1E11ේሶ࡮ಽሶ♽ߩࠪࡘ࡟࡯࠺ࠖࡦࠟ࡯ᣇ⒟ᑼߩ⸃ᴺ. III.
ᣂߒ޿ࠨࡦࡊ࡝ࡦࠣᴺߣ⸘▚⚿ᨐ(㊂ൻ⎇*, JST-CREST**, ੩ᄢ㒮Ꮏ***) ٤ਛㄞ ඳ*,**, ⍹Ꮉ ᢕਯ***, ਛ᎑ ᶈਯ*,**,⑳ߚߜ߇ឭ᩺ߒߡ޿ࠆ৻⥸ߩේሶ࡮ಽሶ♽ߩࠪࡘ࡟࡯࠺ࠖࡦࠟ࡯ᣇ⒟ᑼ(SE)ߩ⸃ᴺ[1]ߦ߅޿ߡ
ߪ‫ޔ‬LSE ᴺࠍណ↪ߒ[2]‫߇ࠣࡦ࡝ࡊࡦࠨޔ‬ᔅⷐߢ޽ࠆ‫ޕ‬ᓥ᧪ߩࡕࡦ࠹ࠞ࡞ࡠ࡮ࡔ࠻ࡠࡐ࡝ࠬᴺ [3]ߢߪ‫࡞ࡊࡦࠨޔ‬ὐߩ transferability ߇ߥ޿‫⚿ޔ‬ᨐ߇ randomness error ࠍ฽߻ߚ߼ㅪ⛯ᕈ߇ߥ޿‫ޔ‬╬ߩᰳ㒱߇޽ߞߚ‫ߦߎߎޕ‬ᣂߚߦឭ᩺ߔࠆࡠ࡯ࠞ࡞࡮ࠨࡦࡊ࡝ࡦࠣᴺߪ‫ࡠ࡞ࠞ࠹ࡦࡕޔ‬ᴺߦࠃࠄߥ޿ rational sampling ߢ޽ࠅ‫ ޔ‬atomic shell sampling ࠍ unit sampling ߣߔࠆ local ߢ߆ߟtransferable ߥࠨࡦࡊ࡝ࡦࠣᴺߢ޽ࠆ‫ߩߎޕ‬ᣇᴺߢߪ‫ޔ‬ේሶࠨࡦࡊ࡝ࡦࠣ‫ޔ‬ಽሶࠨࡦࡊ࡝ࡦࠣߣ޿ߞߚᄙ㔚ሶࠨࡦࡊ࡝ࡦ߽ࠣ⥋ߞߡ◲නߢ޽ࠅ‫ ߽ࠄࠇߘޔ‬transferable ߢ޽ࠆ‫ࡦࠨߩߎߦࠄߐޕ‬ࡊ࡝ࡦࠣߪㅪ⛯ߥ⚿ᨐࠍਈ߃ࠆߩߢ‫⚿ޔ‬ᨐߩᄌಽ߿ᓸಽ߇น⢻ߢ޽ࠅ‫ޔ‬ᓥߞߡ‫ޔ‬SE ߣห╬ߢ޽ࠆ‫ޟ‬H㧞ਸ਼ࠛ࡜࡯ᦨዊߩේℂ‫ࠍޠ‬ታ↪⊛ߦ૶߁ߎߣ߇น⢻ߦߥࠆ‫ޕ‬LSE ᴺߩ਎⇇ߢߪ‫ޟߦ⥸৻ޔ‬ᄌಽේℂ‫߇ޠ‬ᚑࠅ┙ߚߥ޿ߎߣ߆ࠄ‫ߩߎޔ‬੐ታߪᄢ߈ߥ⑔㖸ߢ޽ࠅ‫ޔ‬ᄢ߈ߥน⢻ᕈࠍᗧ๧ߔࠆ‫ޕ‬ߎߩࠃ߁ߦ‫ޔ‬local, transferable, ㅪ⛯‫◲ޔ‬න‫ޔ‬㜞ㅦ‫ߤߥޔ‬ᄢ߈ߥࡔ࡝࠶࠻ࠍᜬߟߎߩᣇᴺߪ‫ޔ‬II ߢㅀߴߚ FATM(from atom to molecule)ᴺߣߣ߽ߦ‫ޔ‬ൻቇߩේℂߦࡑ࠶࠴ߒߡ޿ߡ‫ޔ‬SE ࠍൻቇߩ࡟ࡌ࡞ߢ⸃ߎ߁ߣߔࠆᚒ‫ߦ࠴࡯ࡠࡊࠕߩޘ‬ㆡߒߡ޿ࠆ‫ ߦࠄߐޕ‬I ߢㅀߴߚ෻ኻ⒓ൻࠕ࡞ࠧ࡝࠭ࡓߣߣ߽ߦ‫ߩࠄࠇߎޔ‬ᣇᴺߪ‫ޔ‬SE ࠍታ↪࡟ࡌ࡞ߢ⸃ߊߚ߼ߦᔅⷐߥ⸘▚ᤨ㑆ࠍ⍴❗ߔࠆߎߣߦ߽ߟߥ߇ࠆ‫ࠣࡦ࡝ࡊࡦࠨߩߎޔߪߢߎߎޕ‬ᴺߦߟ޿ߡ⺑᣿ߔࠆߣߣ߽ߦ‫ޔ‬I, II ߢㅀߴߚᣇᴺ߽ㆡ↪ߒߡ‫ޔ‬ᓧࠄࠇߚ⚿ᨐࠍ✚ว⊛ߦႎ๔ߔࠆ‫ޕ‬[ᣇᴺ]㧦SE ߩ⸃ߢ޽ࠆᱜ⏕ߥᵄേ㑐ᢙ߇! # $ ci"i(1)iߦࠃߞߡ⴫ࠊߐࠇࠆߣߒࠃ߁‫ޔᤨߩߎޕ‬㑐ᢙ "i ࠍ complement function ߣ๭߱‫ޕ‬complement ߪ‫ޟ‬วࠊߖߡቢోߦߥࠆࠃ߁ߥ৻ⷐ⚛‫ࠍޠ‬ᗧ๧ߒ‫" {ޔ‬i }߇ᱜ⏕ߥᵄേ㑐ᢙ! ࠍᒛߞߡ޿ࠆߎߣ߆ࠄฬߠߌࠄࠇߚ‫ޕ‬ᵈᗧߔߴ߈ߪߎࠇߪ޿ࠊࠁࠆ‫ޟ‬ቢో♽‫ߪߣޠ‬㆑޿‫ޔ‬ᵈ⋡ߒߡ޿ࠆ♽ߩᱜ⏕ߥᵄേ㑐ᢙ! ࠍ⸥ㅀߔࠆߩߺߢ޽ࠆ‫߇ߜߚ⑳ޕ‬ឭ᩺ߒߡ޿ࠆේሶ࡮ಽሶ♽ߩࠪࡘ࡟࡯࠺ࠖࡦࠟ࡯ᣇ⒟ᑼߩ⸃ᴺߦ߅޿ߡߪ‫ޔߕ߹ޔ‬SE ߩ⸃ߩචಽ᧦ઙߣ߽⸒߃ࠆ(1)ᑼࠍ᭴ᚑߔࠆ complementfunction ࠍ scaled Schrödinger equation (SSE)ࠍၮߦ‫ߡߞ૶ࠍࡦࠕ࠾࠻࡞ࡒࡂߩ♽ޔ‬૞ࠅ਄ߍ(Hamiltonian generated complement function (HGCF))‫ޔ‬ᰴߦ‫✢ޔ‬ᒻᄌᢙ ci ࠍ SE ߩᔅⷐ᧦ઙࠍ૶ߞߡ⸘▚ߔࠆ‫ޕ‬FC(free complement)ᴺ, PC(piecewise complement) ᴺ╬߇ߘࠇߢ޽ࠆ‫ⶄޕ‬㔀ߥ৻⥸ߩේሶ࡮ಽሶߢߪ "i ߩࡂࡒ࡞࠻࠾ࠕࡦⓍಽ╬ࠍ⸘▚ߢ߈ߥ޿ߩߢ‫ޔ‬ᚒ‫ ߪޘ‬SE ߩዪᚲᕈࠍ೑↪ߒߡ‫ޔ‬ฦࠨࡦࡊ࡝ࡦࠣὐߢ SE ߇ᚑࠅ┙ߞߡ޿ࠆߣ޿߁ᔅⷐ᧦ઙࠍ૶ߞߡଥᢙ ci ࠍ⸘▚ߔࠆ
(LSE ᴺ)‫ޕ‬ᱜ⏕ߥᵄേ㑐ᢙߩߣ߈ local energy H! / ! ߪ૗ಣߢ߽ E ߢ޽ࠆ߆ࠄ‫ࡦࠨ߽ߘ߽ߘޔ‬ࡊ࡝ࡦࠣߪኈᤃߥߪߕߢ޽ࠅ‫ⶄޔ‬㔀ߐߪ⃻ታߦߪ(1)ᑼߢή㒢୘ߩ complement function ࠍ฽߼ࠆߎߣߪߢ߈ߥ޿ߚ߼ local energy ߇ߘߩ⒟ᐲߦᔕߓߡᝄേߔࠆߎߣ߆ࠄ᧪ࠆ‫ޕ‬એ਄ߩࠃ߁ߦ‫ޔ‬

