Hindawi Publishing CorporationJournal of Atomic, Molecular, and Optical PhysicsVolume 2012, Article ID 569876, 6 pagesdoi:10.1155/2012/569876

Research Article

Energies, Fine Structures, and Hyperfine Structures ofthe 1s22snp 3

P (n = 2–4) States for the Beryllium Atom

Chao Chen

School of Physics, Beijing Institute of Technology, Beijing 100081, China

Correspondence should be addressed to Chao Chen, chen chao@tsinghua.org.cn

Received 12 July 2012; Revised 15 August 2012; Accepted 27 August 2012

Academic Editor: Derrick S. F. Crothers

Copyright © 2012 Chao Chen. This is an open access article distributed under the Creative Commons Attribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Energies and wave functions of the 1s22snp 3P (n = 2–4) states for the beryllium atom are calculated with the full-core plus

correlation wave functions. Fine structures and hyperfine structures are calculated with the first-order perturbation theory. Forthe 1s22s2p 3P state, the calculated energies, fine structure, and hyperfine structure parameters are in good agreement with thelatest theoretical and experimental data in the literature; it is shown that atomic parameters of the low-lying excited states forthe beryllium atom can be calculated accurately using this theoretical method. For the 1s22snp 3P (n = 3, 4) states, the presentcalculations may provide valuable reference data for future theoretical calculations and experimental measurements.

1. Introduction

In recent years, studies of energies, fine structures, andhyperfine structures of the low-lying excited states for theberyllium atom [1–10] have been of great interest to spectro-scopists because there are many strong optical transitionssuitable for spectral and hyperfine structure measurements.On the other hand, studies of the low-lying excited statesfor the beryllium atom play an important role in developingthe excited state theory of multielectron atoms and betterunderstanding the complicated correlation effects betweenelectrons. The fine structure comes from the spin-orbit,spin-other-orbit, and spin-spin interactions. The hyperfinestructure of atomic energy levels is caused by the interactionbetween the electrons and the electromagnetic multipolemoments of the nucleus. The leading terms of this inter-action are the magnetic dipole and electric-quadrupolemoments. The fine and hyperfine structure is sensitive to thecorrelation effects among electrons. Experimentally, someproperties of the atomic nucleus can be obtained by investi-gating the hyperfine structure of the atomic energy levels.The nuclear electric-quadrupole moment, which is difficultto measure directly with nuclear physics techniques, can bedetermined using the measured hyperfine structure and theaccurate theoretical results.

The 1s22s2p 3P state of the beryllium atom is of interest

since it is the lowest excited state in which hyperfine effects

can occur, and the ground state has no hyperfine splittingbecause it is J = 0. It is generally a very demanding task tocalculate hyperfine structure accurately. Polarization of theclosed shells in the 1s2 core, due to the Coulomb interactionwith open shells, can have a large effect on the hyperfinestructure. Up till now, the most sophisticated theoreticalcalculations of the hyperfine structure parameters for the1s22s2p 3

P state of the Be atom have been carried out usinglinked-cluster many-body perturbation (LC MBPT) theory[5, 6], Hartree-Fock and CI allowing all SD excitations tocorrelation orbitals of Slater type by Beck and Nicolaides [7],as well as multiconfiguration Hartree-Fock (MCHF) method[8, 9]. Experimentally, the magnetic dipole and electric-quadrupole hyperfine constants have been determined veryaccurately with the atomic-beam magnetic-resonance tech-nique [10] for the 1s22s2p 3

P state in beryllium. To the bestof our knowledge, few results on energies, fine structures,and hyperfine structures have been investigated for the1s22snp 3

P (n ≥ 3) states of the beryllium atom due to therestriction of resolution from experiments and the numericalunsteadiness in theoretical calculations.

An elegant and complete variation approach, namely,the full core plus correlation (FCPC) method, has beendeveloped by Chung [11, 12]. This method has been suc-cessfully applied to three- and four-electron systems, withthe 1s2 core. Many elaborate calculations, especially for

2 Journal of Atomic, Molecular, and Optical Physics

the dipole polarizabilities [13], quadrupole and octupolepolarizabilities [14], and total atomic scattering factors [15],show that FCPC wave functions have a reasonable behav-ior over the whole configuration space for three-electronsystems. This method has also been used to calculate thehyperfine structure of the 1s2ns 2S and 1s2np

