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Calculation of the Zeeman effect in the n2S12 , n2P12 , and n2P32 (n = 2,3,4, and 5) states of the lithium atom
13 July 1998 PHYSICS LETTERS A
Physics Letters A 244 (1998) 120-126
Calculation of the Zeeman effect in the n2S1p, IZ~Z$,~,and n2Q2 ( rz = 2, 3, 4, and 5 ) states of the lithium atom X.-X. Guan”, Z.-W. Wangavb a Institute of Atomic and Molecular Physics, Jilin Universily Chungchun 130023. Jib,
China b Department of Mathematics and Physics, Wuyi University, Jiangmen 529020, Guangdong, China
Received 3 November 1997; revised manuscript received 20 April 1998; accepted for publication 20 April 1998 Communicated by B. Fricke
The gJ-factors of the n2S1p n2P1p, and n2P3,2 (n = 2, 3, 4, and 5) states of the lithium atom are calculated by using the full-core plus correlation method. The relativistic correction to order cy2(a.u.) and the effect of the motion of the center of mass are treated by using first-order perturbation theory. For the 22S1,2 state, our result, 2.00230101, agrees more closely with experimental data, 2.00230100(64), than previous theoretical results. The gJ-factor values of other low-lying excited states for Li are also given. @ 1998 Published by Elsevier Science B.V. PACS: 31.20.Di; 31.3O.J~
1. Introduction
The detailed Zeeman structure of the atomic system in an external magnetic field has received considerable attention of experimentalists and theorists. The fine-structure intervals for the excited states of atoms can be derived through investigating the behaviour of atoms in the magnetic field. In the past two decades, many high-precision experimental studies have been carried out for the helium and lithium atoms. The values of the fine-structure splitting in the He 33P state and Li 2*P, and 3*P states were measured to very high accuracy by using the technique of level crossing spectroscopy [ l-71. The precise values of gJ-factors for the corresponding states are very important for measurements with such a high accuracy. On the other hand, the multiconfiguration Hartree-Fock (MCHF) method has been extensively applied for a long time [ 81. There are several types of HF wave functions, including the restricted Hartree-Fock (RHF), the unrestricted Hartree-Fock (UHF), the spinextended Hartree-Fock (SEHF) [ 9,101, the spin-optimized Hartree-Fock (SOHF) [ IO], and the maximally paired Hartree-Fock [ 111. They were used to study the relativistic effect on the g,-factor of Li and other light atoms. For Li and other systems, ab initio calculations of the gl-factor were performed recently by Lindroth and Ynnerman [ 121, who used the relativistic wave function obtained in the coupled-cluster approximation. Recently, Yan and Drake [ 1] have considered the relativistic correction to the gJ-factor of the excited states for the helium atom, by using variational wave functions constructed from doubled Hylleraas-type basis sets. The experimental values of the fine-structure splitting for He 33P~ states have been reanalyzed by using improved 037%9601/98/$19.00 @ 1998 Published by Elsevier Science B.V. All tights reserved. PIISO375-9601(98)00332-6
X.-X. Guan. Z.-M! Wang/Physics Letters A 244 (1998) 120-126
121
gJ-factors. In other words, the accurate theoretical calculation for gf-factors is necessary to extract a more accurate fine-structure splitting. More recently, Anthony and Sebastian have used an approximately relativistic theory of bound states which ensures PoincarC invariance of the atomic system to relative order a2 to derive the Zeeman Hamiltoni~, and calculated the gj-factors of the 9 and P-states for the helium atom [ 133. Lithium is the simplest atomic species, besides helium. It has one ls2-core and a non-paired nl electron in its ground and singly excited states. Recently, a full-core plus correlation (FCPC) method [ 141 has been used to calculate the energy and fine-structure of the excited states for lithium-like systems to high precision [ 15-171. In the FCPC method, a large predete~ined multi~on~guration interaction (CI) ls2-core wave function is frozen as a single term in the total three-electron wave function. The effect of the valence electron is accounted for by multiplying the core wave function with a linear combination of single-particle Slater orbitals. The relaxation of the core and other possible correlation are described by another large CI wave function. In this work, we present the gJ-factors results of the n2St/2, n2Pt,2, and n2Psi2 (n = 2, 3, 4 and 5) states for Li, using the FCPC wave functions. The relativistic correction up to order CX*(a.u.) and the effect of the motion of center of mass are evaluated with the first-perturbation theory. The computation in this work is carried out in atomic units (au.). The nuclear mass is taken from Ref. [ IS]. The isotope used is 7Li. A brief account of the theory is given in Section 2. The calculated results and discussion are presented in Section 3. A conclusion is given in Section 4.
