Abstract

For an arbitrary electron configuration l1n1l2n2l3n3, analytical expressions are derived for the energy-matrix coefficients of the electrostatic-interaction parameters Fk(li,lj) and Gk(li,lj) and of the spin–orbit parameters ζ(li). A computer program is described which calculates decimal values of these coefficients for any configuration with less than five open subshells, starting from only a table of the terms, parents, and coefficients of fractional parentage for each open subshell lini. Given also a set of values of the parameters, the program evaluates and diagonalizes the energy matrices to obtain the eigenvalues and eigenvectors of the states belonging to this configuration. If so specified, the program does the above for two configurations of opposite parity, differences eigenvalues to obtain wavelengths for dipole transitions, and from the eigenvectors computes line strengths and (given an absolute value for the reduced dipole matrix element) transition probabilities.

© 1968 Optical Society of America

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References

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  1. J. C. Slater, Phys. Rev. 34, 1293 (1929).
    [Crossref]
  2. E. U. Condon and G. H. Shortley, The Theory of Atomic Spectra (Cambridge Univ.-Press, 1935).
  3. J. C. Slater, Quantum Theory of Atomic Structure (McGraw-Hill Book Co., New York, 1960), 2 Vols.
  4. Except where indicated to the contrary, in this paper lengths are measured in Bohr radii and energies in Rydbergs.
  5. (II)G. Racah, Phys. Rev. 62, 438 (1942); (III)Phys. Rev. 63, 367 (1943); (IV)Phys. Rev. 76, 1352 (1949).
    [Crossref]
  6. For a bibliography up to 1960 see Ref. 3, Ch. 24 and Appendix 21.
  7. F. R. Innes and C. W. Ufford, Phys. Rev. 111, 194 (1958); A. P. Yutsis, Ya. A. Vizbaraite, R. I. Karaziya, A. Yu. Savukinas, and A. A. Bandzaitis, Lietuvos Fisikos Rinkinys IV, 198 (1964); B. W. Shore, Phys. Rev. 139, A1042 (1965); U. Fano, Phys. Rev. 140, A67 (1965).
    [Crossref]
  8. C. W. Nielson and G. F. Koster, Spectroscopic Coefficients for the pn, dn, and fn Configurations (MIT Press, Cambridge, Mass., 1963).
  9. R. D. Cowan and K. L. Andrew, J. Opt. Soc. Am. 55, 502 (1965).
    [Crossref]
  10. M. Rotenberg, R. Bivins, N. Metropolis, and J. Wooten, The 3−j and 6−j Symbols (Technology Press. MIT, Cambridge, Mass., 1959).
  11. G. Racah, J. Opt. Soc. Am. 50, 408 (1960). The abbreviations LSLK, etc. used in the present paper are not unambiguous; they are intended only as suggestive abbreviated lists of the various intermediate quantum numbers, for reference purposes within this paper.
  12. A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton Univ. Press, Princeton, New Jersey, 1957).
  13. F. R. Innes, Phys. Rev. 91, 31 (1953), Eq. (13).
    [Crossref]
  14. Reference 3, Sec. 14–2.
  15. F. Rohrlich, Astrophys. J. 129, 441, 449 (1959).
    [Crossref]
  16. B. G. Wyboume, Spectroscopic Properties of the Rare Earths (John Wiley & Sons, Inc., New York, 1965).
  17. Reference 10, Eqs. (3.1) and (2.7).
  18. The author is indebted to J. D. Louck for pointing out this relation to him.
  19. Reference 2, Eqs. 54(8) and 94(14), and Ref. 9, Eqs. (23)–(24).
  20. F. Rohrlich, Phys. Rev. 74, 1372 (1948).
    [Crossref]
  21. J. Sugar (private communication).
  22. R. D. Cowan and N. J. Peacock, Astrophys. J. 142, 390 (1965); Astrophys. J. 143, 283 (1966). N. J. Peacock, R. D. Cowan, and G. A. Sawyer, in Proc. of the Seventh International Conference on Phenomena in Ionized Gases (Beograd, 1966), Vol. II, p. 599; B. C. Fawcett, N. J. Peacock, and R. D. Cowan, Proc. Phys. Soc. (London) 1B, 295 (1968).
    [Crossref]
  23. R. D. Cowan, Astrophys. J. 147, 377 (1967).
    [Crossref]
  24. C. J. Humphreys, E. Paul, R. D. Cowan, and K. L. Andrew, J. Opt. Soc. Am. 57, 855 (1967).
    [Crossref]
  25. L. Å. Svensson and J. O. Ekberg, Arkiv Fysik 37, 65 (1968).

