | 8 | 24 | ||||
32 | 4 | 2 | 9 |
Name | Dimension | Function |
---|---|---|
mtjj_C | The arrays for the atomic states of with any occupation of subshells | |
MT | 63 | the arrays for subshells (see Table 4) |
mtjj2_C | The arrays for the atomic states of in case | |
MT9 | 6 | The arrays for subshell |
MT11 | 189 | The arrays for subshells |
trk_C | The data of the orbitals for interacting subshells | |
BD1 | 3 | The array B (see Section 3.3) for the first subshell of the ket function |
BD2 | 3 | The array B for the second subshell of the ket function |
BK1 | 3 | The array B for the first subshell of the bra function |
BK2 | 3 | The array B for the second subshell of the bra function |
ID1 | 7 | The array I (see Section 3.3) for the first subshell of the ket function |
ID2 | 7 | The array I for the second subshell of the ket function |
IK1 | 7 | The array I for the first subshell of the bra function |
IK2 | 7 | The array I for the second subshell of the bra function |
BD3 | 3 | The array B (see Section 3.3) for the third subshell of the ket function |
BD4 | 3 | The array B for the fourth subshell of the ket function |
BK3 | 3 | The array B for the third subshell of the bra function |
BK4 | 3 | The array B for the fourth subshell of the bra function |
ID3 | 7 | The array I (see Section 3.3) for the third subshell of the ket function |
ID4 | 7 | The array I for the fourth subshell of the ket function |
IK3 | 7 | The array I for the third subshell of the bra function |
IK4 | 7 | The array I for the fourth subshell of the bra function |
ribojj_C | 63 | The arrays for performing summation in two spaces |
(quasi-spin and angular momentum j) for subshells | ||
(for example, see expression (13)) | ||
ribojj9_C | 6 | The arrays for performing summation in two spaces |
(quasi-spin and angular momentum j) for subshells | ||
ribojj11_C | 189 | The arrays for performing summation in two spaces |
(quasi-spin and angular momentum j) for subshells |
ASF Expansions | Number of | Speed-Up | ||||
---|---|---|---|---|---|---|
CSF | for | for | for | With Different | Actual | |
GRASP92 [54] and GRASP(New) | GRASP92 | GRASP(New) | Number of | |||
4SDT (J = 0) | 2149 | 3606 | 756,023 | 1,530,086 | 2.7 | 5.4 |
4SDT (J = 1) | 5786 | 14,017 | 4,070,156 | 8,188,130 | 3.9 | 7.8 |
4SDT (J = 2) | 8016 | 21,356 | 7,018,885 | 14,077,044 | 3.9 | 7.8 |
4SDT (J = 3) | 8378 | 21,342 | 7,634,136 | 15,290,955 | 3.7 | 7.4 |
4SDT (J = 4) | 7284 | 15,971 | 6,111,074 | 12,260,139 | 4.1 | 8.2 |
4SDT (J = 5) | 5349 | 9435 | 3,810,165 | 7,656,054 | 3.6 | 7.2 |
4SDT (J = 6) | 3370 | 4556 | 1,836,602 | 3,706,544 | 3.3 | 6.6 |
4SDT (J = 7) | 1788 | 1789 | 693,761 | 1,412,443 | 3.0 | 6.0 |
5SD (J = 0) | 468 | 621 | 75,192 | 150,455 | 1.9 | 3.8 |
5SD (J = 1) | 1134 | 2324 | 395,450 | 792,560 | 2.1 | 4.2 |
5SD (J = 2) | 1609 | 3704 | 697,651 | 1,395,839 | 2.4 | 4.8 |
5SD (J = 3) | 1584 | 3441 | 721,907 | 1,444,095 | 2.3 | 4.6 |
5SD (J = 4) | 1361 | 2500 | 558,223 | 1,117,681 | 2.3 | 4.6 |
5SD (J = 5) | 920 | 1361 | 314,909 | 632,306 | 1.8 | 3.6 |
5SD (J = 6) | 559 | 644 | 141,328 | 284,102 | 1.7 | 3.4 |
5SD (J = 7) | 259 | 226 | 44,137 | 89,398 | 1.3 | 2.6 |
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Gaigalas, G. A Program Library for Computing Pure Spin–Angular Coefficients for One- and Two-Particle Operators in Relativistic Atomic Theory. Atoms 2022, 10, 129. https://doi.org/10.3390/atoms10040129
Gaigalas G. A Program Library for Computing Pure Spin–Angular Coefficients for One- and Two-Particle Operators in Relativistic Atomic Theory. Atoms. 2022; 10(4):129. https://doi.org/10.3390/atoms10040129
Chicago/Turabian StyleGaigalas, Gediminas. 2022. "A Program Library for Computing Pure Spin–Angular Coefficients for One- and Two-Particle Operators in Relativistic Atomic Theory" Atoms 10, no. 4: 129. https://doi.org/10.3390/atoms10040129