An efficient calculation of the energy levels of the carbon groupNengwu Zheng, Dongxia Ma, Ruyi Yang, Tao Zhou, Tao Wang, and Song Han

Citation: The Journal of Chemical Physics 113, 1681 (2000); doi: 10.1063/1.481969 View online: http://dx.doi.org/10.1063/1.481969 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/113/5?ver=pdfcov Published by the AIP Publishing Articles you may be interested in The effect of energy level offset between Ir dopant and carbazole hosts on the emission efficiency Appl. Phys. Lett. 97, 023309 (2010); 10.1063/1.3462931 The lowest energy states of the group-IIIA–group-VA heteronuclear diatomics: BN, BP, AlN, and AlP from fullconfiguration interaction calculations J. Chem. Phys. 125, 124311 (2006); 10.1063/1.2335446 2-methyl furan: An experimental study of the excited electronic levels by electron energy loss spectroscopy,vacuum ultraviolet photoabsorption, and photoelectron spectroscopy J. Chem. Phys. 119, 3670 (2003); 10.1063/1.1590960 Electron excitation energies using a consistent third-order propagator approach: Comparison with fullconfiguration interaction and coupled cluster results J. Chem. Phys. 117, 6402 (2002); 10.1063/1.1504708 Using the symmetric quasiminimal residuals method to accelerate an inexact spectral transform calculation ofenergy levels and wave functions J. Chem. Phys. 114, 6485 (2001); 10.1063/1.1356005

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JOURNAL OF CHEMICAL PHYSICS VOLUME 113, NUMBER 5 1 AUGUST 2000

This a

ARTICLES

An efficient calculation of the energy levels of the carbon groupNengwu Zheng,a) Dongxia Ma, Ruyi Yang, Tao Zhou,b) Tao Wang, and Song HanDepartment of Chemistry, USTC, Hefei, Anhui 230026, People’s Republic of China

~Received 29 November 1999; accepted 4 May 2000!

Energy levels of many-valence-electron atomic~ionic! systems play an important role in manyscientific disciplines. Many theoretical methods to calculate the energy levels of Rydberg states ofatoms and ions have been provided, and the carbon group atoms have been treated. In this paper, wecomputed single excited states of carbon group atoms with high precision. The accuracy of most ofour results that have been reached are less than 1 cm21. This is quite good among all the presenttheoretical methods. ©2000 American Institute of Physics.@S0021-9606~00!30629-8#

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I. INTRODUCTION

In recent years, more and more attention has beento the high Rydberg states and the energy levels of atomions because of their wide usages in the fields of laser deopment, plasma diagnosis, etc. Many experimental teniques and theoretical methods focusing on Rydberg sthave been reported such as theR-matrix method,1–3 quantumdefect theory~QDT!,4–6 and density functional theory.7–9

For the carbon group, the experimental data of theergy levels are reported by Moore,10 Martin and RomualdZalubas,11 Sugar and Musgrove,12 Fuhr, Martin, Musgrove,Sugar, and Wiese13 et al. The abundance of experimentdata made it possible to explore the regularity of the Rydbseries theoretically. In 1986, Ginter and Ginter5 calculatedthe J53° energy levels of thempnd and msmp3 configu-rations in SnI, GeI, and SiI by using a perturbed coupledchannel model, and involving theLS- j j frame transforma-tion. As for the highly excited states the calculated resuaccounted well with the experimental data. But in low ecited states, the error between calculated value and exmental value is quite large. For the first time Green and K3

employed the eigenchannelR-matrix approach combinedwith the j j -LS frame transformation and energy-independtransformation of the generalized engensystem to dealthe even-parity bound spectrum of Si whoseJ50. In thatarticle he treated two Rydberg series 3s23p1/2np1/2 and3s23p3/2np3/2. Their results coincide well with the expermental energy levels. But it cannot deal with carbon groatoms systematically. Robicheaux and Green6 first applied anearlyab initio description to calculate the bound-state proerties of the atoms in the carbon group for states with oparity andJ50 – 3. The rapid energy dependence of the3D +

quantum defect near the threshold is the main reasondiscrepancies between the calculated scattering parametethis article and those of previous experiments by using L

a!Author to whom correspondence should be addressed.b!Electronic mail: tzhou@mail.ustc.edu.cn

1680021-9606/2000/113(5)/1681/7/$17.00

rticle is copyrighted as indicated in the article. Reuse of AIP content is sub

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Fano plots.14 This energy dependence brings some probleinto theLS- j j frame transformation for theJ51 – 3 symme-tries.