1968 (1)

L. Å. Svensson and J. O. Ekberg, Arkiv Fysik 37, 65 (1968).

1967 (2)

1965 (2)

R. D. Cowan and K. L. Andrew, J. Opt. Soc. Am. 55, 502 (1965).
[Crossref]

R. D. Cowan and N. J. Peacock, Astrophys. J. 142, 390 (1965); Astrophys. J. 143, 283 (1966). N. J. Peacock, R. D. Cowan, and G. A. Sawyer, in Proc. of the Seventh International Conference on Phenomena in Ionized Gases (Beograd, 1966), Vol. II, p. 599; B. C. Fawcett, N. J. Peacock, and R. D. Cowan, Proc. Phys. Soc. (London) 1B, 295 (1968).
[Crossref]

1960 (1)

G. Racah, J. Opt. Soc. Am. 50, 408 (1960). The abbreviations LSLK, etc. used in the present paper are not unambiguous; they are intended only as suggestive abbreviated lists of the various intermediate quantum numbers, for reference purposes within this paper.

1959 (1)

F. Rohrlich, Astrophys. J. 129, 441, 449 (1959).
[Crossref]

1958 (1)

F. R. Innes and C. W. Ufford, Phys. Rev. 111, 194 (1958); A. P. Yutsis, Ya. A. Vizbaraite, R. I. Karaziya, A. Yu. Savukinas, and A. A. Bandzaitis, Lietuvos Fisikos Rinkinys IV, 198 (1964); B. W. Shore, Phys. Rev. 139, A1042 (1965); U. Fano, Phys. Rev. 140, A67 (1965).
[Crossref]

1953 (1)

F. R. Innes, Phys. Rev. 91, 31 (1953), Eq. (13).
[Crossref]

1948 (1)

F. Rohrlich, Phys. Rev. 74, 1372 (1948).
[Crossref]

1942 (1)

(II)G. Racah, Phys. Rev. 62, 438 (1942); (III)Phys. Rev. 63, 367 (1943); (IV)Phys. Rev. 76, 1352 (1949).
[Crossref]

1929 (1)

J. C. Slater, Phys. Rev. 34, 1293 (1929).
[Crossref]

Andrew, K. L.

Bivins, R.

M. Rotenberg, R. Bivins, N. Metropolis, and J. Wooten, The 3−j and 6−j Symbols (Technology Press. MIT, Cambridge, Mass., 1959).

Condon, E. U.

E. U. Condon and G. H. Shortley, The Theory of Atomic Spectra (Cambridge Univ.-Press, 1935).

Cowan, R. D.

C. J. Humphreys, E. Paul, R. D. Cowan, and K. L. Andrew, J. Opt. Soc. Am. 57, 855 (1967).
[Crossref]

R. D. Cowan, Astrophys. J. 147, 377 (1967).
[Crossref]

R. D. Cowan and N. J. Peacock, Astrophys. J. 142, 390 (1965); Astrophys. J. 143, 283 (1966). N. J. Peacock, R. D. Cowan, and G. A. Sawyer, in Proc. of the Seventh International Conference on Phenomena in Ionized Gases (Beograd, 1966), Vol. II, p. 599; B. C. Fawcett, N. J. Peacock, and R. D. Cowan, Proc. Phys. Soc. (London) 1B, 295 (1968).
[Crossref]

R. D. Cowan and K. L. Andrew, J. Opt. Soc. Am. 55, 502 (1965).
[Crossref]

Edmonds, A. R.

