張小景，張永慧，張現(xiàn)周

(河南師范大學(xué)物理與電子工程學(xué)院，新鄉(xiāng)453007)

1 Introduction

At present，researches on hydrogen atom and hydrogen-like ions have been very mature using Dirac equation. Many methods［1-4］have been used to calculate the atomic properties，especially the basis sets expansion methods［5-7］. The ground energies of atoms have been accurately computed. In these studies，it usually regards the interaction between electron and nucleus as Coulomb interaction between two point charges ideally［8］. The Point Charge Distribution Model (PCDM)is the simplest nuclear model，which owns simple formalism for charge distribution，and in which the dynamic equation can be easily solved analytically. However，the PCDM has its limitation. For example，the Coulomb potential contains singularities which lead the wavefunctions to divergence as r →0.It is well known that the nucleus has a finite volume，so its charge also has a certain spatial distribution.Especially the ground state is sensitive to the shape of nucleus in solving the Dirac equation. The problem can be eliminated by using a finite nuclear charge distribution. Therefore， nucleus charge distribution should be considered in calculations in order to acquire high accuracy.

There are three finite nuclear models being commonly used at present:the Homogeneous Sphere Charge Distribution Model (SCDM)，the Gaussian Charge Distribution Model (GCDM)and the Fermi Charge Distribution Model (FCDM). The formalisms of the SCDM and GCDM are simpler than that of the FCDM，although the FCDM is better in describing heavier nuclei. Refs. ［3，9，10］have presented some results of hydrogen -like system and show that the potential，induced by the SCDM，is uniform within the sphere. So the GCDM is the most appropriate one for analyzing the atomic properties in the near nuclear region. In addition，the GCDM is not only appropriate in atomic calculations，but also applicable in molecular calculations.

The potentials in different models have different influences on physical quantity evaluation，especially in the near nuclear region. In the present paper，it is found that when Z is 100，the relative correction between the PCDM and GCDM is up to 10-3. This will affect the results of other physical quantities，such as oscillator strength，polarizability，hyperpolarizability，and so on. Previous work［11，12］show that the nuclear charge distribution effect should not be neglected for heavy atoms where the electron is nearby the nucleus.However，these work rarely involve hydrogen - like systems. The corresponding work that with Gaussian basis sets have been done in Ref.［6］which gives stable eigenstates with positive energy only and in Refs.［11，13］which focus on multi - electron atoms. In Ref. ［5］Gaussian basis sets is also used in calculations.

Moreover，B-splines have many desirable properties［14，15］:it is made of a relatively small number of basis functions，which is complete，with a negligible local independence;the functions approximated by the basis are continuously adjustable，results are numerically stable，and it is flexible due to the freedom distribution of knots. It has been widely used in atomic and molecular calculations［7，16-18］. Refs. ［19，20］ show the results of polarizabilities using B -splines，which is more precise than any previous calculation. B-splines are also applied in hydrogenicdonor states investigation［21］，and have been successfully adopted to calculate the long - range problems between electron and nucleus. To our knowledge，in the GCDM，there are few work adopting B splines to calculate the short - range interaction of hydrogen -like systems. We will study the energy levels and wavefunctions of hydrogen-like systems based on the B-splines basis sets in the GCDM.

Owing to the complexity of the potential energy in the GCDM，compared with that in the PCDM，we solve the Dirac equation numerically only. Notre Dame (ND )［7］and Dual Kinetic Balance(DKB)［22，23］sets based on B -splines basis set are adopted to solve the Dirac equation high precisely.From Ref. ［24］，we know that 1s state is most sensitive to the finite nuclear charge distribution effects，therefore we only compute and discuss the behavior of energy and wavefunction in the near nucleus region.

The structure of the paper is listed as follows:In Sec.2 we make a detailed description for the theoretical method，and present the numerical results and investigate the physical factors behind the data in Sec.3. Then，we conclude in Sec.4. All the work is done in atomic units (a. u. )，unless otherwise stated.