In the summary mentioned above, we can find thatmethod of theR-matrix combined with thej j -LS frametransformation and multichannel quantum defect are notisfying in two aspects though the accuracy of the resultsquite good. First, they all need a large number of parameand a large amount of work. Second, though the quandefect theory is very good when describing the highly ecited states, it deviates terribly for the low excited stawhich must be especially noted.

In this paper, within the frame of the weakest bouelectron potential model~WBEPM! theory, we introducedthe concept of spectrum-level-like series, and calculatedenergy levels of the carbon group atoms. The method andresults are precise. Most of the calculated values haveerror of less than 1 cm21. Compared with the errors of thexperimental data, we can say that our results are satisfy

II. THEORY AND METHOD

A. WBEPM theory

The WBEPM theory was suggested by one of tauthors.15–21 In the theory we suppose the electrons of tsystem can be divided into weakest bound electronnonweakest-bound electrons according to the ability of beexcited or being ionized. Take the ground configurationSi~@Ne]3s23p2) as an example, the two 3p electrons havethe largest chemical activity and they are undistinguishaBut in fact, they can only be excited or ionized one by onSo we can label the one that is first excited or ionized asweakest bound electron, and all the others nonweakest boelectrons. Thus, in any system we can separate the elecinto weakest bound electron and nonweakest bound etrons. This is necessary for this whole theory. The nonweest bound electrons and the nucleus can be treated as a wion core, and the weakest bound electron lies in the aver

1 © 2000 American Institute of Physics

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TABLE I. Some of the spectrum-level-like series and their perturbing levels.

Spectrum-level-like series Foreign perturbing levels n Energy levels~cm21!

C I: @He]2s22pnd(n>3) 3D1+ @He#2s22pnd 1F1

+ 4 83 947.43

Si I: @Ne#3s23pnd(n>3) 3P0+ @Ne#3s23pns 3P0

+ 4 39 683.163

@Ne#3s23pns3P0+ 5 54 245.020

@Ne#3s23pns (1/2,1/2)0+ 6 59 221.11

GeI: @Ar#4s24pnd(n>6) (3/2,5/2)3+ @Ar#4s24pnd (3/2,3/2)3

+ 6 60 886.24

@Ar#4s24pnd (3/2,3/2)3+ 7 62 370.79

@Ar#4s24pnd (3/2,3/2)3+ 8 63 168.53

@Ar#4s24pnd (3/2,3/2)3+ 9 63 715

SnI: @Kr#5s25pnd(n>5) 3P1+ @Kr#5s25pnd 1P1

+ 5 50 125.9

@Kr#5s25pnd 3P1+ 6 56 244.0

@Kr#5s25pnd 1P1+ 8 60 397.0

PbI: @Xn#6s26pns(n>7) 3P1+ @Xn#6s26pns 3P1

+ 7 52 499.53

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potential of the ion core. Thus, the many-valence-electsystem can be treated as a single-electron system.

Reference 15 represented the potential function forweakest bound electron by

V~r i !5A

r i1

B

r i2 ~1!

with

A[2Z8e2, ~2!

B[@d~d11!12dl#h2

8p2m. ~3!

HereZ8 andd are undetermined coefficients,l is the angularquantum number of the weakest bound electron, andh isPlank’s constant.

By solving the single-electron Schro¨dinger equation,

F2\2

2m¹ i

21V~r i !Gw i5e iw i , ~4!

the wave function of the weakest bound electron cangiven by

w i~r ,u,f!5R~r !Yl ,m~u,f! ~5!

in which Yl ,m(u,f) is the same spheric harmonics as thatthe hydrogen atom or hydrogenlike ions systems, and

R~r !5A expS 2Z8r

n8a0D r l 8Ln2 l 21

2l 811 S 2Z8r

n8a0D ~6!

with

s indicated in the article. Reuse of AIP content is sub

129.119.67.237 On: Mon,

n

e

e

f

n82n5 l 82 l 5d, ~7!

and A is the normalization constant. The eigenvalue of tHamiltonian is

e i52RZ82

~n1d!2 . ~8!

