2P114
ࢱࣥࣃࢡ㉁඲㟁Ꮚィ⟬࡟࠾ࡅࡿ Schwegler-Challacombe ࣇ࢛ࢵࢡ஺᥮㡯ィ⟬ຍ㏿ࡢ᳨ド㸦ᮾ኱⏕◊ 1㸪ᮾ኱㝔ᕤ 2㸧‫ۑ‬㜿㒊 ᩄᙪ 1㸪ᖹ㔝 ᩄ⾜ 1㸪▼ᕝ ೺ኴ㑻 2㸪బ⸨ ᩥಇ 1࠙ࡣࡌࡵ࡟ࠚ
ᐦᗘỗ㛵ᩘἲ࡟ࡼࡿ㔞Ꮚ໬Ꮫィ⟬࡟࠾࠸࡚ࠊSCF ィ⟬ࢆᏳ඲࡟ࠊ㧗㏿࡟཰᮰࡬࡜ᑟࡃᢏ⾡ࡣࠊ
HOMO-LUMO ࢠࣕࢵࣉࡀ⊃࠸ࢱࣥࣃࢡ㉁ࡢࡼ࠺࡞ᕧ኱ศᏊ࡛ࡣ≉࡟㔜せ࡛࠶ࡿࠋᐦᗘỗ㛵ᩘἲࡢ஺᥮
┦㛵ỗ㛵ᩘ࡟ࡣࡉࡲࡊࡲ࡞ỗ㛵ᩘࡀᥦ᱌ࡉࢀ࡚࠾ࡾࠊ୰࡛ࡶ B3LYP[1]࡞࡝ࡢࣁ࢖ࣈࣜࢵࢻỗ㛵ᩘࡣࡑࡢィ⟬⢭ᗘࡢ㧗ࡉ࠿ࡽࠊ⌧ᅾᶆ‽ⓗ࡟⏝࠸ࡽࢀ࡚࠸ࡿࠋࣁ࢖ࣈࣜࢵࢻỗ㛵ᩘࡣ Fock ࡢ஺᥮㡯ࢆྵࢇ࡛࠾ࡾࠊィ⟬࡟ࡣ 4 ୰ᚰ 2 㟁Ꮚ✚ศࡀᚲせ࡛࠶ࡿࠋࡑࡢ₇⟬㔞ࡣᇶᗏ㛵ᩘࡢ⥲ᩘࢆ N ࡜ࡍࡿ࡜ O(N4)࡜
࡞ࡿࡢ࡛ࠊᕧ኱ศᏊࡢィ⟬࡛ࡣィ⟬⢭ᗘࢆಖࡕ࡞ࡀࡽ₇⟬㔞ࢆ኱㔞࡟๐ῶࡍࡿ᪉ἲࡀᚲ㡲࡛࠶ࡿࠋFockࡢ஺᥮㡯ࡣࠊࡑࡢᑐ⛠ᛶ࠿ࡽ RI ἲࡢࡼ࠺࡞ᜏ➼ศゎἲ࡛ィ⟬ࢧ࢖ࢬ౫Ꮡᛶࢆୗࡆࡿࡇ࡜ࡀ࡛ࡁ࡞࠸ࡓ
ࡵࠊFock ࡢ஺᥮㡯ࡢ≀⌮ⓗᛶ㉁࡟╔┠ࡋࡓ⊂≉ࡢ㧗㏿ィ⟬ἲࡀᥦ᱌ࡉࢀ࡚࠸ࡿࠋ
ᮏ◊✲࡛ࡣࠊ₇⟬㔞ࢆ⌮ㄽୖ O(N)ࡲ࡛๐ῶࡉࡏࡿࡇ࡜ࡀ࡛ࡁࡿ Schwegler ࡜ Challacombe ࡢ᪉ἲ[2]࡟╔┠ࡋࡓࠋࡓࡔࡋࡇࡢ᪉ἲࡣィ⟬⢭ᗘࢆ᏶඲࡟ࡣಖドࡏࡎࠊࡲࡓィ⟬ࡍࡿ⣔࡟ᛂࡌ࡚⮬࡛᫂࡞࠸」ᩘࡢࣃ࣓࣮ࣛࢱࢆ㐺ษ࡟タᐃࡍࡿᚲせࡀ࠶ࡿࠋᮏ◊✲࡛ࡣ Schwegler-Challacombe ࡢ Fock ஺᥮㡯ィ⟬ຍ㏿ἲ࡟ࡘ࠸᳨࡚ドࢆ⾜ࡗࡓࠋ࢝ࢵࢺ࢜ࣇ㜈್ࡸ஺᥮㡯ࡢ㊥㞳౫Ꮡᛶࢆ㏆ఝࡍࡿᐃᩘ࡞࡝ࠊタᐃࡀᚲせ࡞ࣃ࣓࣮ࣛࢱ࡟ࡘ࠸࡚ SCF ィ⟬ࡢ཰᮰࡟୚࠼ࡿᙳ㡪ࢆㄪ࡭ࠊࢱࣥࣃࢡ㉁ࡢ඲㟁Ꮚィ⟬࡟㐺ࡋࡓࣃ࣓࣮ࣛࢱタᐃࡢ᳨ウࢆ⾜ࡗࡓࠋ࠙Schwegler-Challacombe ࡢࣇ࢛ࢵࢡ஺᥮㡯ィ⟬ຍ㏿ἲࠚ
Fock ࡢ஺᥮㡯ࡣࠊᐦᗘ⾜ิ P ࡜ 4 ୰ᚰ 2 㟁Ꮚ✚ศࢆ⏝࠸࡚௨ୗࡢࡼ࠺࡟⾲ࢃࡉࢀࡿࠋ