2. Theory
The relativistic correction to the Zeeman Hamiltonian of many-electron systems was derived from the Breit equation by reducing it to the large components [7,19]. The correction from the finite nuclear mass should be also considered. In this work the term of order ~/~ will be included. The total Zeeman Hamiltonian, including the relativistic correction terms up to order o* and the nuclear motion correction of order m/M, can be expressed as
Ht = -a2psB
*C
(1; + 2si) (-%VF> ,
(3)
i=l
H5 = -tpsB.i:('i
Xpj). i+j
Here a! is the tine structure constant, B is the external magnetic field, and pa is the Bohr magneton. He is the lowest-order Zeeman effect, where gr. = 1 - m/M. The small departure of gr, from unity arises from the motion
122
X.-X. Gun,
Z.-R Wang/Physics
Letters A 244 (1998) 120-126
of the nucleus. L = xi Zi and S = Ci si are the total orbital angular momentum and total spin, respectively. Hi is analogous to the relativistic increase of mass. Hz, Hs, and H4 come from the reduction of the Breit equation and they include terms analogous to spin-orbit, spin-other-orbit, and orbit-orbit interactions. Hs includes the correction from the motion of the center of mass where M is the atomic mass. If the gyromagnetic ratio of the free electron, gs, is taken to order I_Y~, namely gs=2(l+&0.32848(;)2+...)
=2.002319304377(9),
(8)
the higher order correction terms of 0( (Ye) and 0( cu2m/M) will be also contained in the final numerical results. According to the FCPC method [ 14-171, the wave function for the three-electron system can be written as (9) where A is an antisymmetrization operator. @lsls is a predetermined ls2-core wave function which is represented by a CI basis set, @1s1s(1,2)
kd&ev(-Pm
= AcC
-
w2)Y1(1.2)x(lr2)
.
(10)
kn,l
The angular part is X(1,2) = c
(ZWZ -m10,0)Sm(~lr(P1)S-m(~2,(P2).
(11)
m ,y( 1,2) is a two-electron singlet spin function. The linear and nonlinear parameters in Eq. ( 10) are determined by optimizing the energy of the He-like 131s system. The factor @“l(3) multiplied with @tsts is a linear combination of the Slater orbitals for the valance electron. The second term on the right-hand side of Eq. (9) describes the core relaxation and the intershell correlation in the three-electron system. The basis function is chosen as
@n(i).l(i)(17293)
(12)
= (Pn(i)(R)$fCR^)XSMs.
The radial and angular basis functions are PPn(i)(R)
(13)
=fi~~eXp(-P,~j)
j=l
and I$$W)
=c
(Zlml’2m21’12ml2)
(‘12ml2’3m3I’M)
fiXj,C’jI
9
(14)
j=l
“tj
where Z(i) represents the set of Ii, Z2,112and 13. This angular component is simply denoted as li=
[(~1,~2)~12,~31
(15)
3
where 1,~ and 13 coupling into L is implicitly implied. Using a similar notation, the doublet spin function can be represented by X&U,=
[(~1,~2)~12,~31
.
(16)
X.-X. &an, Z.-W Wang/Physics Letters A 244 (1998) 120-126
123
It has two possible spin doublets, x1 = [(~l,0,)0,~31 x2=
T
(17)
[(si,s2)1~s31.
(18)
In the present work, these FCPC wave functions for the n*Si,z, n’P,p and n2P3/2 (n = 2, 3, 4, and 5) states of Li are obtained by using the variational method [ 14-171. The mass polarization effect is neglected in variational calculations, so this effect is not included in the FCPC wave functions. In Refs. [ 14-1’71, the mass polarization effect on the energy is taken into account using first-order ~~urbation theory. The expectation value of the Zeeman Hamiltonian Hz is calculated by using our FCPC wave functions. In the LSJ coupling scheme, the gl-factor of the system is given by (19) Here g$@ is obtained from Eq. (211,
and gy’ are obtained from Eqs. (3)-(7),
s:” =&(LSJMI~H,~LSJMJ)
respectively, (i=1,2,3,4,
and5).