B. The principle of our calculation

The authors have defined the concept of spectrum-lelike series in Ref. 18. A spectrum-level-like series is a sercomposed by energy levels with same spectral level symin a given electronic configuration series of a system.19 Nowtake the Si atom as an example, we know that the elecconfiguration of Si in the ground state is@Ne#3s23p2. Ex-citing the weakest bound electron to various energy levwill produce various Rydberg series of configurations suas @Ne#3s23pns(n>4), @Ne#3s23pnp(n>4), and so on.Each configuration will go on to produce several spectrterms which further some spectral levels. So we definedconcept of spectrum-level-like series to classify the enelevels. According to the concept just define@Ne#3s23pns(n>4) (1/2,1/2)1

0 is a spectrum-level-like series of the Si atom.

The coupling of the angular momentum of the atomsthe group IVA elements is very complicated. We can labthe spectral level symbol asnl@K#J in general no matterwhich frame it is coupled from. Then we can expressenergy of a level in a spectrum-level-like series as follow

6

TABLE II. Parameters gotten from Eqs.~12! and ~15! of the series listed in Table I.

Parameters C Si Ge Sn Pb

a0 0.059 52 234.546 71 1.651 50 1.877 74 4.825 3a1 20.034 95 2150.782 93 48.484 49 215.209 67 0.057 23a2 23.462 35 2598.203 47 23 612.458 62 2202.779 43 1.495 17a3 23.067 62 24 210.639 97 188 954.748 9 5 837.271 00 22.584 27b1 0.000 00 28.135 51 0.028 49 20.077 85 0.000 14b2 20.071 40 0.001 87 20.005 19b3 0.001 59 0.000 26 20.000 09b4 0.000 02

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TABLE III. A comparison between calculated results and experimental values~in cm21! @He#2s22pnd(n>3) 3D1

+ series of CI ~limit is 90 878.3 cm21!.

n Tcalc Texpta Deviation n Tcalc Texpt Deviation

3 78 318.248 4 78 318.25 20.0016 27 90 732.224 64 83 848.830 3 83 848.83 0.0003 28 90 742.853 55 86 397.818 8 86 397.80 0.0188 29 90 752.399 86 87 777.190 0 87 777.17 0.0200 30 90 761.005 77 88 606.647 7 88 606.8 20.1523 31 90 768.790 78 89 143.574 4 89 143.4 0.1744 32 90 775.855 99 89 510.840 1 89 510.9 20.0599 33 90 782.287 6

10 89 773.029 8 89 773.2 20.1702 34 90 788.159 211 89 966.701 2 89 966.8 20.0988 35 90 793.533 912 90 113.798 7 36 90 798.466 213 90 228.138 9 37 90 803.003 514 90 318.771 4 38 90 807.186 715 90 391.823 9 39 90 811.051 816 90 451.565 5 40 90 814.630 217 90 501.043 8 41 90 817.949 618 90 542.481 7 42 90 821.034 519 90 577.531 5 43 90 823.906 320 90 607.441 8 44 90 826.584 321 90 633.170 3 45 90 829.085 522 90 655.461 8 46 90 831.425 223 90 674.902 3 47 90 833.617 024 90 691.958 0 48 90 835.673 025 90 707.003 6 49 90 837.604 326 90 720.342 9 50 90 839.420 7

aThe experimental valuesTexpt were selected from Ref. 13.

TZ,@K#J~n!5EZ,@K#J

~n!1Tlim i t5Tlim i t2RZ82

~n1d!2 , ~9!

whereR is the mass-effect-modified Rydberg constant,Z isthe nuclear charge, andTlim i t is the ionic limit.

In order to simplify Eq.~12! which have two variousparameters~Z8 andd!, we can do a little transformation,

s indicated in the article. Reuse of AIP content is su

129.119.67.237 On: Mo

Z8

n1d5

Znet

n2dn. ~10!

Then we have

TZ,@K# j~n!5Tlim i t2

RZnet2

~n2dn!2 , ~11!

