3P127‫֩ࢠݪ‬ϞσϧʹΑΔ He-like ΠΦϯ‫ࢠݪ‬ͷ૬ର࿦తిࢠ૬ؔΤωϧΪʔ΁ͷӨ‫ڹ‬††ʢ‫۝‬େӃཧ, ໊ࢢେ ʣɹ ˓౉ล঵߂, த໺੖೭, ‫ؘ‬࿬༸

Correlation energy / hartreeʲংʳHe ‫ͱࢠݪ‬౳ిࢠΛ࣋ͭ He-like ΠΦϯ‫ࢠݪ‬ͷిࢠ૬ؔʹ͍ͭͯ‫ܧ‬ଓ͠‫ڀݚ‬Λଓ͚͍ͯΔɽ͜͜Ͱͷిࢠ૬ؔ͸ Schrödinger ํఔࣜ·ͨ͸ Dirac–Coulomb ํఔࣜͷղͱͦΕͧΕʹରԠ͢
Δ Hartree–Fock ۙࣅղͱͷશΤωϧΪʔࠩͰ͋Δɼిࢠ૬ؔΤωϧΪʔΛࢦ͢ɽඇ૬ର࿦తిࢠ૬ؔΤωϧΪʔ͸ɼCa18+ Ҏ߱ɼ‫ࢠݪ‬൪߸͕େ͖͘ͳΔʹͭΕͯ΄΅Ұఆ
1 ҰํɼPestka, Karwowski ʹΑΔ૬ର࿦త Hylleraas-CI(configuration interaction) ‫ܭ‬ͷ஋ΛͱΔɽ
ࢉ 2 ΛॳΊɼͦͷ‫ޙ‬ͷ‫ ࢉܭ‬3,4 ʹΑͬͯ΋ɼ૬ର࿦తిࢠ૬ؔΤωϧΪʔ͸ɼ͋Δ‫ࢠݪ‬൪߸Ҏ߱ɼ‫ࢠݪ‬൪߸ͷ૿Ճʹରͯ͠ిࢠ૬ؔΤωϧΪʔͷ‫ݮͳܹٸ‬গ͕֬ೝ͞ΕͨɽൃදऀΒ΋̐੒෼೾ಈؔ਺Λ༻͍ͨ૬ର࿦త CI ‫ ࢉܭ‬5 Λߦ͕ͬͨɼ૬ର࿦త Hylleraas-CI ‫ࢉܭ‬ΑΓ΋େ͖͘‫ݮ‬গ͢Δ݁ՌΛಘͨɽʢਤ̍ͷ Rel. CI(NVPA)ʣ
ઌ ͷ ใ ࠂ Ͱ ͸ ɼ͜ ͷ ૬ ର ࿦ త ૬ ؔ Τ ω ϧ-0.04
Ϊʔ͕େ͖͘‫ݮ‬গ͍ͯ͠Δ‫ݪ‬Ҽ͕ no-virtualRel. CI (VPA)
Non-rel.
pair ۙࣅ (NVPA) ʹΑΔࣄΛ໌Β͔ʹ͠ɼஶऀ-0.05
Β͕໌‫͕͍ͳ͍ͯ͠ه‬૬ର࿦త Hylleraas-CI‫ࢉܭ‬͸ NVPA ΛԾఆ͍ͯ͠ͳ͍‫( ࢉܭ‬VPA) ʹ
Rel. Hylleraas-CI
6-0.06
ରԠͨ͠΋ͷͰ͋Δࣄ͕ղ͔ͬͨɽ
૒ํͷ‫͍͓ͯʹࢉܭ‬ɼ༻͍͍ͯΔ‫֩ࢠݪ‬Ϟσϧʹҧ͍͕͋ΔɽຊൃදͰ͸ɼ‫֩ࢠݪ‬Ϟσ
-0.07
ϧʹΑΔిࢠ૬ؔΤωϧΪʔ΁ͷӨ‫͍ͭʹڹ‬
Rel. CI (NVPA)
ͯใࠂ͢Δɽ
-0.08
ʲ‫ํࢉܭ‬๏ʳ̐੒෼૬ର࿦త CI(VPA) ‫ࢉܭ‬Λ
020
406080
Atomicnumberߦ ͏ ͨ Ί ɼલ ճ ֦ ு ͠ ͨ ̐ ੒ ෼ Dirac–Fock–ਤ̍ɽిࢠ૬ؔΤωϧΪʔ
Roothaan ๏ͱ૬ର࿦త CI ‫ࢉܭ‬ϓϩάϥϜ 7 Λ༻͍ͨɽ‫֩ࢠݪ‬Ϟσϧʹ͸ɼઌͷใࠂͷҰ༷ՙి‫ٿ‬໛‫ͱܕ‬૬ର࿦త Hylleraas-CI ‫ͱࢉܭ‬ಉ͡఺ిՙ‫ܕ‬Λ༻͍ͨɽ

