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Electric quadrupole transitions in LiI, BeII, BIII
J Quont. Spectrosc. Radial. Transfer. Vol. IS, pp. 159-162. Pergamon Press 1975. Printed in Great Britain.
ELECTRIC QUADRUPOLE TRANSITIONS IN LiI, BeII, BIII SANKAR SENGUPTA Department of Physics, Jadavpur University, Calcutta 32, InaLa (Received 3 July 1974) Abstract-Theoretical values of the weighted oscillator strengths for different electric quadrupole transitions of LiI, Be11 and BIIJ have been calculated using analytic Hartree-Fock wave functions. An appropriate hypervirial check has been made to assess the accuracy of the values. Some interesting features of the hypervirial operator are discussed.
1. INTRODUCTION ELECTRICdipole
transitions account for the overwhelming majority of spectral lines, Naturally, the calculation of electric dipole transitions has been the subject of main interest in atomic spectroscopy. Recently, however, BOGGARD and ORR’~’ have shown that the measurement of the quadrupole transition matrix elements might be feasible from the study of quadrupolar birefringence in absorption region. In order to compare with the experimental values of the quadrupole transition matrix elements which may be available soon, several theoretical estimates for the same quantity have already been made. The calculation of such matrix elements for atomic lithium has been reported by BOGGARD and ORR”’ using a Coulomb approximation, and by BOYLE and MURRAY”’ using Weiss’s function as well numerical H.F. function of MAYERS.‘~’The accuracy of these values is difficdt to assess because these values depend primarily on the wave functions employed which, in turn, are sensitive to the procedures used to obtain them. Therefore, systematic calculations of these matrix elements with variety of wave functions seem necessary. Recently, MOITRAet ~1.‘~‘” have obtained excellent analytical representations of Hartree-Fock wave functions for the excited states of many-electron systems. These functions are available for different members of the isoelectronic sequence and for highly-excited states. In the present Note, we have used these wave functions to evaluate quadrupole transition matrix elements for LiI, BeII, BIII. Since the previous calculations are primarily restricted to low-lying excited states of atomic lithium, the present calculation provide some new data. Accurate calculation of the binding energy of a state does not imply a similar accuracy in the wave function. A test on the wave functions is a comparison of calculated and experimental oscillator strengths. Since the experimental values are yet to be available, we have checked the accuracy of the wave functions used with the help of an appropriate hypervirial theorem. Hypervirial theorems indicate limitations in the wave functions. They do not, however, allow us to speculate whether the true value lies between the two estimates or not. The results are given in Section III. Some interesting properties of the hypervirial operator are also discussed in this section. 159
SANKARSENGLJPTA
160
2. THEORY
The weighted oscillator strength@’ for electric quadrupole lower level a is given by
radiation from an upper level b to a
where g is the statistical weight of the lower level. The quantity Sba, which is the theoretical measure of the line intensities, is known as the line strength and is given by &a =
Q (‘I is the quadrupole component
il(aIQ’2’lb)12;
(2)
of the electric multipole-moment Q(“’
=
operator
C e r”(j)C,‘“‘(j).
For members of the lithium isoelectronic sequence the closed shell, the line strength is given”’ by
which have single valence electron outside
Sba = $ e*(21 + 1)C2(1, 2, I’ :000)(a~r*~b)2;
(4)
here C(1,2,1’ :OOO)is the usual Clebsch-Gordon coefficient and la) and lb) are the lower and upper valence states corresponding to the angular momenta E’ and 1, respectively. In order to ascertain the accuracy of the wave functions, we have used the hypervirial theorem given by BOYLE and MURRAY.“’ These authors have shown that electric field-gradient polarizability and quadrupole transition probability calculations are performed consistently when the wave functions satisfy the following relation: (aJr2P2(cos tl)jb) = -(alMlb>(&
- Eb)-‘,
(9
where M = 2rP2(cos 0) -$ - 3 sin 0 cos r9-$
3. RESULTS
1.