A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton Univ. Press, Princeton, New Jersey, 1957).

Ekberg, J. O.

L. Å. Svensson and J. O. Ekberg, Arkiv Fysik 37, 65 (1968).

Humphreys, C. J.

Innes, F. R.

F. R. Innes and C. W. Ufford, Phys. Rev. 111, 194 (1958); A. P. Yutsis, Ya. A. Vizbaraite, R. I. Karaziya, A. Yu. Savukinas, and A. A. Bandzaitis, Lietuvos Fisikos Rinkinys IV, 198 (1964); B. W. Shore, Phys. Rev. 139, A1042 (1965); U. Fano, Phys. Rev. 140, A67 (1965).
[Crossref]

F. R. Innes, Phys. Rev. 91, 31 (1953), Eq. (13).
[Crossref]

Koster, G. F.

C. W. Nielson and G. F. Koster, Spectroscopic Coefficients for the pn, dn, and fn Configurations (MIT Press, Cambridge, Mass., 1963).

Metropolis, N.

M. Rotenberg, R. Bivins, N. Metropolis, and J. Wooten, The 3−j and 6−j Symbols (Technology Press. MIT, Cambridge, Mass., 1959).

Nielson, C. W.

C. W. Nielson and G. F. Koster, Spectroscopic Coefficients for the pn, dn, and fn Configurations (MIT Press, Cambridge, Mass., 1963).

Paul, E.

Peacock, N. J.

R. D. Cowan and N. J. Peacock, Astrophys. J. 142, 390 (1965); Astrophys. J. 143, 283 (1966). N. J. Peacock, R. D. Cowan, and G. A. Sawyer, in Proc. of the Seventh International Conference on Phenomena in Ionized Gases (Beograd, 1966), Vol. II, p. 599; B. C. Fawcett, N. J. Peacock, and R. D. Cowan, Proc. Phys. Soc. (London) 1B, 295 (1968).
[Crossref]

Racah, G.

G. Racah, J. Opt. Soc. Am. 50, 408 (1960). The abbreviations LSLK, etc. used in the present paper are not unambiguous; they are intended only as suggestive abbreviated lists of the various intermediate quantum numbers, for reference purposes within this paper.

(II)G. Racah, Phys. Rev. 62, 438 (1942); (III)Phys. Rev. 63, 367 (1943); (IV)Phys. Rev. 76, 1352 (1949).
[Crossref]

Rohrlich, F.

F. Rohrlich, Astrophys. J. 129, 441, 449 (1959).
[Crossref]

F. Rohrlich, Phys. Rev. 74, 1372 (1948).
[Crossref]

Rotenberg, M.

M. Rotenberg, R. Bivins, N. Metropolis, and J. Wooten, The 3−j and 6−j Symbols (Technology Press. MIT, Cambridge, Mass., 1959).

Shortley, G. H.

E. U. Condon and G. H. Shortley, The Theory of Atomic Spectra (Cambridge Univ.-Press, 1935).

Slater, J. C.

J. C. Slater, Phys. Rev. 34, 1293 (1929).
[Crossref]

J. C. Slater, Quantum Theory of Atomic Structure (McGraw-Hill Book Co., New York, 1960), 2 Vols.

Sugar, J.

J. Sugar (private communication).

Svensson, L. Å.

L. Å. Svensson and J. O. Ekberg, Arkiv Fysik 37, 65 (1968).

Ufford, C. W.

F. R. Innes and C. W. Ufford, Phys. Rev. 111, 194 (1958); A. P. Yutsis, Ya. A. Vizbaraite, R. I. Karaziya, A. Yu. Savukinas, and A. A. Bandzaitis, Lietuvos Fisikos Rinkinys IV, 198 (1964); B. W. Shore, Phys. Rev. 139, A1042 (1965); U. Fano, Phys. Rev. 140, A67 (1965).
[Crossref]

Wooten, J.

M. Rotenberg, R. Bivins, N. Metropolis, and J. Wooten, The 3−j and 6−j Symbols (Technology Press. MIT, Cambridge, Mass., 1959).