2 Theoretical method

2.1 Gaussian Charge Distribution Model

In the GCDM，the Dirac Hamiltonian for hydrogen-like ions is HD= cα·p + βc2+ V(r)，where V(r)is the nuclear potential. The potential energy can be obtained as，

It is consistent with that in Ref. ［11］by making a transformation ξ=1/2σ2. The parameter of Gaussian charge distribution is defined as

where〈R20〉1/2=(0.836A1/3+0.5%)ξ fm is the nuclear radial size，A is the mass number. Note that，when r is large enough，greater than several σ，the error function approaches 1 and the potential is equivalent to the one in the PCDM;nearly there is no difference between the two models at this region.However，the region close to nucleus is more interesting，where we pay more attention. In the PCDM，the Coulomb potential that electron suffers from nucleus is dramatically decreased near the nucleus，and tends to be negative infinite. Unlike the point -like distribution，the Gaussian one is a continuous distribution，which is smooth and convergent near the nucleus.The potential energy curves of the ion Z = 50 based on the two models in the vicinity of nucleus is presented in Fig.1 (a).

Fig.1 (a)The potential energies in the GCDM and the PCDM near the nucleus with Z = 50;(b)The first and the second derivatives V '(r)and V ″(r)of the potential energy function V (r)in the GCDM

The wavefunction can be written as

where Pnk(r)and Qnk(r)are the large and small components of radial wavefuncion，Ω±km(^r)represents the spherical spinors，n and k=l(l+1)-j(j+1)-1/4 are principal quantum number and angular quantum number，respectively. The matrix style of Pnk(r)and Qnk(r)is

In our work，we will use the ND and DKB basis sets to expand the radial wavefunction Pnk(r)and Qnk(r)，respectively.

The first and second derivations of V(r)in the GCDM are also continuous in the near - nuclear region，shown in Fig.1 (b).As a result，the second derivative of the wavefunction would not have a cusp at the nuclear boundary.

2.2 Numerical approach

In the numerical computation，we adopt the basis set constructed from B-splines. The wavefunction varies rapidly near the nucleus，so we need to put more B splines in this region to achieve a more accurate approximation of wavefunction in terms of B splines. The B splines are“bound”by the knots，so the density of B splines is decided by the density of knots. In our calculation，following deBoor［25］，we adopt the exponential knot distribution in order to put more knots (and then B splines)into this important region. The whole cavity［0，rmax］is divided into n segments.

where γ is a parameter of breakpoint sequence that manipulates the density of the breakpoints. A representative set of B splines is shown in Fig.2. In our calculations，B splines of order 7 are used and the radius of the cavity is 40 a. u.

Fig.2 The twenty B spline curves of order k=7 in interval［10 -6，10 -3］. The parameter γ=15. The knot sequence is distributed exponentially. BS denotes Bspline

The radial wavefunction Ψ(r)is described by the large component Pnk(r)and small component Qnk(r). In numerical evaluation，it can be expanded by a finite basis set. In this paper，both the ND and DKB basis sets are adopted to cross check the results.

In the DKB basis set scenario，several B -splines are removed to ensure the continuity of the wavefunction at the boundary of the cavity. In the ND basis set scenario，we apply the MIT - bag model boundary condition［7］.

3 Results and discussion

Table 1 represents the convergence of the ground energies of the ion Z = 20 as the basis set size N increases，where the radius of the cavity is 5 a. u and γ=6.8955. In our calculations，in which only the ground states of hydrogen-like ions are studied，the radius of the cavity is chosen as r = 40 /Z［15］and the results are obtained at N = 50. Experimental values of hydrogen-like systems with Z ≤28 are reported in Ref. ［26］. We calculate the ground energies of several ions to compare with the experimental measurements. The ND and DKB basis sets are adopted respectively to calculate the energies of these hydrogen-like ions in the PCDM and in the GCDM.The results are presented in Table 2.

The energies in the PCDM and GCDM are nearly equal in principle when the atom is light，both using the ND and DKB basis set. As Z increases，difference appears between the values in the two models，and the difference is more obvious as the atom is heavy enough. Comparing the calculated results withthe experimental ones，one may find that the data in the GCDM are more accurate，since they are more close to the experimental data. From the theoretical point of view，nucleus in the GCDM has a weaker ability attracting the electron，because of the diffusion of its charge to a space extent. So the energy level in the GCDM is bigger than the one in the PCDM，whichagrees with our results very well. For the high - Z scenario，we only calculate energies using the ND basis set in the two models. Some representative values are listed in Table 3.