TABLE IV. A comparison between calculated results and experimental values~in cm21! @Ne#3s23pnd(n>3) 3P0

+ series of SiI ~limit is 66 035.00 cm21!.

n Tcalc Texpta Deviation n Tcalc Texpt Deviation

3 50 602.435 0 50 602.435 20.0000 27 65 884.861 74 56 733.369 9 56 733.369 9 20.0000 28 65 895.377 65 61 960.269 1 61 960.270 20.0009 29 65 904.826 66 63 123.360 9 63 123.35 0.0109 30 65 913.348 17 63 863.756 5 63 863.78 20.0235 31 65 921.059 68 64 358.429 0 64 358.44 20.0110 32 65 928.060 69 64 703.293 1 64 703.23 0.0631 33 65 934.435 8

10 64 952.538 3 64 952.57 20.0317 34 65 940.257 611 65 138.217 5 65 138.24 20.0225 35 65 945.588 112 65 280.115 7 65 280.10 0.0157 36 65 950.481 213 65 390.927 6 65 390.91 0.0176 37 65 954.983 514 65 479.080 7 65 479.16 20.0793 38 65 959.135 315 65 550.339 4 65 550.29 0.0494 39 65 962.972 316 65 608.750 4 65 608.6 0.1504 40 65 966.525 417 65 657.220 3 65 657.2 0.0203 41 65 969.822 018 65 697.879 9 65 697.6 0.2799 42 65 972.886 119 65 732.319 0 65 732.3 0.0190 43 65 975.739 120 65 761.743 1 44 65 978.400 121 65 787.079 7 45 65 980.885 722 65 809.051 7 46 65 983.211 223 65 828.228 9 47 65 985.389 924 65 845.065 7 48 65 987.434 025 65 859.927 7 49 65 989.354 326 65 873.112 0 50 65 991.160 6

aThe experimental valuesTexpt were selected from Ref. 13.

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1684 J. Chem. Phys., Vol. 113, No. 5, 1 August 2000 Zheng et al.

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TABLE V. A comparison between calculated results and experimental values~in cm21! @Ar#4s24pnd(n>6) (3/2,5/2)3

+ series of GeI ~limit is 65 480.60 cm21!.

n Tcalc Texpta Deviation n Tcalc Texpt Deviation

6 61 268.390 0 61 268.39 20.0000 35 65 386.1509 65 386.1 0.05097 62 522.980 0 62 522.98 20.0000 36 65 391.4636 65 391.2 0.26368 63 271.309 7 63 271.31 20.0003 37 65 396.3400 65 396 0.34009 63 789.997 0 63 790 20.0030 38 65 400.8266 65 400.6 0.2266

10 64 144.081 5 64 144 0.0815 39 65 404.9640 65 404.8 0.18411 64 395.707 4 64 396 20.2926 40 65 408.7874 65 408.6 0.187412 64 581.961 5 64 581.6 0.3615 41 65 412.3278 65 412.2 0.12713 64 724.308 0 64 724.4 20.0920 42 65 415.6126 65 415.5 0.112614 64 835.584 7 64 835.7 20.1153 43 65 418.6658 65 418.5 0.165815 64 924.160 6 64 924.1 0.0606 44 65 421.5087 65 421.4 0.10816 64 995.763 0 64 995.6 0.1630 45 65 424.1601 65 424.2 20.039917 65 054.432 5 65 054.1 0.3325 46 65 426.6369 65 426.5 0.13618 65 103.086 2 65 102.7 0.3862 47 65 428.9541 65 428.9 0.05419 65 143.869 3 65 143.2 0.6693 48 65 431.1251 65 431.1 0.02520 65 178.386 3 65 177.72 0.6663 49 65 433.1620 65 433.1 0.0621 65 207.854 8 65 207.2 0.6548 50 65 435.0756 65 435.1 20.024422 65 233.212 0 65 232.5 0.7120 51 65 436.8757 65 436.9 20.024323 65 255.187 8 65 254.5 0.6878 52 65 438.5710 65 438.6 20.029024 65 274.357 5 65 273.93 0.4275 53 65 440.1696 65 440.2420.070425 65 291.179 4 65 290.5 0.6794 54 65 441.6786 65 441.7920.111426 65 306.021 7 65 305.43 0.5917 55 65 443.104727 65 319.183 6 65 318.6 0.5836 56 65 444.453828 65 330.909 6 65 330.3 0.6096 57 65 445.7314 65 445.8520.118629 65 341.401 3 65 341.1 0.3013 58 65 446.9424 65 446.9920.047630 65 350.826 3 65 350.5 0.3263 59 65 448.0913 65 448.1920.098731 65 359.324 4 65 358.8 0.5244 60 65 449.1824 65 449.16 0.0232 65 367.013 5 65 366.7 0.3135 61 65 450.2195 65 450.2920.070533 65 373.993 2 65 373.8 0.1932 62 65 451.2060 65 451.2820.074034 65 380.348 2 65 380.1 0.2482 63 65 452.1453 65 452.2820.1347