Wyboume, B. G.

B. G. Wyboume, Spectroscopic Properties of the Rare Earths (John Wiley & Sons, Inc., New York, 1965).

Arkiv Fysik (1)

L. Å. Svensson and J. O. Ekberg, Arkiv Fysik 37, 65 (1968).

Astrophys. J. (3)

R. D. Cowan and N. J. Peacock, Astrophys. J. 142, 390 (1965); Astrophys. J. 143, 283 (1966). N. J. Peacock, R. D. Cowan, and G. A. Sawyer, in Proc. of the Seventh International Conference on Phenomena in Ionized Gases (Beograd, 1966), Vol. II, p. 599; B. C. Fawcett, N. J. Peacock, and R. D. Cowan, Proc. Phys. Soc. (London) 1B, 295 (1968).
[Crossref]

R. D. Cowan, Astrophys. J. 147, 377 (1967).
[Crossref]

F. Rohrlich, Astrophys. J. 129, 441, 449 (1959).
[Crossref]

J. Opt. Soc. Am. (3)

R. D. Cowan and K. L. Andrew, J. Opt. Soc. Am. 55, 502 (1965).
[Crossref]

G. Racah, J. Opt. Soc. Am. 50, 408 (1960). The abbreviations LSLK, etc. used in the present paper are not unambiguous; they are intended only as suggestive abbreviated lists of the various intermediate quantum numbers, for reference purposes within this paper.

C. J. Humphreys, E. Paul, R. D. Cowan, and K. L. Andrew, J. Opt. Soc. Am. 57, 855 (1967).
[Crossref]

Phys. Rev. (5)

F. R. Innes, Phys. Rev. 91, 31 (1953), Eq. (13).
[Crossref]

F. Rohrlich, Phys. Rev. 74, 1372 (1948).
[Crossref]

J. C. Slater, Phys. Rev. 34, 1293 (1929).
[Crossref]

(II)G. Racah, Phys. Rev. 62, 438 (1942); (III)Phys. Rev. 63, 367 (1943); (IV)Phys. Rev. 76, 1352 (1949).
[Crossref]

F. R. Innes and C. W. Ufford, Phys. Rev. 111, 194 (1958); A. P. Yutsis, Ya. A. Vizbaraite, R. I. Karaziya, A. Yu. Savukinas, and A. A. Bandzaitis, Lietuvos Fisikos Rinkinys IV, 198 (1964); B. W. Shore, Phys. Rev. 139, A1042 (1965); U. Fano, Phys. Rev. 140, A67 (1965).
[Crossref]

Other (13)

C. W. Nielson and G. F. Koster, Spectroscopic Coefficients for the pn, dn, and fn Configurations (MIT Press, Cambridge, Mass., 1963).

For a bibliography up to 1960 see Ref. 3, Ch. 24 and Appendix 21.

E. U. Condon and G. H. Shortley, The Theory of Atomic Spectra (Cambridge Univ.-Press, 1935).

J. C. Slater, Quantum Theory of Atomic Structure (McGraw-Hill Book Co., New York, 1960), 2 Vols.

Except where indicated to the contrary, in this paper lengths are measured in Bohr radii and energies in Rydbergs.

A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton Univ. Press, Princeton, New Jersey, 1957).

M. Rotenberg, R. Bivins, N. Metropolis, and J. Wooten, The 3−j and 6−j Symbols (Technology Press. MIT, Cambridge, Mass., 1959).

B. G. Wyboume, Spectroscopic Properties of the Rare Earths (John Wiley & Sons, Inc., New York, 1965).

Reference 10, Eqs. (3.1) and (2.7).

The author is indebted to J. D. Louck for pointing out this relation to him.

Reference 2, Eqs. 54(8) and 94(14), and Ref. 9, Eqs. (23)–(24).

J. Sugar (private communication).

Reference 3, Sec. 14–2.