Table 1 Convergence of the ground energies of the ion Z = 20 as the basis set size N increases

Table 2 Comparison of the ground energies of hydrogen-like atoms obtained by adopting the ND and DKB basis sets，Theoretical values in the PCDM (TPCDM)and the relevant experimental data

TEPointdenotes the theoretical values of PCDM.△E/EPointis the relative correction of the ground state energy induced by the nuclear charge distribution.The shift increases as the nuclear charge increases.With the increase of the atomic number，the spacial charge density of nucleus increases and the electron is getting closer to nucleus. As a result，the two models have different potential energies，which induces the bigger shift. Therefore the nuclear charge distribution effect should be considered in the high atomic number situation. In such way，the results approach experimental values better. This effect is also demonstrated in wavefunctions.

Fig. 3 shows the behavior of wavefunctions in the near nucleus region. The curves behave steeply in the PCDM，while in the GCDM they are gentle，owing to the difference of potential energy in that area.Besides，the wavefunctions in the GCDM are consistent with that in the PCDM far away from the nucleus，so the former is steeper than the latter between near and far nuclear area.

Fig.3 The ground state wavefunction of ions with Z=1，15，50 and 100 respectively in the PCDM and GCDM

Table 4 compares the energies of hydrogen-like ions，computed by three different methods to solve the Dirac equation:ND，DKB and Desclaux’s Finite -Difference Numerical (FDN)methods. It presents that the DKB method is better than the ND method in accuracy. Our results are closer to the experimental values than the FDN method and have a higher accuracy. Adopting different method affects the precision of the wavefunction. Taking U+91as an example，F(xiàn)ig.4 shows the detailed information in the GCDM between the results calculated by the ND and DKB methods，respectively. The difference is slight，the relative magnitude is only 10-7，which is negligible.

Adopting B -splines in the near nuclear region，the results have been obtained and are more tend to experimental values. Refs. ［19，20］have presented the results of the polarizabilities using B-splines. This implies that the B -splines are not only appropriate for disposing the long-range problems，but also suitable in the short-range facet. It provides a method for disposing the near nuclear region of hydrogen-like ions.

4 Conclusion

Fig.4 The ground state wavefunctions of U +91 obtained in the DKB and ND methods，respectively，and the difference between them:(a)the lager component Pnk;(b)the small component Qnk

Table 3 Ground energies of high-Z hydrogen-like ions in the PCDM and GCDM using the ND basis set

Table 4 The ground energies of hydrogen-like ions obtained by different methods

The ground energies of hydrogen - like system have been calculated in the GCDM and PCDM，respectively. The results computed in the GCDM are closer to the experimental values than those in the PCDM. The relative energy correction increases with the increasing of Z，which can be up to 10-3when Z =100. Compared with the PCDM，the wavefunction is smooth，induced by the Gaussian nucleus potential energy which does not contain any singularities as r→0. It implies that，in the high -Z situation，the PCDM is not appropriate to calculate physical quantities in high accuracy. And a comparison of the DKB and ND method is also presented. The wavefunctions computed by the DKB and ND sets are almost the same.

Besides，it is different from that B-spline functions are usually used to calculate long -range problems，such as the calculations of polarizabilities. In this paper，the B-splines have been successfully applied to the short - range analysis between electron and nucleus in the GCDM. Some extensions of B -spline functions are now still in progress. For example，it plays an important role in the research of the nuclear field shift to stimulate the cusp of the second derivatives at the nuclear boundary for the radial wavefunctions of hydrogen - like ions in the SCDM and to calculate multipole expansion coefficients of the radial wavefunctions in the SCDM and FCDM.

［1］ Desclaux J P. A multiconfiguration relativistic Dirac-Fock program［J］. Comp. Phys. Commun.，1975，9:31.

［2］ Desclaux J P. Relativistic Dirac-Fock expectation values for atoms with Z=1 to Z=120［J］. At. Data Nucl.Data Tables，1973，12:311.

［3］ Hasegawa T，F(xiàn)ujimura N，Matsuoka O. Hydrogenlike atoms in uniformly charged sphere model of atomic nucleus.I. Reference calculations of energy levels［J］. Int. J.Quant. Chem.，1991，39:805.

［4］ Kutzelnigg W. Basis set expansion of the dirac operator without variational collapse ［J］. Int. J. Quant.Chem.，1984，25:107.