2P states (n =

2–5) for the lithium isoelectronic sequence; the results arein good agreement with the Hylleraas calculations and withthe experiment data [16]. As is well known, theoreticalcalculations of the hyperfine structure parameters dependsensitively on the behavior of the wave function in the prox-imity of the nucleus. In addition, core polarization effectsfor the low l states need to be included in the nonrelativisticwave function. It would be interesting to find out whetherthe FCPC wave function can also be successful for calcu-lating hyperfine structure parameters of low-lying excitedstates for the beryllium atom. In this work, the FCPC wavefunctions are carried out on the 1s22snp 3

P (n = 2–4) statesof the beryllium atom. The energies, fine structures, andhyperfine structures are calculated and compared with thedata available in the literature. The purpose of this work is toexplore the capacity of the FCPC wave function to calculatethe atomic parameters of the low-lying excited states for theberyllium atom and provide more reliable theoretical data tostimulate further experimental measurements.

2. Theory

According to the FCPC method [11, 12], the wave functionfor the four-electron 1s22snp 3P state can be written as

Ψ(1, 2, 3, 4)

= A

⎡⎣Φ1s1s(1, 2)Φ2snp(3, 4)

+∑

i

CiΦn(i),l(i)(1, 2, 3, 4)

⎤⎦,

(1)

where A is an antisymmetrization operator. Φ1s1s is apredetermined 1s2-core wave function which is representedby a CI basis set,

Φ1s1s(1, 2) = A∑

k,n,l

Cknlrk1 r

n2 exp

(−βlr1 − ρlr2)Yl(1, 2)χ(1, 2),

(2)

the angular part is

Yl(1, 2) =∑m

〈l,m, l,−m | 0, 0〉Ylm(θ1,ϕ1

)Yl−m

(θ2,ϕ2

).

(3)

χ (1,2) is a two-electron singlet spin function. The linear andnonlinear parameters in (2) are determined by optimizingthe energy of the two-electron core. The factor Φ2snp(3, 4)

represents the wave function of the two outer electrons whichis given by

Φ2snp(3, 4) = A∑

k,n,l

dknlrk3 r

n4 exp

(−λlr3 − ηlr4)Yl(3, 4)χ(3, 4),

(4)

the angular part is

Yl(3, 4) =∑m

〈l,m, l + 1,−m | 0, 0〉Ylm(θ3,ϕ3

)Yl+1−m

(θ4,ϕ4

).

(5)

The latter wave function of (1) describes the core relaxationand the intrashell electron correlation in the four-electronsystem. It is given by

Φn(i),l(i)(1, 2, 3, 4) = ϕn(i),l(i)(R)YLMl(i) (Ω)χSSZ , (6)

where

ϕn(i),l(i)(R) =4∏

j=1

rnj

j exp(−αjr j

). (7)

A different set of αj is used for each l(i). The angular part is

YLMl(i)

(R)=∑mj

〈l1l2m1m2 | l12m12〉

× 〈l12l3m12m3 | l123m123〉

× 〈l123l4m123m4 | LM〉4∏

j=1

Yljmj

(Ω j

).

(8)

To simplify notation, this angular function is simplydenoted as

l(i) = [(l1, l2)l12, l3]l123, l4, (9)

with the understanding that l123 and l4 couple into L, thetotal orbital angular momentum. There are three possiblespin functions for the 1s22snp 3

P state, namely,

χ1 = [(s1, s2)0, s3]12

, s4,

χ2 = [(s1, s2)1, s3]12

, s4,

χ3 = [(s1, s2)1, s3]32

, s4.

(10)

For the radial basis functions of each angular-spin com-ponent, a set of linear and nonlinear parameters is chosen.These parameters are determined in the energy optimiza-tion process. For each set of l1, l2, l3, and l4, we try all possiblel(i) and χ and keep the ones which make significant contri-bution to the energy in (1).

The fine structure perturbation operators [1, 2] are givenby

HFS = Hso + Hsoo + Hss, (11)

Journal of Atomic, Molecular, and Optical Physics 3

where the spin-orbit, spin-other-orbit, and spin-spin opera-tors are

HSO = Z

2c2

4∑

i=1

⇀l i · ⇀sir3i

,

HSOO = − 12c2

4∑

i, j=1i /= j

[1r3i j

(⇀ri − ⇀

rj)×⇀pi

]·[⇀si + 2

⇀sj]

,

HSS = 1c2

4∑

i, j=1i< j

1r3i j

⎡⎢⎣⇀si · ⇀sj −

3(⇀si · ⇀ri j

)(⇀sj · ⇀ri j

)

r2i j

⎤⎥⎦.