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Figures (1)

Fig. 1
Fig. 1

Simplified block diagram of a computer program for calculating coefficient and dipole matrix elements, energy levels, and spectra of arbitrary atomic electron configurations.

Tables (1)

Tables Icon

Table I Problem sizes, and IBM 7030 computing times for L S representation.a

Equations (63)

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H b b = E av δ b b + i j k [ f k F k ( l i , l j ) + g k G k ( l i , l j ) ] + i d i ζ ( l i ) ,
l 1 n 1 l 2 n 2 l q n q
F k ( l i , l j ) = 0 0 2 r k r > k + 1 × R i ( r 1 ) R j ( r 2 ) R i ( r 1 ) R j ( r 2 ) r 1 2 r 2 2 d r 1 d r 2 ,
G k ( l i , l j ) = 0 0 2 r < k r > k + 1 × R i ( r 1 ) R j ( r 2 ) R i ( r 2 ) R j ( r 1 ) r 1 2 r 2 2 d r 1 d r 2 ,
ζ ( l i ) = 1 2 α 2 0 R i 2 ( r ) ( d V / d r ) r d r ,
{ [ ( ( α 1 L 1 S 1 ) L 1 S 1 , α 2 L 2 S 2 ) L 2 S 2 , α q L q S q ] L q S q } T q M q .
{ [ ( α 1 L 1 S 1 J 1 ) T 1 , ( α 2 L 2 S 2 J 2 ) ] T 2 , ( α q L q S q J q ) } T q M q .
T L S , J T = i > 1 [ L i , S i , T i 1 , J i ] 1 2 { L i 1 L i L i S i 1 S i S i T i 1 J i T i } ,
[ x ] 2 x + 1 , [ x , y , z ] ( 2 x + 1 ) ( 2 y + 1 ) ( 2 z + 1 ) , etc .
L S L K : ( { [ ( L q 1 ) L q 1 , L q ] L q , ( S q 1 ) S q 1 } K , S q ) T q M q ,
J T T K : { [ ( J q 1 ) T q 1 , L q ] K , S q } T q M q ,
L S T K : ( { [ ( L q 1 ) L q 1 , ( S q 1 ) S q 1 ] T q 1 , L q } K , S q ) T q M q ,
L S T L K S : ( { [ ( L q 2 ) L q 2 , ( S q 2 ) S q 2 ] T q 2 , ( L q 1 L q ) L } K , ( S q 1 S q ) S ) T q M q ,
L S T L S T : { [ ( L q 2 ) L q 2 , ( S q 2 ) S q 2 ] T q 2 , [ ( L q 1 l q ) L , ( S q 1 S q ) S ] } T } T q M q .
T L S , L S L K = ( 1 ) L q + S q 1 + S q + T q [ K , S q ] 1 2 { K S q T q S q L q S q 1 } ,
T J T , J T T K = ( 1 ) T q 1 + L q + S q + T q [ K , J q ] 1 2 { K S q T q J q T q 1 L q } ,
T L S L K , L S T K = ( 1 ) L q + T q 1 + L q + S q 1 [ L q , T q 1 ] 1 2 { L q S q 1 K T q 1 L q L q 1 } ,