(6)
AND DISCUSSIONS
Our results are presented in Tables 1 and 2. In Table 1, we have compared the values of the quadrupolar part of the operator [equation (3)l with the corresponding values of the hypervirial form. The excellent agreement between the two sets of values clearly indicates the accuracy of the values presented, as well as of the wave functions used. It also suggests that the effect of spin-orbit interaction on the radial wave functions may not be significant in the calculation of quadrupole oscillator strengths. In Table 2, we have presented the value of gf for transitions obeying the selection rules A/ = 0 and Ahl= 2, respectively. Our values of gf for 2s -+ 3d and 2p -+3p transitions of Li agree well with the values obtained by BOGGARDand ORF?’ and BOYLE
Electric quadrupole transitions in LiI, BeII, BIB
161
and MURRAY.(~) It is not possible to make an elaborate comparison since sufficient data are not available. Since the formalism of MOITRAet ~1.“’ did not yield excited s-states, only 2s-3d transition has been included with the s-function is given by CLEMENTI.“’ The hypervirial operator has an interesting property. For a given pair of participating states, the quantity (a [MIb) is independent of atomic number 2. This result can be demonstrated with hydrogenic radial functions. The first part of M is controlled by the operator r(a/ar). Therefore, with increasing Z, a large increase in velocity is counterbalanced by a proportionate diminution in the positional coordinate. The remaining part of M is controlled by overlap of two normalized radial functions and is automatically independent of Z. This hydrogenic feature is also evident in the present calculation, as is shown in Table 1. As the result, once equation (5) is satisfied for the first member of an isoelectronic series it will be possible to predict the values of the transition matrix elements for the other members of the series knowing only the energy difference between the states involved in the transition. Table
1. Comparison of the quadrupolar parts of the operator corresponding values of the hypervirial form
Transition 22P-32P LiI Be11 BIII 22P-42P LiI Be11 BIB 32P-42P LiI Be11 BIII 3*DA2D LiI Be11 BIII 32D-52D LiI Be11 BIB 42D-52D LiI Be11 BIII 4’F - SF LB Be11 BIB 4’F-6’F LiI Be11 BIII 5ZF-62F LiI Be11 BIB
(alMlb)(a.u.)
(alMJb)/(E,
- Eb)(a.u.)
with the
(a-IQ’2’lb)(a.u.)
-0.6870 -0.6908 -0.6891
-9.5023 -2.3636 - 1.0501
-9.6255 -2.3629 - 1.0545
-0.2860 -0.2883 -0.2883
-2.9365 -0.7330 -0.3266
-3.0123 -0.7337 -0.3265
-1.1050 -1.1009 -1.1010
-44.0239 -109065 -4.8611
-43.7178 - 10.8955 -4.8591
-06416 -06444 -06436
-26.4047 -6.6228 -2.9417
-26.6681 -6.6244 -2.9421
-0.2509 -0.2516 -0.2511
-7.0494 - 1.7687 -0.7845
-7.1871 -1.7938 -0.7910
-0.9669 -0.9685 -0.9671
-85.5699 -21.3329 -9.5472
-85.9857 -216400 -9.5570
-0.7001 -0.7082 -0.7085
-61.9518 - 15.7383 -6.9942
-62.0257 -15.7631 -6.9966
-0.2435 -0.2603 -0.2595
-13.9993 -3.7462 - 1.6603
-14.0102 -3.7508 - 1.6680
-1.0427 -1.0389 -1.0401
- 170.9403 -42.4051 -18.9125
-169.6528 -42.1328 -18.8991
SANKARSENGUPTA
162
Table 2. Calculated gf values for LiI, Bell, and BIB for various transitions
gf X 10'(au.) Transition
LiI
Be11
BIB
22P-32P 22P-42P 32p42P
0.09328 0.0223 1 0.0805 I 0.19029 0.04346 0.19894 0.07446 0.00275 0.06092 0.28570 0.13887 0.11099 0.46527 o#I915 0.00338 0.57916 0.00178 0.38657
0.37144 0.08723 0.32581 0.75381 0.17291 0.79576 0.21113 0.04404 0.24342 I.09887 0.55400 0.45573 I .84748 0.02504 0.00806 I.39177 0.06575 I.21936
0.83688 0.36404 1.24079 1.69650 0.38269 1.77054 0.47448 0.09906 0.55436 2.49923 1.24618 0.95716 4.13109 0.05350 0.01644 2.97953 0.13981 2.44419
32D-42D 3'0-5'0 4=D-S2D 4'F-52F 42F-62F 5ZF-62F 22P-42F 22P-52F 2ZP-62F 3’P--4*F 32P-52F 3’P+i2F 4’P-5’F 42P-62F 2=S-3’0
Acknowfedgement-The
author is indebted to U.G.C. for granting financial assistance for computation.
REFERENCES M. P. BOGGARDand B. J. ORR, Molec. Phys. 14, 557 (1%8). L. L. BOYLEand L. J. MURRAY,J. Phys. B2, 433 (1969). D. F. MAYERS,private communication (1974). R. K. MOITRA,P. K. MUKHERJEE and S. SENGUPTA, Int. J, Q. Chem. 4, 465 (1970). R. K. MOITRAand P. K. MUKHERIEE, Int. J. Q. Chem. 6, 211 (1972). B. SHOREand D. MENZEL,Principles of Atomic Spectra, p. 445. Wiley, New York (1968). I. B. LEVINSON and A. A. NIKITIN,Handbook for Theoretical Computation of Line Intensities in Atomic Spectra. Israel programme for scientific translation, Jerusalem (1%5). 8. E. CLEMENTI,Tables of Atomic Functions, IBM, New York (1965).