(12)

To calculate the fine structure splitting, the LSJ couplingscheme is used:

ΨLSJJZ =∑

M,SZ

〈LSMSZ | JJZ〉ΦLSMSZ . (13)

The fine structure energy levels are calculated by first-order perturbation theory

(ΔEFS)J =⟨ΦLSJJZ

∣∣∣Hso + Hsoo + Hss

∣∣∣ΦLSJJZ

⟩. (14)

For an N-electron system, the hyperfine interactionHamiltonian can be represented as follows [17, 18]:

Hh f s =∑

k=1

T(k) ·M(k), (15)

where T(k) and M(k) are spherical tensor operators of rankk in the electronic and nuclear spaces, respectively. The k =1 term represents the magnetic-dipole interaction betweenthe magnetic field generated by the electrons and nuclearmagnetic dipole moments, and the k = 2 term the electricquadrupole interaction between the electric-field gradientfrom the electrons and the nonspherical charge distributionof the nucleus. The contributions from higher-order termsare much smaller and can often be neglected.

In the nonrelativistic framework, the electronic tensoroperators, in atomic units, can be written as

T(1) = α2

2

4∑

i=1

[2glr−3

i l(1)i −√10gs

{s(1)i C(2)

i

}(1)r−3i

+8π3gss

(1)i δ(ri)

],

T(2) = −4∑

i=1

r−3i C(2)

i ,

(16)

where gl = (1 − me/M) is the orbital electron g factor, andgs = 2.0023193 is the electron spin g factor. M is the nuclear

mass. The tensorC(2)i is connected to the spherical harmonics

Ylm(i) by

C(l)m =

√4π2l

+ 1Ylm. (17)

The hyperfine interaction couples the electronic angularmomenta J and the nuclear angular momenta I to a totalangular momentum F = I + J. The uncoupling and couplinghyperfine constants are defined in atomic units as [17, 18]:

aC =⟨γLSMLMS

∣∣∣∣∣∣

N∑

i=1

8πδ3(ri)s0(i)

∣∣∣∣∣∣γLSMLMS

⟩

(Fermi contact),

aSD =⟨γLSMLMS

∣∣∣∣∣∣

N∑

i=1

2C(2)0 (i)s0(i)r−3

i

∣∣∣∣∣∣γLSMLMS

⟩

(Spin dipolar

),

al =⟨γLSMLMS

∣∣∣∣∣∣

N∑

i=1

l0(i)r−3i

∣∣∣∣∣∣γLSMLMS

⟩

(orbital),

bq =⟨γLSMLMS

∣∣∣∣∣∣

N∑

i=1

2C(2)0 (i)r−3

i

∣∣∣∣∣∣γLSMLMS

⟩

(electric quadrupole

),

AJ = μII

1

[J(J + 1)(2J + 1)]1/2

⟨γJ∥∥∥T(1)

∥∥∥γJ⟩

,

AJ−1,J = μII

1

[J(2J − 1)(2J + 1)]1/2

⟨γJ − 1

∥∥∥T(1)∥∥∥γJ⟩

,

BJ = 2Q[

2J(2J − 1)(2J + 1)(2J + 2)(2J + 3)

]1/2⟨γJ∥∥∥T(2)

∥∥∥γJ⟩

,

(18)

where ML = L and MS = S. In these expressions, μI isthe nuclear magnetic moment and Q is the nuclear electricquadrupole moment. I is the nuclear spin, and J is the atomicelectronic angular moment.

3. Results and Discussions

In order to achieve accurate calculation results for variousproperties of the low-lying excited states for the berylliumatom, the choice of basis function with sufficiently highquality is critical and it is our major concern. The sevenl components (0,0), (1,1), (2,2), (3,3), (4,4), (5,5), (6,6)altogether 159 terms are used for the 1s2 core. The Φ2snp

in (1) has four angular components, l is summed from 0to 3 with the angular components (0,1), (1,2), (2,3), and(3,4), and the number of terms in Φ2snp ranges from 36 to15. Most of the other correlation effects are included in (6),which accounts for the intershell as well as the intrashellcorrelations. Many relevant angular and spin couplings areimportant for the energy, these basis functions are triedto include in Φn(i),l(i) (1, 2, 3, 4) with significant energycontribution. For each set of orbital angular momenta l1, l2,l3, and l4, there could be several ways to couple this set intothe desired total orbital angular momentum. In this work,

4 Journal of Atomic, Molecular, and Optical Physics

Table 1: Nonrelativistic energies of the 1s22snp 3P (n = 2–4) states for the beryllium atom (in a.u.).