T L S , L S T L K S = ( 1 ) L q 2 + S q 2 + T q 2 + L q + S q T q + L q 1 + S q 1 + L q + S q + L S [ L q 1 , S q 1 , L q , S q , L , S , K , T q 2 ] 1 2 × { L q 2 L q 1 L q 1 L q L q L } { S q 2 S q 1 S q 1 S q S q S } { K S T q S q L q S q 2 } { L q S q 2 K T q 2 L L q 2 } ,
T L S T L K S , L S T L S T = ( 1 ) T q 2 + L + S + T q [ K , T ] 1 2 { K S T q T T q 2 L } .
( l C ( k ) l ) = ( 1 ) l [ l , l ] 1 2 ( l l k 0 0 0 ) ,
( l n α L S U ( k ) l n α L S ) = δ S S n [ L , L ] 1 2 ( 1 ) l + L + k α ¯ L ¯ S ¯ ( 1 ) L ¯ ( l n α L S l n 1 α ¯ L ¯ S ¯ ) { l l k L L L ¯ } ( l n 1 α ¯ L ¯ S ¯ l n α L S ) ,
( l n α L S V ( k 1 ) l n α L S ) = n ( 3 2 [ L , S , L , S ] ) 1 2 ( 1 ) l + L + k × α ¯ L ¯ S ¯ ( 1 ) L ¯ + S ¯ + S 1 2 ( l n α L S l n 1 α ¯ L ¯ S ¯ ) { l l k L L L ¯ } { 1 2 1 2 1 S S S ¯ } ( l n 1 α ¯ L ¯ S ¯ l n α L S ) .
f k ( l i , l i )
s < t n C s ( k ) · C t ( k ) s < t u s ( k ) · u t ( k ) = 1 2 [ ( u s ( k ) ) ( u t ( k ) ) s u s ( k ) · u s ( k ) ] .
f k = 1 2 ( l C ( k ) l ) 2 { [ L ] 1 { α L ( α L S U ( k ) α L S ) × ( α L S U ( k ) α L S ) } n [ l ] 1 δ α α } .
1 2 n ( n 1 ) ( l C ( k ) l ) 2 / { ( 2 l + 1 ) ( 4 l + 1 ) } .
d ( l )
d ( l ) = ( 1 ) L + S + J { l ( l + 1 ) ( 2 l + 1 ) } 1 2 × { L S J S L 1 } ( α L S V ( 11 ) α L S ) .
f k ( l i , l )
f k = ( b | ( s = 1 n i C s ( k ) ) · | ( t = 1 n j C t ( k ) ) b ) = ( l i C ( k ) l i ) ( l j C ( k ) l j ) ( b | U i ( k ) · U j ( k ) | b )
f k ( l i , l j ) = ( l i C ( k ) l i ) ( l j C ( k ) l j ) I ( k ) ,
I ( k ) = ( 1 ) L j 1 + L j + L j { L j 1 L j L j L j L j 1 k } × [ m = i + 1 j 1 ( 1 ) L m 1 + L m + L m + k [ L m , L m ] 1 2 { L m 1 L m L m L m k L m 1 } ] × [ δ i 1 + ( 1 δ i 1 ) ( 1 ) L i 1 + L i + L i + k [ L i , L i ] 1 2 { L i 1 L i L i k L i L i } ] × ( l i n i α i L i S i U ( k ) l i n i α i L i S i ) ( l j n j α i L j S j U ( k ) l j n j α j L j S j ) .
g k ( l i , l j )
1 2 r ( 1 ) r [ r ] { l i l i r l i l i k } [ ( U i ( r ) · U j ( r ) ) + 4 ( U i ( r ) · U j ( r ) ) ( S i ( 1 ) · S j ( 1 ) ) ] .
g k ( l i , l j ) = 1 2 ( l i C ( k ) l j ) 2 r ( 1 ) r [ r ] { l i l i r l i l i k } [ I ( r ) + 4 I ( r 1 ) ] ,