［5］ Visser O，Aerts P J C，Hegarty D，et al. The use of gaussian nuclear charge distributions for the calculation of relativistic electronic wavefunctions using basis set expansions［J］. Chem. Phys. Lett.，1987，134:34.

［6］ Hegarty D，Aerts P J C. Variational solutions of the Dirac equation for atoms and molecules ［J］. Phys.Scripta，1987，36:432.

［7］ Johnson W R，Blundell S A，Sapirstein J. Finite basis sets for the Dirac equation constructed from B splines［J］. Phys. Rev. A，1988，37:307.

［8］ Antonio L A Fonseca，Daniel L Nascimento，F(xiàn)abio F Monteiro，et al. A variational approach for numerically solving the two - component radial dirac equation for one-particle systems［J］. J. Mod. Phys.，2012，3(4):350.

［9］ Ishikawa Y，Koc K，Schwarz W H E. The use of Gaussian spinors in relativistic electronic structure calculations:the effect of the boundary of the finite nucleus of uniform proton charge distribution ［J］. Chem.Phys.，1997，225:239.

［10］ Matsuoka O，F(xiàn)ujimura N，Hasegawa T. Hydrogenlike atoms in uniformly charged sphere model of atomic nucleus.II. Application of basis-set expansion method［J］. Int.J. Quant. Chem.，1991，39:813.

［11］ Visscher L，Dyall K G. Dirac-Fock atomic electronic structure calculations using different nuclear charge distributions［J］. At. Data Nucl. Data Tables，1997，67:207.

［12］ Dirk A，Markus R，Juergen H. A comparative study of finite nucleus models for low -lying states of few -electron high - Z atoms ［J］. Chem. Phys. Lett.，2000，320:457.

［13］ Dyall K G，F(xiàn)aegri K. Finite nucleus effects on relativistic energy corrections ［J］. Chem. Phys. Lett.，1993，201:27.

［14］ Bachau H，Cormier E，Decleva P，et al. Applications of B - splines in atomic and molecular physics ［J］.Rep. Prog. Phys.，2001，64:1815.

［15］ Sapirstein J，Johnson W R. The use of basis splines in theoretical atomic physics ［J］. J. Phys. B，1996，29:5213.

［16］ Bian X B，Qiao H X，Shi T Y. B -Spline with symplectic algorithm method for solution of time -dependent schr?dinger equations ［J］. Chin. Phys. Lett.，2006，23:2403.

［17］ Song H W，Li Y. Calculation of radial matrix elements and anticrossings for highly excited Stark states using the B - spline approach ［J］. Phys. Rev. A，2008，78:062504.

［18］ Grant I P. B - spline methods for radial Dirac equations［J］. J. Phys. B，2009，42:055002.

［19］ Zhang Y H，Tang L Y，Zhang X Z，et al. Relativistic quadrupole polarizability for the ground state of hydrogen-like ions ［J］. Chin. Phys. Lett.，2012，29:063101.

［20］ Tang L Y，Zhang Y H，Zhang X Z，et al. Computational investigation of static multipole polarizabilities and sum rules for ground - state hydrogenlike ions［J］. Phys. Rev. A，2012，86:012505.

［21］ Kang Sh，Li J，Shi T Y. Investigation of hydrogenicdonor states confined by spherical quantum dots with B-splines［J］. J. Phys. B，2006，39:3491.

［22］ Shabaev V M，Tupitsyn I I，Yerokhin V A，et al. Dual kinetic balance approach to basis-set expansions for the Dirac equation［J］. Phys. Rev. Lett.，2004，93:130405.

［23］ Beloy K，Derevianko A.Application of the dual - kinetic - balance sets in the relativistic many - body problem of atomic structure［J］. Comp. Phys. Commun.，2008，179:310.

［24］ Zheng S D，Li B W，Li J G，et al. The influences of the finite nuclear size effects on the energy levels and wavefunctions of hydrogen-like ions［J］. Acta Phys.Sin.，2009，58(3):1556 (in Chinese)

［25］ deBoor D C. A practical guide to splines［M］. New York:Springer–Verlag，2001:87.

［26］ Weast R C. Handbook of chemicstry and physics［M］.Florida:CRC Press，1981:1815.