2P112
ಽഀ⛔ᴦ(DC)-CPHFᴺߩ㐿⊒ߣࠛࡀ࡞ࠡ࡯൨㈩⸘▚߳ߩᔕ↪
㧔ᣧᄢవㅴℂᎿ1, ಽሶ⎇2, ᣧᄢℂᎿ⎇3㧕٤࿖ቯ෹㓉1㧘ዊᨋᱜੱ1,2㧘ਛ੗ᶈᏈ1,3‫ޤ⸒✜ޣ‬ಽഀ⛔ᴦ(DC)ᴺߪ SCF ⸘▚ߩ linear-scaling ᴺߩ৻ߟߢ޽ࠆ‫ޕ‬ᒰ⎇ⓥቶߢߪ DC ᴺࠍ Hartree-Fock(HF)⸘▚ߦㆡ↪ߒ‫ޔ‬post-HF ⸘▚ᴺ߳᜛ᒛߔࠆᚻᴺࠍឭ᩺ߒߡ߈ߚ‫࡯ࠡ࡞ࡀࠛߩࠄࠇߎޔߦࠄߐޕ‬⸘▚ᴺࠍ㊂ሶൻቇ⸘▚ࡄ࠶ࠤ࡯ࠫ GAMESS ߩ౏㐿 [1]ߦታⵝߒߡ߈ߚ߇‫ޔ‬MD ⸘▚߿᭴ㅧᦨㆡൻ
⸘▚ߦᔅⷐߢ޽ࠆ⸃ᨆ⊛ࠛࡀ࡞ࠡ࡯൨㈩ߩᬌ⸛ߪⴕߞߡߎߥ߆ߞߚ‫ޕ‬ߎࠇ߹ߢߦᣢߦ‫ޔ‬DC-SCF ߦၮߠߊ⸃ᨆ⊛ࠛࡀ࡞ࠡ࡯൨㈩ߩ⴫ᑼߪ Yang ࠄߦࠃࠅឭ᩺ߐࠇߡ޿
ࠆ[2]‫⴫ߩߎޔߒ߆ߒޕ‬ᑼߪㅢᏱߩ SCF ࠛࡀ࡞ࠡ࡯൨㈩ߩᑼࠍ DC ⊛ߦ᳞߼ࠆ߽ߩߢ޽ࠅ‫ޔ‬DC-SCFᴺߦኻߔࠆ෩ኒߥࠛࡀ࡞ࠡ࡯൨㈩ࠍ᳞߼ࠆ߽ߩߢߪߥ޿‫⎇ᧄޕ‬ⓥߢߪ‫ޔ‬DC-SCF ࠛࡀ࡞ࠡ࡯ࠍ⋥ធᓸಽߔࠆߎߣߦࠃࠅ‫ޔ‬ᱜ⏕ߥ DC-SCF ࠛࡀ࡞ࠡ࡯൨㈩ߩ⴫ᑼࠍዉ޿ߚ‫⴫ߩߎޕ‬ᑼߢߪ‫ޔ‬coupledperturbed HF (CPHF)ᣇ⒟ᑼߩ⸃߇ᔅⷐߣߥࠆ‫ᧄߢߎߘޕ‬ᚻᴺࠍታⵝߒߡࠕ࠮ࠬࡔࡦ࠻ࠍⴕߞߚ‫ޕ‬߹ߚ‫ߩߎޔ‬ᣇᴺࠍ಴⊒ὐߣߒߡ‫ޔ‬Yang ࠄߩᣇᴺߣߪ೎ߩ CPHF ⸘▚ࠍ↪޿ߥ޿ㄭૃᴺ߽㐿⊒ߒߚ‫ޕ‬‫ޣ‬DC-HF ࠛࡀ࡞ࠡ࡯൨㈩ߩ⸘▚ᴺ‫ޤ‬
HF ࠛࡀ࡞ࠡ࡯൨㈩ߪ‫ޔ‬1, 2 㔚ሶⓍಽߩ൨㈩ࠍ฽߻ Hellmann-Feynman 㗄ߣ‫ޔ‬ኒᐲⴕ೉ D ߩ൨㈩ࠍ฽߻ Pulay 㗄(P term)߆ࠄᚑࠆ‫ޕ‬DC ᴺߢߪኒᐲⴕ೉ߦㄭૃࠍ฽߻ߚ߼‫ޔ‬
Pulay 㗄ߦᏅ⇣߇↢ߓࠆ‫ޕ‬Yang ࠄ[2]ߪ‫ޔ‬ㅢᏱߩ HF ࠛࡀ࡞ࠡ࡯൨㈩ߩ⴫ᑼ߆ࠄ‫ޔ‬એਅߩࠃ߁ߦㄭૃ⊛ߦ Pulay 㗄ࠍ᳞߼ࠆᚻᴺࠍឭ᩺ߒߚ‫ޕ‬! #DDC "!
#S "F ( % &2Tr ' W DC
P term $ 2Tr '(             (1)
qq##)
*)
*
ߎߎߢWDCߪࠛࡀ࡞ࠡ࡯㊀ߺઃ߈ኒᐲⴕ೉ߢ޽ࠅ‫ޔ‬DCᴺߦ߅ߌࠆኒᐲⴕ೉߆ࠄߩ㘃ផߢಽഀⴕ೉pࠍ↪޿ߡએਅߩࠃ߁ߦਈ߃ࠄࠇࠆ‫ޕ‬+,W./DC 2 3 W./- 2 3 p./
3 f 0 1 F & 1 i- C.-iC/-i1 i-            (2)--i(1)ᑼߪ‫ޔ‬ㅢᏱߩ HF ᴺߢߪ෩ኒߦᱜߒ޿ᄌᒻߦߥߞߡ޿ࠆ߇‫ޔ‬DC-HF ᴺߢߪಽഀⴕ೉ߩᓇ㗀ߦࠃࠅㄭૃߣߥߞߡߒ߹޿‫ޔ‬ᱜߒߊߪએਅߩࠃ߁ߥᑼߣߥࠆ‫ޕ‬!
5 #C ! #D- - "#C - 6 "P term 2 23 3 'F ( 2 23 3 3 ' 3 ni 8 .i C/-i 7 C.-i / i 9F/.- (     (3)8#q 9; (*- .4S (- ) ) #q- .4S (- ) / '* ..
) i : #qߎߎߢniߪ゠㆏භ᦭ᢙ‫ޔ‬S(!)ߪㇱಽ♽!ߩਛᄩ㗔ၞߩAOࠍ⴫ߔ‫ߩߎޕ‬ᑼߩD-<ߦߪಽഀⴕ೉ߪ฽߹ࠇߡ޿ߥ޿‫ޕ‬ㇱಽ♽ߩMOଥᢙߩ൨㈩ࠍએਅߩࠃ߁ߦหߓㇱಽ♽ߩMOߢዷ㐿ߔࠆ‫ޕ‬#C.-i#qMO(- )% 3 U -pi,q C.- p [i 4 occ(- )]               (4)pDC ᴺߢߪ㕖ᢛᢙߩභ᦭ᢙߣߥࠆߎߣ߽޽ࠆ߇‫ޔ‬CPHF ᣇ⒟ᑼࠍዉߊ㓙ߦߪ‫◲ޔ‬ଢߩߚ߼ߦࡈࠚ࡞ࡒḰ૏ࠍႺߦභ᦭゠㆏ߣ઒ᗐ゠㆏ߦಽഀߔࠆㄭૃࠍⴕ߁ߎߣߦߔࠆ‫(ޔࠅࠃߦࠇߎޕ‬4)ᑼߩዷ㐿ଥᢙߩ߁ߜභ᦭-઒ᗐㇱಽߪ‫ޔ‬એਅߩ CPHF ᣇ⒟ᑼࠍ⸃޿ߡᓧࠄࠇࠆ‫ޕ‬