I ( r 1 ) = ( 1 ) L j 1 + S j 1 + L j + S j + L i + S j { L j 1 L j L j L j L j 1 r } { S j 1 S j S j S j S j 1 1 } × [ m = i + 1 j 1 ( 1 ) L m 1 + S m 1 + L m + S m + L m + S m + r + 1 [ L m , S m , L m , S m ] { L m 1 L m L m L m r L m 1 } { S m 1 S m S m S m 1 S m 1 } ] × [ δ i 1 + ( 1 δ i 1 ) ( 1 ) L i 1 + S i 1 + L i + S i + L i + S i + r + 1 [ L i , S i , L i , S i ] 1 2 { L i 1 L i L i r L i L i } { S i 1 S i S i 1 S i S i } ] × ( l i n i α i L i S i V ( r 1 ) l i n i α i L i S i ) ( l j n j α i L j S j V ( r 1 ) l j n j α j L j S j ) .
1 2 n i n j ( l i C ( k ) l j ) 2 / { ( 2 l i + 1 ) ( 2 l j + 1 ) } .
D S 1 2 ( b P ( 1 ) b ) ,
P ( 1 ) = i r i
l 1 k l 2 n l 3 m 1 l 1 k l 2 n 1 l 3 m ,
D ( { L 1 S 1 , l 2 n L 2 S 2 } L 2 S 2 , l 3 m 1 L 3 S 3 , L 3 S 3 T 3 P ( 1 ) { L 1 S 1 , l 2 n 1 L 2 S 2 } L 2 S 2 , l 3 m L 3 S 3 , L 3 S 3 T 3 ) = Z ( { L 1 S 1 , ( L 2 S 2 l 2 s 2 ) L 2 S 2 } L 2 S 2 , L 3 S 3 , L 3 S 3 r { L 1 S 1 , L 2 S 2 } L 2 S 2 , { L 3 S 3 , l 3 s 3 } L 3 S 3 , L 3 S 3 ) ,
Z = δ α 1 L 1 S 1 L 1 S 1 , α 1 L 1 S 1 L 1 S 1 δ S 3 S 3 ( 1 ) L 3 S 3 T 3 1 ( l 2 n α 2 L 2 S 2 l 2 n 1 α 2 L 2 S 2 ) ( l 3 m 1 α 3 L 3 S 3 l 3 m α 3 L 3 S 3 ) × ( n m [ T 3 , T 3 ] ) 1 2 { L 3 T 3 S 3 T 3 L 3 1 } .
D = Z ( 1 ) L 1 + S 1 + L 2 + S 2 + l 2 + s 2 + L 2 + S 2 [ L 2 , S 2 , L 2 , S 2 ] 1 2 { L 1 L 2 L 2 l 2 L 2 L 2 } { S 1 S 2 S 2 s 2 S 2 S 2 } × ( { ( L 1 S 1 , L 2 S 2 ) L 2 S 2 , l 2 s 2 } L 2 S 2 , L 3 S 3 , L 3 S 3 r { L 1 S 1 , L 2 S 2 } L 2 S 2 , { L 3 S 3 , l 3 s 3 } L 3 S 3 , L 3 S 3 ) .
LS ( 1 ) l 2 s 2 L 3 S 3 L 2 S 2 L S [ L 2 , S 2 , L , S ] 1 2 { L 2 L 3 L 3 L l 2 L 2 } { S 2 S 3 S 3 S s 2 S 2 } × ( 1 ) L 2 + S 2 + L 3 + S 3 + l 3 + s 3 + L 3 + S 3 [ L 3 , S 3 , L , S ] 1 2 { L 2 L 3 L L L 3 L 3 } { S 2 S 3 S s 3 S 3 S 3 } × ( { L 2 S 2 , L 3 S 3 } LS , l 2 s 2 , L 3 S 3 r { L 2 S 2 , L 3 S 3 } LS , l 3 s 3 , L 3 S 3 ) .
( 1 ) L L 3 l 3 1 [ L 3 , L 3 ] 1 2 { l 2 L 3 L L 3 l 3 1 } P 23 ,
P 23 ( l 2 r l 3 ) = ( 1 ) l 2 [ l 2 , l 3 ] 1 2 ( l 2 1 l 3 0 0 0 ) 0 r R l 2 R l 3 r 2 d r .