1. 2. 3. 4. 5. 6. 7.
ELECTRIC QUADRUPOLE TRANSITIONS IN LiI, BeII, BIII SANKAR SENGUPTA Department of Physics, Jadavpur University, Calcutta 32, InaLa (Received 3 July 1974) Abstract-Theoretical values of the weighted oscillator strengths for different electric quadrupole transitions of LiI, Be11 and BIIJ have been calculated using analytic Hartree-Fock wave functions. An appropriate hypervirial check has been made to assess the accuracy of the values. Some interesting features of the hypervirial operator are discussed.
1. INTRODUCTION ELECTRICdipole
transitions account for the overwhelming majority of spectral lines, Naturally, the calculation of electric dipole transitions has been the subject of main interest in atomic spectroscopy. Recently, however, BOGGARD and ORR’~’ have shown that the measurement of the quadrupole transition matrix elements might be feasible from the study of quadrupolar birefringence in absorption region. In order to compare with the experimental values of the quadrupole transition matrix elements which may be available soon, several theoretical estimates for the same quantity have already been made. The calculation of such matrix elements for atomic lithium has been reported by BOGGARD and ORR”’ using a Coulomb approximation, and by BOYLE and MURRAY”’ using Weiss’s function as well numerical H.F. function of MAYERS.‘~’The accuracy of these values is difficdt to assess because these values depend primarily on the wave functions employed which, in turn, are sensitive to the procedures used to obtain them. Therefore, systematic calculations of these matrix elements with variety of wave functions seem necessary. Recently, MOITRAet ~1.‘~‘” have obtained excellent analytical representations of Hartree-Fock wave functions for the excited states of many-electron systems. These functions are available for different members of the isoelectronic sequence and for highly-excited states. In the present Note, we have used these wave functions to evaluate quadrupole transition matrix elements for LiI, BeII, BIII. Since the previous calculations are primarily restricted to low-lying excited states of atomic lithium, the present calculation provide some new data. Accurate calculation of the binding energy of a state does not imply a similar accuracy in the wave function. A test on the wave functions is a comparison of calculated and experimental oscillator strengths. Since the experimental values are yet to be available, we have checked the accuracy of the wave functions used with the help of an appropriate hypervirial theorem. Hypervirial theorems indicate limitations in the wave functions. They do not, however, allow us to speculate whether the true value lies between the two estimates or not. The results are given in Section III. Some interesting properties of the hypervirial operator are also discussed in this section. 159
SANKARSENGLJPTA
160
2. THEORY
The weighted oscillator strength@’ for electric quadrupole lower level a is given by
radiation from an upper level b to a
where g is the statistical weight of the lower level. The quantity Sba, which is the theoretical measure of the line intensities, is known as the line strength and is given by &a =
Q (‘I is the quadrupole component
il(aIQ’2’lb)12;
(2)
of the electric multipole-moment Q(“’
=
operator
C e r”(j)C,‘“‘(j).
For members of the lithium isoelectronic sequence the closed shell, the line strength is given”’ by
which have single valence electron outside
Sba = $ e*(21 + 1)C2(1, 2, I’ :000)(a~r*~b)2;
(4)
here C(1,2,1’ :OOO)is the usual Clebsch-Gordon coefficient and la) and lb) are the lower and upper valence states corresponding to the angular momenta E’ and 1, respectively. In order to ascertain the accuracy of the wave functions, we have used the hypervirial theorem given by BOYLE and MURRAY.“’ These authors have shown that electric field-gradient polarizability and quadrupole transition probability calculations are performed consistently when the wave functions satisfy the following relation: (aJr2P2(cos tl)jb) = -(alMlb>(&
- Eb)-‘,
(9
where M = 2rP2(cos 0) -$ - 3 sin 0 cos r9-$
3. RESULTS
1.