(21)
3. Results and discussion The same seven angular component 222 terms are used in the wave function for the Is*-core. For the ls2ns states, we used the wave function with nine terms in the a,,, of Eq. (9). The number of terms in the @,(i),/(i) of Eq. (9) ranges from about 390 to 58 1 terms. For the 1s2np states, we used the same number of terms for the Qp,, in Eq. (9) and about 1004-l 176 terms in the ~~(i),~(~)of Eq. (9). The linear and nonlinear parameters are individually optimized in the energy minimization process [ 14-171. As expected, in order to obtain a more accurate FCPC wave function (9), a larger number of terms is needed with decreasing n (principal quantum number) due to stronger correlation. The expectation values of the Zeeman Hamiltonian Hz for the ls2ns and ls2np states are calculated by using our FCPC wave functions obtained above. Their convergence is rather fast. As a typical example, the convergence of the expectation values for HI and HZ in the ls22s state is shown in Table 1. It is seen from this table that both (Ht) and (HZ) converge to five digits. From the ls*ns states, because L = 0 and the ranks of both H4 and HS are unity, all elements of their reduced matrix must be zero. Only Hi, Hz, and Hs con~ibute to gijz-values of these states with 2S symme~y. In other words, for the 2S1,2 states, the gJ-factor in the present treatment is the same for the isotopes of lithium. Obviously, this conclusion no longer holds for the *PJ states. Our calculated expectation values of HI, Hz, and Hs and gi/z-value for 1~~2s state are given in Table 2. In the literature, most calculations of the 22Si~z gl/z-factor for Li were carried out by using various HF approximations [20,21]. These theoretical results and experimental data are also included in Table 2 for comparison. It appears that our gip-values agree with the experimental data [22] to eight digits. It can be seen clearly from this table that most results of Hegstrom [ 20f fall outside the experimental error bounds. Our calculated g1/2- factor is in excellent agreement with the experimental value recommended by Arimondo et al. Using spin-extended HF and coupled-cluster approximation, Veseth [ 211, Lindroth and Yneerman [ 121 obtained the results, 2.00230086, 2.00230158(20),
124
X.-X. Guan, Z-W Wang/Physics
L.&em A 244 (1998) 120-126
Table i Convergence of the expectation values for Ht, Hz in the 1~~2s state for Li Angular component
No. of term
- WI )”
Vf2jb
core+2s 1(0,0)0,01 Ito, l)l, 11 [(0,2)2.21
9 41 123 93 11 50 32 35 22 34 34 34
0.10853082 0.00901022 0.09416941 0.00040683 -0.~22956 o.OOoO936O 0.00003094 0.00002436 0.~1472 0.~1265 -0.00000946 0.00000580
0.17415461 0.00343575 0.00228714 0.00921272 -0.~1~1 o.OWO4714 0.00001533 0.00001385 o.OOOOO784 0.~21 -0.00000567 O.OOOOQ28.5
20 20 32
0.00000425 0.~265 o.OOOOOO95
0.00000203 0.~147 O.OOOOOO88
590
0.12206818
0.18016614
I(O, O)O,Ol’ t(o,3)3.31 t(o,4)4,41 [(1,2)1,11 [(0,1)1,11” 1(0,5)5,51 I(l.O)l, 11
[ (0,6)6,61 [(l,2)1*IIC ((2,3)1,11 I(O, 2% 21C
Total
a In (2a2&Z). b In (a*&&). c In these angular components the spins of the first two electrons couple into a triplet. Table 2 Comparison of the valuesa of (Hi),
(Hz),
(ff3), gl,q of tbe ls22s state for Li
Reference
Metbodb
(HI)
(H2)
(H3)
a/2
Hegstrom Hegstrom Hegstram Hegstrom Veseth Lindroth et al. this work
R-HF U-HF MP-HF SO-SCF SE-HF CCSD FCPC
-2.2240 -2.5283 -2.4173 -2.4930 -2.6125
1.8392 1.8968 1.8740 1.8881 1.9134
-1.1274 -1.1359 -1.1303 -1.1324 -1.1392
-2.6001
I,.9188
-1.1469
2.tW230416 2.00230161 2.00230255 2.00230192 2.00230086 2.~30158(20) 2.00230101
Arimondo et al.
experiment
2.00230100(64)
a Tbe values of (Ht), (Hz}, (Hs) are in 10-5~nBf2. b R-HF: restricted Hartme-Fock, UHF: unrestricted Hartree-Fock, MP-HF: maximally paired Hartree-Fock, SE-HF: spin-extended Hattree-Fock, SO-SCF: spin-optimized self-consistent field, CCSD: coupled-cluster single-, double- excitation approximation.
Their gl/z-factor for the 22S in Li also agree with experiment. Our calculated results of the excited states ls*ns (y1= 3, 4 and 5) for Li are given in Table 3. No definitive comparison with experimental data can be made because we are not aware of any experimental data for the g-factors of these states. From Table 1, it is noted that the convergence character of (HI) and (Ha) is different from that of the energy [ 14-173. For the latter case, the contribution from the second term in the total wave function (9) is only about a part in lo4 of the energy. In the present case, however, the contribution to (IIt) and (Hz) from these partial waves belonging to the second term in Eq. (9) is much larger. For example, their contribution to (HI) is about a tenth of the {HI) value. Especially the wave functions reflecting the core-pol~zation effect, for instance, [ (O,O)O,l ] and [ (0,l) 1,l ] , provide about 10% of total contribution. The contribution from other partial wave functions is only about 1% of the total amount. respectively.