D = δ α 1 L 1 S 1 L 1 S 1 , α 1 L 1 S 1 L 1 S 1 δ S 3 S 3 ( 1 ) L 1 + S 1 + L 2 + S 2 + L 2 + S 2 + S 2 + S 3 + L 3 + 1 2 T 3 × ( n m [ L 2 , S 2 , L 2 , S 2 , L 2 , S 2 , L 3 , S 3 , L 3 , L 3 , T 3 , T 3 ] ) 1 2 × { L 3 T 3 S 3 T 3 L 3 1 } { L 1 L 2 L 2 l 2 L 2 L 2 } { S 1 S 2 S 2 s 2 S 2 S 2 } { S 3 S 3 S 2 S 2 s 2 S 3 } { l 2 L 2 L 2 l 3 L 3 L 3 1 L 3 L 3 } × ( l 2 n α 2 L 2 S 2 l 2 n 1 α 2 L 2 S 2 ) ( l 3 m 1 α 3 L 3 S 3 l 3 m α 3 L 3 S 3 ) P 23 .
l 1 n 1 l 2 n 2 l q 2 k l q 1 n l q m 1 l 1 n 1 l 2 n 2 l q 2 k l q 1 n 1 l q m
D ( α 1 L 1 S 1 , α 2 L 2 S 2 , L 2 S 2 T 2 r α 1 L 1 S 1 , α 2 L 2 S 2 , L 2 S 2 T 2 ) = δ S 2 S 2 ( 1 ) l 1 + L 1 + L 1 + S 1 + S 2 + L 2 T 2 × ( n m [ L 1 , S 1 , L 2 , S 2 , L 2 , L 2 , T 2 , T 2 ] ) 1 2 { L 2 T 2 S 2 T 2 L 2 1 } { S 2 S 2 S 2 S 1 s 1 S 2 } { l 1 L 1 L 1 l 2 L 2 2 L 2 1 L 2 L 2 } × ( l 1 n α 1 L 1 S 1 l 1 n 1 α 1 L 1 S 1 ) ( l 2 m 1 α 2 L 2 S 2 l 2 m α 2 L 2 S 2 ) P 12 .
D ( α 1 L 1 S 1 , l 2 , L 2 S 2 T 2 r α 1 L 1 S 1 , l 2 , L 2 S 2 T 2 ) = δ α 1 L 1 S 1 , α 1 L 1 S 1 δ S 2 S 2 ( 1 ) L 1 + S 2 + l 2 + T 2 [ L 2 , L 2 , T 2 , T 2 ] 1 2 { L 2 T 2 S 2 T 2 L 2 1 } { l 2 l 2 1 L 2 L 2 L 1 } P l 2 l 2 .
D ( l S J r l s J ) = ( 1 ) l + s + J [ J , J ] 1 2 { l J s J l 1 } P l l .
( D β ) = ( T I 3 ) ( D LS ) ( T T 3 ) ,
tr [ ( D ) ( D ) ] = i [ ( D ) ( D ) ] i i = i [ j D j i D j i ] = D j i 2 S .
S β = tr [ ( D β ) ( D β ) ] = tr [ ( T T 3 ) ( D LS ) ( T T 3 ) ( T T 3 ) ( D LS ) ( T T 3 ) ] = tr [ ( T T 3 ) ( D LS ) ( D LS ) ( T T 3 ) ] = tr [ ( D LS ) ( D LS ) ] = S LS ,
( S ) / P 2 = n m { [ L 1 , S 1 , L 2 , S 2 , L 3 , S 3 ] / [ l 2 , s 2 , l 3 ] } × ( l 2 n α 2 L 2 S 2 l 2 n 1 α 2 L 2 S 2 ) 2 × ( l 3 m 1 α 3 L 3 S 3 l 3 m α 3 L 3 S 3 ) 2 .
( S ) / P 2 = 2 [ i = 1 q 2 A ( l i n i ) ] · B ( l q 1 n ) · B ( l q m ) ,
A ( l n ) = [ L , S ]
B ( l n ) = n ( [ L , S ] ) / [ l , s ] .
A ( l n ) = ( 4 l + 2 n ) = ( 4 l + 2 ) ! n ! ( 4 l + 2 n ) ! ,
B ( l n ) = ( 4 l + 1 ) ! / { ( n 1 ) ! ( 4 l + 2 n ) ! } ;
| T q T q | 1 ( T q = T q = 0 not allowed ) ,
g f = ( 8 π 2 m c a 0 2 / 3 h ) S σ = 3.0376 · 10 6 S σ ,
g A = ( 64 π 4 e 2 a 0 2 / 3 h ) S σ 3 = 2.0260 · 10 6 S σ 3 sec 1 = 0.66698 g f σ 2 sec 1 ,