aThe experimental valuesTexpt were selected from J. Sugar and A. Musgrove, J. Phys. Chem. Ref. Dat22,1213 ~1993!.

s

nd

te

-

rst

theurbleing

oris

ect-u-omr

Ti5Tlim i t2RZnet

2

~n2dn!2 , ~12!

in which Znet is the net charge of nuclear andZnet51 foratoms.dn equal to the quantum defect in QDT here.

Risberg suggested a formula for unperturbed levels a

dn~en!5a11a2en1a3en21¯

5a11a2

~n2dn!1

a3

~n2dn!2 1¯ ~13!

by employingen5(n2dn)22. One can solve this equatiothrough iteration method. But it is obviously complicateMartin simplified this formula as follows:

dn~en!5a11a2

~n2d0!2 1a3

~n2d0!4 1a4

~n2d0!6 ~14!

in which d0 is the quantum defect of the lowest excited stain a given series.

The perturbation is not involved in Martin formula. Enlightened by Ritz, White, and Langer, we derive

dn5(i 50

3

aien2i1(

j 51

Nbj

en2e j~15!

in which

s indicated in the article. Reuse of AIP content is sub

129.119.67.237 On: Mon

.

s

en51

~n2d0!2 , ~16!

e j5Tlim i t2Tt,perturb

RZnet2 . ~17!

N is the number of foreign perturbing levels.The data ofen , e i , and dn can all be obtained from

experimental values. Then we got the parametersai and bj

through the least-square fitting of this equation with the fiseveral experimental data.

III. RESULTS AND DISCUSSION

We have calculated the energy levels of atoms ofgroup IVA elements in this paper, and we listed some of oresults here. The foreign perturbing levels are listed in TaI; their parameters are listed in Table II; the correspondenergy, levels calculated from Eq.~12! combined with Eq.~15! and experimental data are listed in Tables III–VII fcomparison. It is obvious that the accuracy of our resultsvery good. Comparing the multichannel quantum deftheory~MQDT! andR-matrix method, our calculation is simpler. The superiority of our method to them lies in the accracy in the low excited states. Some results derived frthe Martin formula directly are listed in Table VIII focomparison.

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TABLE VI. A comparison between calculated results and experimental values~in cm21! @Kr#5s25pnd(n>5) 3P1

+ series of SnI ~limit is 63 483.2 cm21!.

n Tcalc Texpta Deviation n Tcalc Texpt Deviation

5 48 982.000 2 48 982.0 0.0002 33 63 364.535 66 55 074.002 1 55 074.0 0.0021 34 63 371.967 07 57 934.776 4 57 934.7 0.0764 35 63 378.721 98 60 051.313 2 60 051.4 20.0868 36 63 384.880 09 60 947.577 7 60 948.4 20.8223 37 63 390.509 5