X.-X. Gum, Z.-W Wang/Physics
Letters A 244 (1998) 120-126
125
Table 3 The various corrections
to the g-factor(in
IO-“peB/2),
and the g ,/2-values
of the. excited states Is*ns (n = 3, 4, and 5) for Li.
n
3
4
5
(HI)
-0.90161 0.74594 -0.46723
-0.45314 -0.39466 -0.25 184
-0.27157 0.24374 -0.15712
(Hz) (k)
2.00231144
2.00231618
2.0023 I306
RI/?
Table 4 The various relativistk COmCtiOnS to the g-factors. the g-values as 10h (Hi) li = 1. 2, 3, and 5). IO7 (&f, and 1056g~
24 12 2p3J2
3&z 3&f2 4p,/2 4Q,/2 5PljZ 5412
-5.5826 - 10.0800 -2.3291 -4.3206 -1.2614 -2.3791 -0.7872 -
1so30
- 14.363 5.6800 -6.2459 2.4802 -3.4684
1.3787 -2.2014 -0.8769
8.8651 -3.7195 7.6890 -2.005 I 2.249 1
- 1.0538 I.3701 -0.5745
of the lsZnp states (n = 2, 3, 4, aad 5) for ‘Li; the results ;LTedisplayed
-0.9981
-61.4888
-7.26691
-0.4990 - I .2060 -0.6030
-30.7444
-3.89058 -0.26261 -0.47155 -0.31024
-0.0345 -0.0173 0.2636 -0.1318
-1.6195 -0.8097 -0.6182 -0.3091 -0.5398 -0.2699
0.66582089 I .33406753 0.66589094 1.33410172 0.66589046 1.33410407
-0.23650 -0.21847
0.66S89138
-0.14837
1.33410495
The results of our calculation of various relativistic corrections to the g-factors and the gj-values of the n2P1,2 and n2P3,2 states (n = 2, 3, 4, and 5) for Li are given in Table 4. For the 22Pr,2 and 22P3,2 states,
our predicted values, 0.66582089 and 1.33406753, fall well within the experimental uncertainty, 0.6668(20), 1.335( 10) [22]. We hope that accurate measurements can be carried out for these states of Li so that a critical comparison with theory can be made.
4. Conclusion In this work, we aim at examining whether an accurate prediction of the g-factor is possible for the ground and excited states of the lithium atom by using the FCPC method, which was successfully used to calculate the ionization potential, excitation energy and fine structure of lithium-like systems. These results agree well with experiment and recent theoretical values [ 231. Our calculated g-value of the ground state agrees closely with the experimental data. The results of the excited states, d&,2 (n = 3, 4, and 5), n2P+ and n2P3,2 (n = 3, 4, and 5), are also obtained by using the same method. We hope that these results will stimulate interests in making precision experimental measurements for these states. Recently we have calculated the dipole pol~izabilities of lithium-like systems for Z = 3-50 [ 241, the ionization potentials of lithium-like systems for 2 = 2 I-30 [ 25 ] and the hyperfine structure of lithium-like systems [26] by using the FCPC method. These results indicate that the FCPC method is very effective for calculating the structure and property of systems with a ls2-core.
Acknowledgement
This work is supported by the National Natural Science Foundation of China and the State Education Commission of China. The authors express their gratitude to Professor Kwong. T. Chung for his hospitality.
126
X.-X. Guan, Z.-W Wang/Physics
Letters A 244 (1998) 120-126
References [I] [2] 131 [4] [5] [6] f7] 181
Z.-C. Yan, G.W.F. Drake, Phys. Rev. A 50 (1994) R1980. D.-H. Yang, H. Metculf, Phys. Rev. A 32 (1985) 2249 . D.-H. Yang, P McNichoil, H. Metcalf, Phys. Rev. A 33 ( 1986) 1725. S.A. Lewis, F.M.J. Pichanick, V.W. Hughes, Phys. Rev. A 2 ( 1970) 86. M.L. Lewis, VW. Hughes, Phys. Rev. A 11 (1975) 383. R.C. Isler, S. Marcus, R. Novick, Phys. Rev. 187 (1969) 66. M.L. Lewis, V.W. Hughes, Phys. Rev. A 8 ( 1973) 2845. C.F. Fischer, The Hartree-Fock Method for Atoms: A Numerical Approach (Wiley, New York, 1977); 3481. [9] W.A. Goddard III, Phys. Rev. 157 (1967) 93. [lo] U. Kaldor, F.E. Harris, Phys. Rev. 183 (1969) 1. [I l] W.A. Goddard 111, Phys. Rev. 169 (1968) 120. f 121 E. Lindroth, A. Ynnerman, Phys. Rev. A 47 (1993) 961. [ 131 J.M. Anthony, K.J. Sebastian, Phys. Rev. A 48 (1993) 3792; A 51 (1995) 868. [ 141 K.T. Chung, Phys. Rev. A 44 (1991) 5421. (151 Z.-W. Wang, X.-W, Zhu, K.T. Chung, Phys. Rev. A 46 (1992) 6914. f16] Z.-W. Wang, X.-W. Zhu, K.T. Chung, J. Phys. B 25 (1992) 3915. 1171 Z.-W. Wang, X.-W. Zhu, K.T. Chung, Phys. Ser. 47 (1993) 65. [ 181 AH. Wapstra, G. Audi, Nucl. Phys. A 432 ( 1985) 1. [ 191 W. Perl. Phys. Rev. 91 (1953) 852. [ 201 R.A. Hegstrom, Phys. Rev. A 11 f 1975) 421. [21] V. Veseth, Phys. Rev. A 22 (1980) 803. [22] E. Arimondo, M. Inguscio,, P. Violino, Rev. Mod. Phys. 49 (1977) 31. [23] Z.-C. Yan, G.W.F. Drake, Phys. Rev. Lett. 79 (1997) 1646. 1241 Z.-W. Wang, K.T. Chung, J. Phys. B 27 (1994) 855. 1251 Z.-W. Wang, Z.-M. Ge, Phys. Ser. T 73 (1997) 53. [26] X.-X. Guan, Z.W. Wang, Europ. Phys. J. D (1998) in press.