(6)
AND DISCUSSIONS
Our results are presented in Tables 1 and 2. In Table 1, we have compared the values of the quadrupolar part of the operator [equation (3)l with the corresponding values of the hypervirial form. The excellent agreement between the two sets of values clearly indicates the accuracy of the values presented, as well as of the wave functions used. It also suggests that the effect of spin-orbit interaction on the radial wave functions may not be significant in the calculation of quadrupole oscillator strengths. In Table 2, we have presented the value of gf for transitions obeying the selection rules A/ = 0 and Ahl= 2, respectively. Our values of gf for 2s -+ 3d and 2p -+3p transitions of Li agree well with the values obtained by BOGGARDand ORF?’ and BOYLE
Electric quadrupole transitions in LiI, BeII, BIB
161
and MURRAY.(~) It is not possible to make an elaborate comparison since sufficient data are not available. Since the formalism of MOITRAet ~1.“’ did not yield excited s-states, only 2s-3d transition has been included with the s-function is given by CLEMENTI.“’ The hypervirial operator has an interesting property. For a given pair of participating states, the quantity (a [MIb) is independent of atomic number 2. This result can be demonstrated with hydrogenic radial functions. The first part of M is controlled by the operator r(a/ar). Therefore, with increasing Z, a large increase in velocity is counterbalanced by a proportionate diminution in the positional coordinate. The remaining part of M is controlled by overlap of two normalized radial functions and is automatically independent of Z. This hydrogenic feature is also evident in the present calculation, as is shown in Table 1. As the result, once equation (5) is satisfied for the first member of an isoelectronic series it will be possible to predict the values of the transition matrix elements for the other members of the series knowing only the energy difference between the states involved in the transition. Table
1. Comparison of the quadrupolar parts of the operator corresponding values of the hypervirial form
Transition 22P-32P LiI Be11 BIII 22P-42P LiI Be11 BIB 32P-42P LiI Be11 BIII 3*DA2D LiI Be11 BIII 32D-52D LiI Be11 BIB 42D-52D LiI Be11 BIII 4’F - SF LB Be11 BIB 4’F-6’F LiI Be11 BIII 5ZF-62F LiI Be11 BIB
(alMlb)(a.u.)
(alMJb)/(E,
- Eb)(a.u.)
with the
(a-IQ’2’lb)(a.u.)
-0.6870 -0.6908 -0.6891
-9.5023 -2.3636 - 1.0501
-9.6255 -2.3629 - 1.0545
-0.2860 -0.2883 -0.2883
-2.9365 -0.7330 -0.3266
-3.0123 -0.7337 -0.3265
-1.1050 -1.1009 -1.1010
-44.0239 -109065 -4.8611
-43.7178 - 10.8955 -4.8591
-06416 -06444 -06436
-26.4047 -6.6228 -2.9417
-26.6681 -6.6244 -2.9421
-0.2509 -0.2516 -0.2511
-7.0494 - 1.7687 -0.7845
-7.1871 -1.7938 -0.7910
-0.9669 -0.9685 -0.9671
-85.5699 -21.3329 -9.5472
-85.9857 -216400 -9.5570
-0.7001 -0.7082 -0.7085
-61.9518 - 15.7383 -6.9942
-62.0257 -15.7631 -6.9966
-0.2435 -0.2603 -0.2595
-13.9993 -3.7462 - 1.6603
-14.0102 -3.7508 - 1.6680
-1.0427 -1.0389 -1.0401
- 170.9403 -42.4051 -18.9125
-169.6528 -42.1328 -18.8991
SANKARSENGUPTA
162
Table 2. Calculated gf values for LiI, Bell, and BIB for various transitions
gf X 10'(au.) Transition
LiI
Be11
BIB
22P-32P 22P-42P 32p42P
0.09328 0.0223 1 0.0805 I 0.19029 0.04346 0.19894 0.07446 0.00275 0.06092 0.28570 0.13887 0.11099 0.46527 o#I915 0.00338 0.57916 0.00178 0.38657
0.37144 0.08723 0.32581 0.75381 0.17291 0.79576 0.21113 0.04404 0.24342 I.09887 0.55400 0.45573 I .84748 0.02504 0.00806 I.39177 0.06575 I.21936
0.83688 0.36404 1.24079 1.69650 0.38269 1.77054 0.47448 0.09906 0.55436 2.49923 1.24618 0.95716 4.13109 0.05350 0.01644 2.97953 0.13981 2.44419
32D-42D 3'0-5'0 4=D-S2D 4'F-52F 42F-62F 5ZF-62F 22P-42F 22P-52F 2ZP-62F 3’P--4*F 32P-52F 3’P+i2F 4’P-5’F 42P-62F 2=S-3’0
Acknowfedgement-The
author is indebted to U.G.C. for granting financial assistance for computation.
REFERENCES M. P. BOGGARDand B. J. ORR, Molec. Phys. 14, 557 (1%8). L. L. BOYLEand L. J. MURRAY,J. Phys. B2, 433 (1969). D. F. MAYERS,private communication (1974). R. K. MOITRA,P. K. MUKHERJEE and S. SENGUPTA, Int. J, Q. Chem. 4, 465 (1970). R. K. MOITRAand P. K. MUKHERIEE, Int. J. Q. Chem. 6, 211 (1972). B. SHOREand D. MENZEL,Principles of Atomic Spectra, p. 445. Wiley, New York (1968). I. B. LEVINSON and A. A. NIKITIN,Handbook for Theoretical Computation of Line Intensities in Atomic Spectra. Israel programme for scientific translation, Jerusalem (1%5). 8. E. CLEMENTI,Tables of Atomic Functions, IBM, New York (1965).
1. 2. 3. 4. 5. 6. 7.