10 61 558.270 9 61 556.3 1.9709 38 63 395.669 111 61 972.072 8 61 973.2 21.1272 39 63 400.409 812 62 266.989 9 62 267.3 20.3101 40 63 404.775 613 62 484.311 9 62 484.0 0.3119 41 63 408.805 114 62 648.783 0 62 651.8 23.0170 42 63 412.531 915 62 776.081 1 62 776.2 20.1189 43 63 415.985 616 62 876.523 8 62 878.1 21.5762 44 63 419.192 317 62 957.111 9 62 956.4 0.7119 45 63 422.174 818 63 022.721 8 63 025.9 23.1782 46 63 424.953 719 63 076.829 0 63 080.3 23.4710 47 63 427.547 120 63 121.961 9 63 122.5 20.5381 48 63 429.971 021 63 159.993 6 63 160.4 20.4064 49 63 432.240 022 63 192.334 8 63 191.6 0.7348 50 63 434.367 023 63 220.063 5 63 220.7 20.6365 51 63 436.363 624 63 244.015 1 63 244.3 20.2849 52 63 438.240225 63 264.844 2 63 267.9 23.0558 53 63 440.006 226 63 283.069 9 63 283.5 20.4301 54 63 441.670 227 63 299.108 2 63 297.5 1.6082 55 63 443.239 928 63 313.295 0 56 63 444.722 229 63 325.904 4 57 63 446.123 630 63 337.161 6 58 63 447.449 831 63 347.253 1 59 63 448.706 132 63 356.334 3 60 63 449.897 3

aThe experimental valuesTexpt were selected from C. E. Moore, Atomic Energy Levels, Nat. Stand. Ref. DSer., Nat. Bur. Stand.~U.S.! 1971, 74.

TABLE VII. A comparison between calculated results and experimental values~in cm21! @Xn#6s26pns(n>7) 3P1

+ series of PbI ~limit is 59 821.0 cm21!.

n Tcalc Texpta Deviation n Tcalc Texpt Deviation

7 35 287.227 4 35 287.24 20.0126 34 59 692.085 98 48 686.885 7 48 686.87 0.0157 35 59 700.488 59 53 511.611 3 53 511.34 0.2713 36 59 708.095 6

10 55 719.971 6 55 720.52 20.5484 37 59 715.004 511 56 942.254 0 56 942.26 20.0060 38 59 721.298 112 57 689.254 2 57 688.73 0.5242 39 59 727.047 313 58 179.050 6 58 179.3 20.2494 40 59 732.313 314 58 517.525 4 58 517.67 20.1446 41 59 737.148 615 58 761.169 8 58 761 0.1698 42 59 741.598 916 58 942.370 2 58 941.8 0.5702 43 59 745.704 217 59 080.778 4 59 080.4 0.3784 44 59 749.499 118 59 188.881 3 45 59 753.014 219 59 274.921 8 46 59 756.276 320 59 344.519 7 47 59 759.309 121 59 401.612 5 48 59 762.133 722 59 449.026 3 49 59 764.768 623 59 488.831 3 50 59 767.230 524 59 522.572 8 51 59 769.534 225 59 551.422 5 52 59 771.692 926 59 576.282 2 53 59 773.718 627 59 597.855 3 54 59 775.622 028 59 616.696 5 55 59 777.412 729 59 633.248 4 56 59 779.099 530 59 647.867 4 57 59 780.690 231 59 660.843 1 58 59 782.192 132 59 672.413 0 59 59 783.611 533 59 682.772 9 60 59 784.954 5

aThe experimental valuesTexpt were selected from C. E. Moore, Atomic Energy Levels, Nat. Stand. Ref. DSer., Nat. Bur. Stand.~U.S.! 1971, 208.

s indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

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1686 J. Chem. Phys., Vol. 113, No. 5, 1 August 2000 Zheng et al.

This a

In a given spectrum-level-like series, a foreign levelthe vicinity will perturb some energy levels if the foreiglevel has the same parity and the same quantum numbJwith the series. The strength of the perturbing is redualong with the growth of the energy difference betweenperturbing level and the perturbed level. The perturbatcontributed to the complication of the atomic spectrum.the ionic limit of the perturbing levels are different from thof the perturbed series, the perturbation is evident in theexcited Rydberg states, because the interval between thelimits are great.

According to QDT, Martin proposed a formula witwhich the energy levels of the Na atom were calculated scessfully and an accuracy of less than 1 cm21 was arrived.22

Though the Martin formula is employed in the transformtion of the parameters in this paper@see Eq.~10!#, our theo-retical method is distinctly different from Martin’s methodor QDT. The reasons are the following:

~a! An alkali atom is a rather simple system constituteda valence electron and a close-shelled kernel. Qtreated the alkali atoms excellently by separatingvalence electron from the atomic core and considerthat the valence electron moves in a spherical poten~Znet51 for neutral atom!. But a many-valence-electron atomic system cannot be transferred into atem similar to the alkali atom in the idea of separatiof valence and core electrons because many valeelectrons in the system are undistinguishable. SoMartin formula cannot be extended in the conceptthe many-valence-electron system. In WBEPM theoby introducing the concept of the bound electron athe idea of separating the weakest bound electron fthe nonweakest bound electrons, we can concludethe weakest bound electron is moving in the potenof the nonweakest bound electrons and nucleus, ansimilar to an alkali system. Thus, we got a unified ne

TABLE VIII. Comparison of Martin’s method and ours~in cm21!