Phys. Rev.
A 41 (1990)
Physics Letters A 244 (1998) 120-126
Calculation of the Zeeman effect in the n2S1p, IZ~Z$,~,and n2Q2 ( rz = 2, 3, 4, and 5 ) states of the lithium atom X.-X. Guan”, Z.-W. Wangavb a Institute of Atomic and Molecular Physics, Jilin Universily Chungchun 130023. Jib,
China b Department of Mathematics and Physics, Wuyi University, Jiangmen 529020, Guangdong, China
Received 3 November 1997; revised manuscript received 20 April 1998; accepted for publication 20 April 1998 Communicated by B. Fricke
The gJ-factors of the n2S1p n2P1p, and n2P3,2 (n = 2, 3, 4, and 5) states of the lithium atom are calculated by using the full-core plus correlation method. The relativistic correction to order cy2(a.u.) and the effect of the motion of the center of mass are treated by using first-order perturbation theory. For the 22S1,2 state, our result, 2.00230101, agrees more closely with experimental data, 2.00230100(64), than previous theoretical results. The gJ-factor values of other low-lying excited states for Li are also given. @ 1998 Published by Elsevier Science B.V. PACS: 31.20.Di; 31.3O.J~
1. Introduction
The detailed Zeeman structure of the atomic system in an external magnetic field has received considerable attention of experimentalists and theorists. The fine-structure intervals for the excited states of atoms can be derived through investigating the behaviour of atoms in the magnetic field. In the past two decades, many high-precision experimental studies have been carried out for the helium and lithium atoms. The values of the fine-structure splitting in the He 33P state and Li 2*P, and 3*P states were measured to very high accuracy by using the technique of level crossing spectroscopy [ l-71. The precise values of gJ-factors for the corresponding states are very important for measurements with such a high accuracy. On the other hand, the multiconfiguration Hartree-Fock (MCHF) method has been extensively applied for a long time [ 81. There are several types of HF wave functions, including the restricted Hartree-Fock (RHF), the unrestricted Hartree-Fock (UHF), the spinextended Hartree-Fock (SEHF) [ 9,101, the spin-optimized Hartree-Fock (SOHF) [ IO], and the maximally paired Hartree-Fock [ 111. They were used to study the relativistic effect on the g,-factor of Li and other light atoms. For Li and other systems, ab initio calculations of the gl-factor were performed recently by Lindroth and Ynnerman [ 121, who used the relativistic wave function obtained in the coupled-cluster approximation. Recently, Yan and Drake [ 1] have considered the relativistic correction to the gJ-factor of the excited states for the helium atom, by using variational wave functions constructed from doubled Hylleraas-type basis sets. The experimental values of the fine-structure splitting for He 33P~ states have been reanalyzed by using improved 037%9601/98/$19.00 @ 1998 Published by Elsevier Science B.V. All tights reserved. PIISO375-9601(98)00332-6
X.-X. Guan. Z.-M! Wang/Physics Letters A 244 (1998) 120-126
121
gJ-factors. In other words, the accurate theoretical calculation for gf-factors is necessary to extract a more accurate fine-structure splitting. More recently, Anthony and Sebastian have used an approximately relativistic theory of bound states which ensures PoincarC invariance of the atomic system to relative order a2 to derive the Zeeman Hamiltoni~, and calculated the gj-factors of the 9 and P-states for the helium atom [ 133. Lithium is the simplest atomic species, besides helium. It has one ls2-core and a non-paired nl electron in its ground and singly excited states. Recently, a full-core plus correlation (FCPC) method [ 141 has been used to calculate the energy and fine-structure of the excited states for lithium-like systems to high precision [ 15-171. In the FCPC method, a large predete~ined multi~on~guration interaction (CI) ls2-core wave function is frozen as a single term in the total three-electron wave function. The effect of the valence electron is accounted for by multiplying the core wave function with a linear combination of single-particle Slater orbitals. The relaxation of the core and other possible correlation are described by another large CI wave function. In this work, we present the gJ-factors results of the n2St/2, n2Pt,2, and n2Psi2 (n = 2, 3, 4 and 5) states for Li, using the FCPC wave functions. The relativistic correction up to order CX*(a.u.) and the effect of the motion of center of mass are evaluated with the first-perturbation theory. The computation in this work is carried out in atomic units (au.). The nuclear mass is taken from Ref. [ IS]. The isotope used is 7Li. A brief account of the theory is given in Section 2. The calculated results and discussion are presented in Section 3. A conclusion is given in Section 4.