100120ʦҰ༷ՙి‫ٿ‬໛‫( ܕ‬Uiform Sphere model:US)ʧ
‫ࢠݪ‬൒‫ʹܘ‬͸ Visscher Βͱಉ͘͡ 2.03952714 A1/3 + 1.39058668 × 10−5 bohrsʢA: ࣭ྔ਺ʣͰ༩͑ɼશͯͷ He-like ΠΦϯ‫ڞʹࢠݪ‬௨ͷ‫ج‬ఈؔ਺ͱͯ࣍͠ͷ 80 ߲͔Β੒Δ even-tempered Ψ΢ε‫جܕ‬ఈؔ਺Λ༻͍ͨɽ

ζiUS = α β i−1(α = 0.00588, β = 1.493320, i = 1, · · · , 80)‫ ͳີݫ‬Dirac–Fock(DF) ղͱͯ͠ Visscher Βͷ਺஋ղΛ‫ج‬४ʹͱΔɽ͜ͷ‫ج‬ఈؔ਺Λ࢖༻ͯ͠ੜͨ͡શΤωϧΪʔͷ‫ࠩޡ‬͸ Z = 116 ͷ͕࣌࠷େͰ 0.03 µ hartrees Ͱ͋Δɽʦ఺ిՙ‫( ܕ‬PointCharge model: PC)ʧઌͣɼ͍͔ͭ͘ͷ He-like ΠΦϯ‫࠷͍ͯͭʹࢠݪ‬దͳ‫ي‬ಓࢦ਺Λ‫ٻ‬Ίɼશͯͷ He-like ΠΦϯ‫ݪ‬ࢠʹ‫ڞ‬௨ͷ‫ي‬ಓࢦ਺ൣғͱͯ͠ 1.6 × 1022 ∼ 4.0 × 10−2 Λܾఆͨ͠ɽDF ͷશΤωϧΪʔͱͷ‫ޡ‬͕ࠩ US ͱಉ͡ఔ౓Ͱ͋Δ༷ʹɼ136 ߲͔Β੒Δ even-tempered Ψ΢ε‫جܕ‬ఈؔ਺Λ࡞੒ͨ͠ɽζiPC = α β i−1(α = 0.04, β = 1.495650, i = 1, · · · , 136)ʲ݁Ռͱߟ࡯ʳVPA ‫Ͱࢉܭ‬͸ෛͷӡಈΤωϧΪʔΛ࣋ͭεϐϊʔϧ΋ߟྀ͢ΔͨΊɼfull CI ‫ܭ‬ࢉͷ‫ྔࢉܭ‬͸๲େʹͳΔɽ‫ࢉܭ‬؆ུԽͷͨΊɼNVPA ‫ ͱࢉܭ‬VPA ‫ࢉܭ‬ͷిࢠ૬ؔΤωϧΪʔࠩ δCE ΛҎԼʹఆٛ͢Δ֤ରশੑ (spd f ) ຖͷిࢠ૬ؔΤωϧΪʔࠩ δCE(s), δCE(p), δCE(d),δCE( f ) ͷ࿨ͱͯ͠‫ٻ‬Ίͨɽίϩϯͷ‫ޙ‬ͷରশੑ͸‫ج͍ͨ༻ʹࢉܭ‬ఈؔ਺ͷରশੑΛࣔ͢ɽ

δCE = CE(VPA) − CE(NVPA) = δCE(s) + δCE(p) + δCE(d) + δCE( f ) + · · ·δCE(s) ≡ CE(VPA:s)−CE(NVPA:s),δCE(p) ≡ CE(VPA:sp) − CE(NVPA:sp) − δCE(s)δCE(d) ≡ CE(VPA:sd) − CE(NVPA:sd) − δCE(s)δCE( f ) ≡ CE(VPA:s f ) − CE(NVPA:s f ) − δCE(s)2 He ‫ ͔ ࢠ ݪ‬Β 116 Uuh114+ Π Φ ϯ ‫ Ͱ · ࢠ ݪ‬શ ͯ ͷ He-like ΠΦ ϯ ‫ ࢉ ܭ ͯ ͍ ͭ ʹ ࢠ ݪ‬Λ ߦ͍ɼ‫ٻ‬Ί֤ͨରশੑຖͷిࢠ૬ؔΤωϧΪʔࠩΛਤ̎ʹࣔ͢ɽPartial CE δCE / hartree