@Ne#3s23pnd(n>3) 3P0+ series of SiI ~limit is 66 035.00 cm21!.

n Texpta Tcalc

b Tcalc2Texpt TMartinc TMartin2Texpt

3 50 602.435 50 602.435 0 20.0000 50 602.435 0.0004 56 733.3699 56 733.369 9 20.0000 56 733.369 9 0.0005 61 960.270 61 960.269 1 20.0009 61 960.27 0.0006 63 123.35 63 123.360 9 0.0109 63 123.35 0.007 63 863.78 63 863.756 5 20.0235 63 711.815 8 2151.9648 64 358.44 64 358.429 0 20.0110 64 131.293 9 2227.1469 64 703.23 64 703.293 1 0.0631 64 460.650 22242.580

10 64 952.57 64 952.538 3 20.0317 64 723.908 6 2228.66111 65 138.24 65 138.217 5 20.0225 64 934.713 9 2203.52612 65 280.10 65 280.115 7 0.0157 65 103.811 52176.28913 65 390.91 65 390.927 6 0.0176 65 240.040 92150.86914 65 479.16 65 479.080 7 20.0793 65 350.502 8 2128.65715 65 550.29 65 550.339 4 0.0494 65 440.767 22109.52316 65 608.6 65 608.750 4 0.1504 65 515.143 7293.45617 65 657.2 65 657.220 3 0.0203 65 576.948 5280.25118 65 697.6 65 697.879 9 0.2799 65 628.733 4268.86719 65 732.3 65 732.319 0 0.0190 65 672.469 1259.831

aTexpt are the experimental values selected from Ref. 13.bTcalc are our calculated results.cTMartin are the results calculated with Martin formula directly.

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theoretical method to deal with single or manvalence-electron system and calculate the atomicergy levels. Therefore, we employed Martin formulatransformation of the parameters in this paper reasably. That is to say, Eqs.~10! and~14! are reasonable

~b! The spectral terms and atomic energy levels of alkatom are very simple. But they are complicated fmany-valence-electron system because a configuramay have many spectral terms, and one spectral tmay have many atomic energy levels. This is the sond difficult point in extending Martin formula tomany-valence-electron systems. Within the frameWBEPM theory, by introducing the concept ospectrum-level-like series, we haveEZ,@K#J

5e

52(RZ82/n82) @see Eqs.~8! and~9!#. So we can clas-sify the levels and explore the regularity of them.

~c! The energy levels of carbon group atoms are perturlevels. The results are not good if we treat them wMartin formula directly~see Table VIII!. The perturba-tion cannot be ignored. So we derived the express~15! to calculate the energy levels.

~d! In WBEPM theory, we provided the idea of separatithe weakest bound electron from the nonweakest boelectrons, which simplified the problem greatly. Fany atomic or ionic system there is a weakest bouelectron even if some electrons are undistinguishafrom it. When this electron was excited out of the sytem, one of the residual electrons would becomeweakest bound electron. So each electron mightcome the new weakest bound electron of the new stem. Thus a many-valence-electron problem becomesingle-electron problem.

The excellent results of our calculation testified the crectness of the idea of separation of the weakest bound etron from the non-weakest-bound-electrons.

IV. CONCLUSION

Within the WBEPM theory, we calculated the energlevels of the atoms of the carbon group by introducingconcept of spectrum-level-like series. Results are compawith those from experimental values. The most satisfactaspect of this present method is that it is very simple, precand computationally efficient.

ACKNOWLEDGMENTS

The work is supported by the Anhui Provincial Foundtion of Natural Science Research~Grant No. 98512038! andthe National Natural Science Foundation of China~GrantNo. 59872039!.

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