2. Theory
The relativistic correction to the Zeeman Hamiltonian of many-electron systems was derived from the Breit equation by reducing it to the large components [7,19]. The correction from the finite nuclear mass should be also considered. In this work the term of order ~/~ will be included. The total Zeeman Hamiltonian, including the relativistic correction terms up to order o* and the nuclear motion correction of order m/M, can be expressed as
Ht = -a2psB
*C
(1; + 2si) (-%VF> ,
(3)
i=l
H5 = -tpsB.i:('i
Xpj). i+j
Here a! is the tine structure constant, B is the external magnetic field, and pa is the Bohr magneton. He is the lowest-order Zeeman effect, where gr. = 1 - m/M. The small departure of gr, from unity arises from the motion
122
X.-X. Gun,
Z.-R Wang/Physics
Letters A 244 (1998) 120-126
of the nucleus. L = xi Zi and S = Ci si are the total orbital angular momentum and total spin, respectively. Hi is analogous to the relativistic increase of mass. Hz, Hs, and H4 come from the reduction of the Breit equation and they include terms analogous to spin-orbit, spin-other-orbit, and orbit-orbit interactions. Hs includes the correction from the motion of the center of mass where M is the atomic mass. If the gyromagnetic ratio of the free electron, gs, is taken to order I_Y~, namely gs=2(l+&0.32848(;)2+...)
=2.002319304377(9),
(8)
the higher order correction terms of 0( (Ye) and 0( cu2m/M) will be also contained in the final numerical results. According to the FCPC method [ 14-171, the wave function for the three-electron system can be written as (9) where A is an antisymmetrization operator. @lsls is a predetermined ls2-core wave function which is represented by a CI basis set, @1s1s(1,2)
kd&ev(-Pm
= AcC
-
w2)Y1(1.2)x(lr2)
.
(10)
kn,l
The angular part is X(1,2) = c
(ZWZ -m10,0)Sm(~lr(P1)S-m(~2,(P2).
(11)
m ,y( 1,2) is a two-electron singlet spin function. The linear and nonlinear parameters in Eq. ( 10) are determined by optimizing the energy of the He-like 131s system. The factor @“l(3) multiplied with @tsts is a linear combination of the Slater orbitals for the valance electron. The second term on the right-hand side of Eq. (9) describes the core relaxation and the intershell correlation in the three-electron system. The basis function is chosen as
@n(i).l(i)(17293)
(12)
= (Pn(i)(R)$fCR^)XSMs.
The radial and angular basis functions are PPn(i)(R)
(13)
=fi~~eXp(-P,~j)
j=l
and I$$W)
=c
(Zlml’2m21’12ml2)
(‘12ml2’3m3I’M)
fiXj,C’jI
9
(14)
j=l
“tj
where Z(i) represents the set of Ii, Z2,112and 13. This angular component is simply denoted as li=
[(~1,~2)~12,~31
(15)
3
where 1,~ and 13 coupling into L is implicitly implied. Using a similar notation, the doublet spin function can be represented by X&U,=
[(~1,~2)~12,~31
.
(16)
X.-X. &an, Z.-W Wang/Physics Letters A 244 (1998) 120-126
123
It has two possible spin doublets, x1 = [(~l,0,)0,~31 x2=
T
(17)
[(si,s2)1~s31.