FC-LSE ߿ PC-LSE ╬ࠊࠇࠊࠇߩ⸃ᴺߢߪ‫ࠣࡦ࡝ࡊࡦࠨޔ‬ᴺߦኻߔࠆଐሽᕈߪዊߐ޿ߪߕߢ޽ࠅ‫ޔ‬ᓥߞߡᱜߒ޿ࠨࡦࡊ࡝ࡦࠣᴺࠍ૶ߞߡ޿ࠆ㒢ࠅ‫ࠣࡦ࡝ࡊࡦࠨޔ‬ᴺߩ⚦߆ߥ⹦⚦ߦ޽߹ࠅߎߛࠊࠆᔅⷐ߇ߥ޿ߎߣ߇ಽ߆ࠆ‫ޔߢߎߘޕ‬ᰴߩᤃߒߊ᦭↪ߥᣇᴺࠍዉ౉ߔࠆ‫ޕ‬ Importance sampling ߪ⸘▚⚿ᨐߩ♖ᐲࠍ਄ߍࠆ߁߃ߢ㊀ⷐߥ⠨߃ᣇߢ޽ࠅ[3]‫ߦࠇߎޔ‬ၮߠ޿ߡ‫ޔ‬ኒᐲ㑐ᢙ % ߣ reference function & ࠍዉ౉ߔࠆ‫ޕ‬ਔ⠪ߪ & # % ߢ⚿߫ࠇߡ޿ࠆ‫ޕ‬᥉ㅢߩ
Metropolis sampling ߢߪ % ߣߒߡ♽ߩᵄേ㑐ᢙߩ㧞ਸ਼ࠍߣࠆ߇‫ޔ‬ᵄേ㑐ᢙߪ local ߢ߽transferable ߢ߽ߥ޿ߩߢ‫ߥ߁ࠃߩߘޔ‬ᕈ⾰ࠍᜬߟࠨࡦࡊ࡝ࡦࠣࠍቯ⟵ߔࠆߎߣߪߢ߈ߥ޿‫ޕ‬%ߪනߦࠨࡦࡊ࡝ࡦࠣὐߩಽᏓࠍዉߊኒᐲ㑐ᢙߢ޽ࠅ‫ޔ‬㊂ሶജቇ⊛ߥ㊂ߢ߽૗ߢ߽ߥ޿߆ࠄ‫ޔ‬ᚒ‫ࠍ &ߪޘ‬ᰴߩࠃ߁ߦ‫ޔ‬reference orbital, & a ( ri ) ߩⓍߢᦠߊ‫ޕ‬& (1, 2,..., N ) # & a (1)&b (2) !& n ( N ) .