0.012

0.010

0.008

p0.006

sd0.004

f0.002020406080

100120

Atomic numberਤ ̎ ɽ֤ ର শ ੑ ຖ ͷ ి ࢠ ૬ ؔ Τ ω ϧ Ϊ ʔ ࠩഁઢ͸ US ͷ‫݁ࢉܭ‬ՌͰɼ࣮ઢ͸ PC ͷ‫݁ࢉܭ‬ՌͰ͋Δɽ͍ͣΕͷରশੑʹ͓͍ͯ΋ɼ‫ࢠݪ‬൪߸ͷେ͖͘ͳΔʹͭΕͯ NVPA ʹΑΔӨ‫͕ڹ‬େ͖͘ͳ͍ͬͯΔࣄ͕֬ೝ͞Εͨɽ·ͨ‫ݪ‬ࢠ֩ϞσϧʹΑΔӨ‫ڹ‬͸͋·ΓΈΒΕ͍ͯͳ͍ɽҎ্ͷ݁Ռ͔Β‫ٻ‬Ίͨ૬ର࿦తిࢠ૬ؔ
ΤωϧΪʔΛਤ̏ʹࣔ͢ɽ
Correlation energy / hartree-0.04

CI (VPA)
Hylleraas-CI

US-0.05

PC-0.06

US-0.07

CI (NVPA)
PC-0.08

-0.09

020

406080100120

Atomic numberਤ ̏ ɽి ࢠ ૬ ؔ Τ ω ϧ Ϊ ʔͲͪ Βͷ ‫֩ ࢠݪ‬Ϟ σ ϧͰ ΋ɼNVPA Λআ͍ ͨࣄ ʹΑ Γɼ‫ ࢠݪ‬൪ ߸͕ େ͖ ͳ෦ ෼Ͱ େ͖͘‫ݮ‬গ ͠ ͯ ͍ ͨ ి ࢠ ૬ ؔ Τ ω ϧ Ϊ ʔ ͕ ิ ਖ਼ ͞ Ε ɼ૬ ର ࿦ త Hylleraas-CI ‫ ݁ ࢉ ܭ‬Ռ ʹ ۙ ͍ ‫ ۂ‬ઢ ͕ಘΒΕͨɽ·ͨɼ‫ࢠݪ‬൪߸͕େ͖ͳͱ͜ΖͰɼ֩ϞσϧͷӨ‫͕͋ڹ‬Δࣄ͕֬ೝ͞Εͨɽ૬ର࿦త CI ‫ྀͨ͠ߟͰࢉܭ‬ରশੑ͸ɼ f ‫ي‬ಓ·ͰͰ͋Δɽg ‫ي‬ಓҎ߱Λߟྀͨ͠৔߹ɼPC
Ͱ͋Δ૬ର࿦త Hylleraas-CI ‫݁ࢉܭ‬Ռʹ͍ۙ‫ۂ‬ઢ͕ಘΒΕΔͷ͸ɼUS ͷ‫͋Ͱࢉܭ‬Δͱ‫͑ݴ‬Δɽࠓճͷௐ͔ࠪΒ֩ϞσϧͷӨ‫͕͋ڹ‬Δࣄ͕֬ೝ͞Ε͕ͨɼͦͷࠩ͸ɼDF ղΛ࠶‫͢ݱ‬ΔͨΊʹେ͖ͳ‫ي‬ಓࢦ਺ͷ‫ج‬ఈؔ਺Λଟ͘༻͍ͨͨΊʹੜͨ͡਺஋‫্ࢉܭ‬ͷ‫͑ݴͱࠩޡ‬Δ͔΋͠Εͳ͍ɽ౰೔͸ɼ‫ࢉܭ‬ͷৄࡉͳͲ΋‫ؚ‬Ίͯൃද͢Δɽ
1J. Midtdal and K. Aashanar, Phys. Norv. 2, 99 (1967).
G. Pestka and J. Karwowski, Explicitly Correlated Wavefunctions in Chemistry and Physics, 331 (2003).
3
G. Pestka, H. Tatewaki, and J. Karwowski, Phys. Rev. A 70, 024501 (2004).
4
H. Tatewaki and T. Noro, Chem. Phys. Lett. 399, 480 (2004).
5
Y. Watanabe and H. Tatewaki, J. Chem. Phys. 123, 074322 (2005).
6
Y. Watanabe, H. Nakano, and H. Tatewaki, J. Chem. Phys. 126, 174105 (2007).
7Y. Watanabe and O. Matsuoka, J. Chem. Phys. 116, 9585 (2002).
2