(18)
In the present work, these FCPC wave functions for the n*Si,z, n’P,p and n2P3/2 (n = 2, 3, 4, and 5) states of Li are obtained by using the variational method [ 14-171. The mass polarization effect is neglected in variational calculations, so this effect is not included in the FCPC wave functions. In Refs. [ 14-1’71, the mass polarization effect on the energy is taken into account using first-order ~~urbation theory. The expectation value of the Zeeman Hamiltonian Hz is calculated by using our FCPC wave functions. In the LSJ coupling scheme, the gl-factor of the system is given by (19) Here g$@ is obtained from Eq. (211,
and gy’ are obtained from Eqs. (3)-(7),
s:” =&(LSJMI~H,~LSJMJ)
respectively, (i=1,2,3,4,
and5).
(21)
3. Results and discussion The same seven angular component 222 terms are used in the wave function for the Is*-core. For the ls2ns states, we used the wave function with nine terms in the a,,, of Eq. (9). The number of terms in the @,(i),/(i) of Eq. (9) ranges from about 390 to 58 1 terms. For the 1s2np states, we used the same number of terms for the Qp,, in Eq. (9) and about 1004-l 176 terms in the ~~(i),~(~)of Eq. (9). The linear and nonlinear parameters are individually optimized in the energy minimization process [ 14-171. As expected, in order to obtain a more accurate FCPC wave function (9), a larger number of terms is needed with decreasing n (principal quantum number) due to stronger correlation. The expectation values of the Zeeman Hamiltonian Hz for the ls2ns and ls2np states are calculated by using our FCPC wave functions obtained above. Their convergence is rather fast. As a typical example, the convergence of the expectation values for HI and HZ in the ls22s state is shown in Table 1. It is seen from this table that both (Ht) and (HZ) converge to five digits. From the ls*ns states, because L = 0 and the ranks of both H4 and HS are unity, all elements of their reduced matrix must be zero. Only Hi, Hz, and Hs con~ibute to gijz-values of these states with 2S symme~y. In other words, for the 2S1,2 states, the gJ-factor in the present treatment is the same for the isotopes of lithium. Obviously, this conclusion no longer holds for the *PJ states. Our calculated expectation values of HI, Hz, and Hs and gi/z-value for 1~~2s state are given in Table 2. In the literature, most calculations of the 22Si~z gl/z-factor for Li were carried out by using various HF approximations [20,21]. These theoretical results and experimental data are also included in Table 2 for comparison. It appears that our gip-values agree with the experimental data [22] to eight digits. It can be seen clearly from this table that most results of Hegstrom [ 20f fall outside the experimental error bounds. Our calculated g1/2- factor is in excellent agreement with the experimental value recommended by Arimondo et al. Using spin-extended HF and coupled-cluster approximation, Veseth [ 211, Lindroth and Yneerman [ 121 obtained the results, 2.00230086, 2.00230158(20),
124
X.-X. Guan, Z-W Wang/Physics
L.&em A 244 (1998) 120-126
Table i Convergence of the expectation values for Ht, Hz in the 1~~2s state for Li Angular component
No. of term
- WI )”
Vf2jb
core+2s 1(0,0)0,01 Ito, l)l, 11 [(0,2)2.21
9 41 123 93 11 50 32 35 22 34 34 34
0.10853082 0.00901022 0.09416941 0.00040683 -0.~22956 o.OOoO936O 0.00003094 0.00002436 0.~1472 0.~1265 -0.00000946 0.00000580
0.17415461 0.00343575 0.00228714 0.00921272 -0.~1~1 o.OWO4714 0.00001533 0.00001385 o.OOOOO784 0.~21 -0.00000567 O.OOOOQ28.5
20 20 32
0.00000425 0.~265 o.OOOOOO95
0.00000203 0.~147 O.OOOOOO88
590
0.12206818
0.18016614
I(O, O)O,Ol’ t(o,3)3.31 t(o,4)4,41 [(1,2)1,11 [(0,1)1,11” 1(0,5)5,51 I(l.O)l, 11
[ (0,6)6,61 [(l,2)1*IIC ((2,3)1,11 I(O, 2% 21C
Total
a In (2a2&Z). b In (a*&&). c In these angular components the spins of the first two electrons couple into a triplet. Table 2 Comparison of the valuesa of (Hi),
(Hz),
(ff3), gl,q of tbe ls22s state for Li
Reference
Metbodb
(HI)
(H2)
(H3)
a/2
Hegstrom Hegstrom Hegstram Hegstrom Veseth Lindroth et al. this work
R-HF U-HF MP-HF SO-SCF SE-HF CCSD FCPC
-2.2240 -2.5283 -2.4173 -2.4930 -2.6125
1.8392 1.8968 1.8740 1.8881 1.9134
-1.1274 -1.1359 -1.1303 -1.1324 -1.1392
-2.6001
I,.9188
-1.1469
2.tW230416 2.00230161 2.00230255 2.00230192 2.00230086 2.~30158(20) 2.00230101
Arimondo et al.
experiment
2.00230100(64)
a Tbe values of (Ht), (Hz}, (Hs) are in 10-5~nBf2. b R-HF: restricted Hartme-Fock, UHF: unrestricted Hartree-Fock, MP-HF: maximally paired Hartree-Fock, SE-HF: spin-extended Hattree-Fock, SO-SCF: spin-optimized self-consistent field, CCSD: coupled-cluster single-, double- excitation approximation.