This work Hibberta Weissb

1s22s2p 3P −14.56637 −14.5184 −14.51844

1s22s3p 3P −14.39839 −14.3510 −14.35106

1s22s4p 3P −14.36248 −14.31530aReference [1].

bReference [2].

Table 2: Fine structure splittings νJ–J of the 1s22snp 3PJ (n = 2–4) states for the beryllium atom (in cm−1).

1s22s2p 3PJ 1s22s3p 3PJ 1s22s4p 3PJ

ν2–1 2.36 0.35 0.13

Experimenta 2.35 (.2)

Other theoryb 2.53

ν1–0 0.64 0.092 0.034

Experimenta 0.64 (.1)

Other theoryb 0.71aReference [3].

bReference [4].

for 1s22snp 3P states, the important angular series (l1, l2, l3,l4) are (0, 0, l, (l+1)), (0, 1, l, l), (l, l, 0, 1), and so forth. In bothcases, the value of l is from 0 to 6, as the energy contributionfrom set with l > 6 is small and negligible. In order to getthe high-quality wave function, the number of angular-spincomponents in the Φn(i),l(i) wave functions ranges from 15 to66, and the number of terms in the Φn(i),l(i) of (6) is about790. The linear and nonlinear parameters are individuallyoptimized in the energy minimization process. Using theRayleigh-Ritz variational method, the basic wave function Ψand the corresponding eigenvalue E are determined.

Nonrelativistic energies of the 1s22snp 3P (n = 2–4)

states for the beryllium atom are given in Table 1. As Table 1shows, for the 1s22snp (n = 2, 3) 3P states, the nonrelativisticenergies in this work are lower and better than those ofHibbert and Weiss [1, 2], the improvement ranging from0.0479 a.u. to 0.0473 a.u. Hibbert and Weiss reported a setof large-scale configuration interaction (CI) calculations forthe 1s22snp (n = 2, 3) 3P states, which can give an accurateapproximation for each state, but it may tend to obscure theglobal picture of the spectrum which is so transparent inthe other approach. The work of Hibbert and of Weiss didnot include any intrashell correlation in the 1s shell, as thecalculations were of transitions in the outer subshells. Thecorrelation energy of the 1s shell is almost independent of thenuclear charge and also of the number of additional electronsoutside the 1s shell. For Be, it is about 0.0457 a.u. and thisaccounts for the main difference between earlier work andthe present; more accurate results are presented in Table 1. Ofcourse, for the calculation of hyperfine parameters, correla-tion within the 1s shell is crucial in obtaining accurate hyper-fine parameters, and this has been achieved in the presentwork. For the 1s22s4p 3

P state, the present calculation fromthe FCPC method is also lower than the result of Weiss [2].

If including the effects of the spin-orbit, spin-other-orbit,and spin-spin interactions, the energies of the fine structure

resolved J levels are obtained. In this work, the fine structuresplittings of the triplet states are calculated with theHso,Hsoo,and Hss operators using the first-order perturbation theory.Table 2 gives the fine-structure splittings of the 1s22snp 3PJ(n = 2–4) states for the beryllium atom. The experimentalBe 2s2p 3PJ splittings are 2.35 (J = 2 → 1) and 0.64(J = 1 → 0) cm−1 [3]. They agree with our prediction2.36 and 0.64 cm−1. Although many theoretical studies havebeen done on the BeI excited systems, the published theore-tical fine structure results are scarce. One exception is Laugh-lin, Constantinides, and Victor [4]. They use a model poten-tial calculation and predict the splittings to be 2.53 and0.71 cm−1 for the 1s22s2p 3

PJ state, which should be con-sidered as quite good in view of the simplicity in theircomputation and fall in experimental uncertainties. Pre-sent calculations for this state are more accurate due to cor-relation effect well described in this method. For the experi-ment, the splitting of 1s22s3p 3

PJ (J = 1, 0) is not resolved.But the splitting from the J = 2 state to the J = 1, 0 is deter-mined to be 0.37 cm−1. In this work, the calculated splittingsare 0.35 (J = 2 → 1) and 0.092 (J = 1 → 0) cm−1. Thisimplies that the predicted splitting from J = 2 to the centerof gravity of J = 1 and 0 should be 0.373 cm−1. It agrees withthe experiment. The good agreement with experiment couldbe used as the indication of the accuracy of the wave functionconstructed here. For the 1s22s4p 3PJ state, our calculatedsplittings are hoped to offer reference for further experi-mental measurements.