(2)ߎࠇߪ‫ޔ‬
ᄙ㔚ሶ♽ߩ Hartree Ⓧߦૃߡ޿ࠆߩߢ‫ޔ‬ࠊࠇࠊࠇߩ Local Sampling ᴺߪ Hartree samplingߣ߽๭ߴࠆ‫ޕ‬ᰴߦ‫ޔ‬᭴ᚑⷐ⚛ & a ( ri ) ߣߒߡ atomic shell orbital ࠍ૶߁‫ౕޕ‬૕⊛ߦߪ nodelessߥ Slater 㑐ᢙߢ Clementi ߩ exponent ࠍ૶߁‫ޕ‬node ߪ㔚ሶ᭴ㅧ⺰⊛ߦߪ㊀ⷐߥ㊂ሶജቇ⊛ᕈ⾰ߢ޽ࠅ‫⸥ߩߘޔ‬ㅀߪචಽߥࠨࡦࡊ࡝ࡦࠣὐࠍߣߞߡⴕ߁ߴ߈ߢ޽ࠆ‫ߪࠣࡦ࡝ࡊࡦࠨޕ‬ᭂᐳᮡߢⴕ޿ߘࠇࠍ x,y,z ᐳᮡߦᄌ឵ߒߚ‫ޕ‬ේሶࠨࡦࡊ࡝ࡦࠣߪߎߩ shell sampling ࠍ૶ߞߡ(2)ᑼߩ
᭽ߦ (1s)2(2s)2(2p)6(3s)2 ߣ޿ߞߚౕวߦ᭴ᚑߔࠆ‫ߦࠄߐޕ‬ಽሶࠨࡦࡊ࡝ࡦࠣߪ‫ޔ‬᭴ᚑේሶࠨࡦࡊ࡝ࡦࠣࠍ⹥ᒰߔࠆේሶߩ૏⟎ߦ߅޿ߡ૞ᚑߔࠆ‫ޕ‬ᩭ㑆〒㔌ߩᄌൻߦኻߒߡߔߴߡߩࠨࡦࡊ
࡝ࡦࠣὐ߇ᐔⴕ⒖േߔࠆ߆ࠄ‫⚿▚⸘ޔ‬ᨐߪᩭ㑆〒㔌ߦኻߒߡṖࠄ߆ߦߥࠆ‫ޕ‬[⚿ᨐ]㧦⴫㧝ߦ Li ේሶߩ⚿ᨐ  ࠍ␜ߒߚ‫ޕ‬FC ᴺߩࠝ࡯࠳࡯߇਄State n߇ࠆ⒟‫ޔ‬H-sq error ߪዊߐߊߥࠅ‫ ޔ‬2Sࠛࡀ࡞ࠡ࡯߽ᱜߒ޿୯ߦㄭߠ޿
ߡ޿ࠆ‫ߪࠣࡦ࡝ࡊࡦࠨޕ‬㧞S, 㧞P⁁ᘒߣ߽౒ㅢߢ޽ࠅ‫ޔ‬107 ߦᲧߴ⴫㧝㧚Li ߩ㧞S, 㧞P ⁁ᘒ (15810 sampling points)