+1 & 1 ,U--- ,qiaai

vir + 0 , occ + 0 ,&30 3 3 A- 0 U 0 2 B,ai ,bj

b,q

bj,q0 ,ai            (5)jߎߎߢ‫ޔ‬   Aai- ,,bj0 2

0C.- C/- 3 p>?
C>0 C?0 =4+./ >? , & +.> /? , & +.? /> ,@3./ +- ,>? + 0 ,ia4LB0-,ai, q 2jb(6)4L

occ + 0 ,DAGA - , q- ,q 0C .-a C/-i F F./
1i &p>?
C>0j C?0k S 0jk, q =2+./ >? , & +.> /? ,@C   (7)& S ./
ABAE./ 4L +- ,0j ,k >? 4L + 0 ,3+, 33 3ߢ‫ޔ‬L(!)ߪㇱಽ♽!ߩਛᄩ෸߮ࡃ࠶ࡈࠔ㗔ၞߦ฽߹ࠇࠆ AO ࠍ⴫ߔ‫(ޕ‬5)ᑼߪᐳᮡ q ࠍዪ࿷ൻ㗔ၞౝߦ฽߻ㇱಽ♽ߦኻߒߡหᤨߦ⸃ߊᔅⷐ߇޽ࠆ߇‫ߩߡోޔ‬ዪ࿷ൻ㗔ၞߦኻߒߡ⸃ߊᔅⷐߪߥ޿‫ޕ‬߹ߚ‫(ޔ‬3)ᑼࠍ಴⊒ὐߣߒߚ CPHF ᣇ⒟ᑼߩ⸃ࠍ૶ࠊߥ޿એਅߩㄭૃ⊛ߥ⴫ᑼࠍឭ᩺ߒߚ‫ޕ‬! #D- - "! #S"
P term 2 23 3 '
F ( % &2Tr ' 3 W- S- [ L(- )H S (- )] D- [ S (- )H L (- )] ( - .4S (- ) ) #q) #q ** ..