The hyperfine structure parameters of the 1s22snp3P(n = 2–4) states for the beryllium atom are calculated inthis work: Fermi contact ac, the spin dipolar aSD, the orbitalal, and the electric quadrupole bq. In the present calculation,Q = 0.0530b, μI = −1.177492 nm, I = 1.5 for Be are takenfrom [19]. The hyperfine interaction in the 1s22s2p 3

P statefor the beryllium atom is of interest since it is the lowestexcited state in which hyperfine effects can occur, which has

Journal of Atomic, Molecular, and Optical Physics 5

Table 3: The hyperfine structure parameters (in a.u.) and coupling constants (in MHz) of the 1s22s2p 3P state for the beryllium atom.

Method ac asd al bq A2 A1 Reference

LC MBPTa 9.2319 −0.06490 0.30478 −0.1156 −124.21 [5, 6]

HF + SDCIb 9.2738 −0.06656 0.30014 −0.1097 −124.76 −139.77 [7]

FE MCHFc 9.2349 −0.06564 0.30261 −0.1150 [8]

MCHF 9.2416 −0.06587 0.30329 −0.11570 −124.50 −139.35 [9]

This work 9.2436 −0.06523 0.30201 −0.11588 −124.51 −139.36

Experiment −124.5368 −139.373 [10]aLinked-cluster many-body perturbation theory.

bHartree-Fock and CI allowing all SD excitations to correlation orbitals of Slater type.cFinite-element multiconfiguration Hartree-Fock.

Table 4: The hyperfine structure parameters (in a.u.) and coupling constants (in MHz) of the 1s22snp 3P (n = 3, 4) states for the berylliumatom.

ac asd al bq A2 A1

1s22s3p 3P 12.029 −0.00898 0.04276 −0.01788 −151.59 −153.61

1s22s4p 3P 12.118 −0.00338 0.01610 −0.00686 −151.92 −152.68

been studied over the past four decades [5–10]. Table 3 givesthe hyperfine structure parameters of the 1s22s2p 3

P statefor the beryllium atom through the FCPC wave function tocompare with data in the literature. As can be seen fromTable 3, the present results for hyperfine structure para-meters are better than the earlier theoretical results [5–7] inwhole. The present calculations also agree with the results byFE MCHF (finite-element multiconfiguration Hartree-Fock)method [8] to two significant figures. The calculated Fermicontact term ac in this work differs from the results from thelatest calculation through MCHF method [9] by only 0.07%,and the differences for the other terms are on the order of afew parts in a thousand. This means that the wave functionused in the present work is reasonable and accurate in thefull configuration space. The hyperfine coupling constantsAJ are also listed in Table 3 to compare with results fromother calculations and experiments. Our calculated hyperfinecoupling constants agree perfectly with the experimentalvalue [10] to four significant figures. That is also true for theMCHF calculation of [9]. It is shown that hyperfine structureparameters of the low-lying excited states for the berylliumatom can be calculated accurately using the present FCPCwave function. For the 1s22snp 3

P (n = 3, 4) states, to the bestof our knowledge, there is no report on hyperfine structureparameters in the literature. The present predictions for thehyperfine structure parameters and coupling constants arelisted in Table 4, which may provide valuable reference datafor other theoretical calculations and experimental measure-ments in future.

4. Summary

In this work, energies, fine-structure splittings, and hyperfinestructure parameters of the 1s22snp 3

P (n = 2–4) statesfor the beryllium atom are calculated with the FCPC wavefunctions. The obtained nonrelativistic energies are muchlower than the previous published theoretical values. The cal-culated fine structure splittings are in good agreement with

experiment. For the 1s22s2p 3P state, the calculated hyperfine

structure parameters are in good agreement with the latesttheoretical and experimental data in the literature; it is shownthat hyperfine constants of the low-lying excited states forthe beryllium atom can be calculated accurately using thiskind of wave function. For other states, the present predictedhyperfine structure parameters may provide valuable refer-ence data for future theoretical calculations and experimentalmeasurements.

Acknowledgments

The author is grateful to Dr. Kwong T. Chung for his com-puter code. The work is supported by National Natural Sci-ence Foundation of China and the Basic Research Founda-tion of Beijing Institute of Technology.

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Journal of

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Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Soft MatterJournal of

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AerodynamicsJournal of

Volume 2014

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PhotonicsJournal of

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Journal of

Biophysics

Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

ThermodynamicsJournal of