Their gl/z-factor for the 22S in Li also agree with experiment. Our calculated results of the excited states ls*ns (y1= 3, 4 and 5) for Li are given in Table 3. No definitive comparison with experimental data can be made because we are not aware of any experimental data for the g-factors of these states. From Table 1, it is noted that the convergence character of (HI) and (Ha) is different from that of the energy [ 14-173. For the latter case, the contribution from the second term in the total wave function (9) is only about a part in lo4 of the energy. In the present case, however, the contribution to (IIt) and (Hz) from these partial waves belonging to the second term in Eq. (9) is much larger. For example, their contribution to (HI) is about a tenth of the {HI) value. Especially the wave functions reflecting the core-pol~zation effect, for instance, [ (O,O)O,l ] and [ (0,l) 1,l ] , provide about 10% of total contribution. The contribution from other partial wave functions is only about 1% of the total amount. respectively.
X.-X. Gum, Z.-W Wang/Physics
Letters A 244 (1998) 120-126
125
Table 3 The various corrections
to the g-factor(in
IO-“peB/2),
and the g ,/2-values
of the. excited states Is*ns (n = 3, 4, and 5) for Li.
n
3
4
5
(HI)
-0.90161 0.74594 -0.46723
-0.45314 -0.39466 -0.25 184
-0.27157 0.24374 -0.15712
(Hz) (k)
2.00231144
2.00231618
2.0023 I306
RI/?
Table 4 The various relativistk COmCtiOnS to the g-factors. the g-values as 10h (Hi) li = 1. 2, 3, and 5). IO7 (&f, and 1056g~
24 12 2p3J2
3&z 3&f2 4p,/2 4Q,/2 5PljZ 5412
-5.5826 - 10.0800 -2.3291 -4.3206 -1.2614 -2.3791 -0.7872 -
1so30
- 14.363 5.6800 -6.2459 2.4802 -3.4684
1.3787 -2.2014 -0.8769
8.8651 -3.7195 7.6890 -2.005 I 2.249 1
- 1.0538 I.3701 -0.5745
of the lsZnp states (n = 2, 3, 4, aad 5) for ‘Li; the results ;LTedisplayed
-0.9981
-61.4888
-7.26691
-0.4990 - I .2060 -0.6030
-30.7444
-3.89058 -0.26261 -0.47155 -0.31024
-0.0345 -0.0173 0.2636 -0.1318
-1.6195 -0.8097 -0.6182 -0.3091 -0.5398 -0.2699
0.66582089 I .33406753 0.66589094 1.33410172 0.66589046 1.33410407
-0.23650 -0.21847
0.66S89138
-0.14837
1.33410495
The results of our calculation of various relativistic corrections to the g-factors and the gj-values of the n2P1,2 and n2P3,2 states (n = 2, 3, 4, and 5) for Li are given in Table 4. For the 22Pr,2 and 22P3,2 states,
our predicted values, 0.66582089 and 1.33406753, fall well within the experimental uncertainty, 0.6668(20), 1.335( 10) [22]. We hope that accurate measurements can be carried out for these states of Li so that a critical comparison with theory can be made.
4. Conclusion In this work, we aim at examining whether an accurate prediction of the g-factor is possible for the ground and excited states of the lithium atom by using the FCPC method, which was successfully used to calculate the ionization potential, excitation energy and fine structure of lithium-like systems. These results agree well with experiment and recent theoretical values [ 231. Our calculated g-value of the ground state agrees closely with the experimental data. The results of the excited states, d&,2 (n = 3, 4, and 5), n2P+ and n2P3,2 (n = 3, 4, and 5), are also obtained by using the same method. We hope that these results will stimulate interests in making precision experimental measurements for these states. Recently we have calculated the dipole pol~izabilities of lithium-like systems for Z = 3-50 [ 241, the ionization potentials of lithium-like systems for 2 = 2 I-30 [ 25 ] and the hyperfine structure of lithium-like systems [26] by using the FCPC method. These results indicate that the FCPC method is very effective for calculating the structure and property of systems with a ls2-core.
Acknowledgement
This work is supported by the National Natural Science Foundation of China and the State Education Commission of China. The authors express their gratitude to Professor Kwong. T. Chung for his hospitality.
126
X.-X. Guan, Z.-W Wang/Physics
Letters A 244 (1998) 120-126
References [I] [2] 131 [4] [5] [6] f7] 181
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