2

P

Cf’s

H-sq error

Energy (au)42563.012931x 10-3-7.477477

5465

1.618399 x 10-3-7.478136

6786

5.747508 x 10-4-7.478068-7.478060

4514

2.120178 x 10-3-7.410586-7.410155

ߡ⚂ 1/100 ߢᷣࠎߢ޿ࠆ‫ޕ‬ฝ࿑ߦߪ LiH ಽሶߩၮᐩ⁁ᘒߣ╙৻ബ⿠⁁ᘒߩࡐ࠹ࡦࠪࡖ࡞ࠞ࡯ࡉࠍ␜ߒߚ‫ޕ‬Ṗࠄ߆ߥࠞ࡯ࡉ߇ឬ߆ࠇߡ߅ࠅ‫ߩߘޔ‬ᓸಽ߆ࠄᝄേߩቯᢙ╬ࠍ⸘▚ߔࠆߎߣ߇ߢ߈ࠆ‫ޕ‬[1] H. Nakatsuji, Phys. Rev. Lett. 93, 030403(2004). [2] H. Nakatsuji, H. Nakashima, Y.
Kurokawa, A. Ishikawa, Phys. Rev. Lett. 99,240402 (2007). [3] B. L. Hammond, W. A.
Lester, Jr., P. J. Reynolds, Monte Carlo Methods
in Ab Initio Quantum Chemistry (WorldScientific, Singapore, 1994). [4] E. Clementi, L.
Raimondi, JCP, 38, 2686 (1963).

Exact energy