(8)ߎߎߢ‫ޔ‬S- [ L+- ,HS +- ,] ߪ S- ߩㇱಽⴕ೉(ᰴరߪ L(!)S(!))‫ޔ‬D- [ S +- ,H L +- ,] ߪ D- ߩㇱಽⴕ೉(ᰴరߪ S(!)
L(!))ߢ޽ࠆ‫(ޕ‬8)ᑼߩዉ಴ߦ߅޿ߡߪ‫ޔ‬ㄭૃ⊛ߥㇱಽ♽ߩኒᐲⴕ೉ߩ╬౹ᕈߩᑼ          D ! % D- S - D-                        (9)ࠍ઒ቯߔࠆߎߣ߆ࠄ಴⊒ߒߡ‫ޔ‬ኒᐲⴕ೉ߩᓸಽࠍ೎ߩᑼߢ⟎߈឵߃ࠆㄭૃࠍߒߡ޿ࠆ‫ޕ‬#S - #DI & DD#q#q(10)‫ޣ‬᭴ㅧᦨㆡൻ⸘▚߳ߩᔕ↪‫ޤ‬
਄ㅀߩDC-HFࠛࡀ࡞ࠡ࡯൨㈩ᴺࠍGAMESSߦታⵝߒ‫!ޔ‬-strandဳࠣ࡝ࠪࡦࡍࡊ࠴࠼(gly)10ߩ᭴ㅧᦨㆡൻ⸘▚ߦᔕ↪ߒߚ‫ޕ‬ㇱಽ♽ߪ 1 ᱷၮߣߒ‫ߪࠔࡈ࠶ࡃޔ‬㓞ធߔࠆᏀฝߘࠇߙࠇ 2 ᱷၮࠍ฽߼ߚ‫ޕ‬ၮᐩ㑐ᢙߪ 6-31Gࠍ↪޿ߚ‫ޕ‬Table 1 ߦYangࠄߩᑼ(1)෸߮੹࿁ឭ᩺ߒߚᑼ(8)ߢ᳞߼ߚ᭴ㅧࡄ࡜ࡔ࡯
࠲ߩㅢᏱߩHF⸘▚ߦࠃࠆᦨㆡ᭴ㅧ߆ࠄߩ⺋Ꮕߩ⛔⸘ࠍ␜ߒߚ‫੹ޕ‬࿁ߩᣇᴺߢ᳞߼ߚ᭴ㅧߪ‫ޔ‬Yang
ࠄߩᣇᴺࠃࠅ߽ㅢᏱߩHFᴺ߆ࠄߩ⺋Ꮕ߇ዊߐߊߥߞߡ޿ࠆߎߣ߇⏕⹺ߢ߈ࠆ‫ޔߚ߹ޕ‬DC-HFࠛࡀ࡞ࠡ࡯ߪ੹࿁ߩᣇᴺߢ᳞߼ߚ᭴ㅧߩᣇ߇Yangࠄߩᣇᴺߢ᳞߼ߚ᭴ㅧࠃࠅ߽ 0.312 mhartreeૐߊ‫ࠃޔ‬ࠅ㜞♖ᐲߥ᭴ㅧᦨㆡൻ⸘▚߇ߢ߈ߡ޿ࠆߎߣ߇⏕߆߼ࠄࠇߚ‫ޕ‬
Table 1. Errors of optimized structure of (gly)10 from conventional HF results.
(ME: mean error, MAE: mean absolute error, MaxE: maximum error)

MethodError

MEMAEMaxE
ME
Present (Eq. (8)) MAEMaxEYang (Eq. (1))

Bond length[pm]
0.043
0.073
0.224
-0.0020.010-0.046

Geometrical parameter
Bond angle
Dihedral angle[degree][degree]
0.04
0.22
0.33
1.06
1.20
0.01
0.05
0.16
0.24
0.55

[1] http://www.msg.chem.iastate.edu/gamess/[2] W. Yang and T.-S. Lee, J. Chem. Phys. 103, 5674 (1995).