K pr = å Pqs pq | rs
qsࡇࢀࡣࠊ௨ୗࡢࡼ࠺࡟᭩ࡁ᥮࠼ࡽࢀࡿࠋ

K pr = òò r pr (r, r¢)

1r (r, r¢)drdr¢| r - r¢ |r (r, r¢) = å Pqs rqs (r, r¢), r pr (r, r¢) = g *p (r) g r (r)qsࡇࡇ࡛ g p (r ) ࡣᇶᗏ㛵ᩘ࡛࠶ࡿࠋSchwegler-Challacombe ἲ࡛ࡣࠊࡇࢀࢆ~K pr = òò r pr (r, r¢)1e -|r -r¢|/ l drdr¢| r - r¢ |~࡟ࡼࡾ㏆ఝࡍࡿࠋࡓࡔࡋ l ࡣ⣔࡟౫Ꮡࡍࡿࢫࢣ࣮ࣝࣃ࣓࣮ࣛࢱ࡛࠶ࡾࠊ㏻ᖖࡣ 1.0 ࡜ࡍࡿࠋ K pr ࡀ㜈್

e 1 ࡼࡾࡶᑠࡉ࠸ሙྜ࡟ K pr ࡢィ⟬ࢆ┬␎(࢝ࢵࢺ࢜ࣇ)ࡍࡿࡇ࡜࡟ࡼࡾࠊO(N4)࡛࠶ࡗࡓ 4 ୰ᚰ 2 㟁Ꮚ✚ศࢆࠊ⤯⦕య࡞࡝ࣂࣥࢻࢠࣕࢵࣉࡀᏑᅾࡍࡿ≀㉁࡟ᑐࡋ࡚ࡣࠊ⌮ㄽୖ O(N)ࡲ࡛๐ῶྍ⬟࡛࠶ࡿ࡜ࡋ࡚࠸ࡿࠋ࠙᪉ἲࠚ
Lys-Gly-Arg ࡢ 3 ṧᇶࢆ⏝࠸࡚ SCF ィ⟬ࢆ⾜ࡗࡓࠋィ⟬࡟ࡣ ProteinDF ࢆ⏝࠸ࡓࠋ཰᮰ุᐃ࡟ࡣࠊ඲
-5࢚ࢿࣝࢠ࣮ኚ໬ࡢ⤯ᑐ್ࡀ 1.0 ´10 a.u.௨ୗ࡛ࠊ࠿ࡘ㟁Ꮚᐦᗘ⾜ิせ⣲ࡢኚ໬ࡢ⤯ᑐ್ࡢ᭱኱್ࡀ

1.0 ´10-4 ௨ୗࡢ᮲௳ࢆ⏝࠸ࡓࠋࡲࡎࣃ࣓࣮ࣛࢱタᐃ࡜ࡋ࡚ࠊe1 = 1.0 ´10-10 ࠾ࡼࡧ l = 1.0 ࡜ࡋ࡚ᐇ㦂ࢆ⾜ࡗࡓ࡜ࡇࢁࠊSCF ィ⟬ࡣ཰᮰ࡋ࡞࠿ࡗࡓࠋࡑࡇ࡛ࠊSCF ィ⟬ࢆ཰᮰ࡉࡏࡿࡓࡵࡢ᮲௳࡟ࡘ࠸࡚ࠊ

ձ e 1 ࢆኚ࠼ࡓሙྜ ղ l ࢆኚ࠼ࡓሙྜ ճィ⟬ἲࡢಟṇ࡟ࡘ࠸᳨࡚ウࡋࡓࠋᮏ◊✲࡛⏝࠸ࡓィ⟬ἲࡢಟṇࡣ௨ୗࡢ࡜࠾ࡾ࡛࠶ࡿࠋSchwegler-Challacombe ࡢ᪉ἲ࡛ࡣࠊ4 ୰ᚰ 2㟁Ꮚ✚ศ್ࡢ኱ࡁ࠸ࡶࡢࡲ࡛ᤞ࡚࡚ࡋࡲ࠺ࡶࡢࡢࠊ⌮ㄽ࡟ࡼࢀࡤࡇࢀ࡟ᑐᛂࡍࡿ㟁Ꮚᐦᗘ⾜ิせ⣲ࡀ༑ศ࡟ᑠࡉࡃ࡞ࡾࠊFock ࡢ஺᥮㡯ࡢせ⣲࡟ࡣᙳ㡪ࢆ୚࠼࡞࠸࡜⪃࠼ࡽࢀ࡚࠸ࡓࠋ࡜ࡇࢁࡀࠊୖグࡢ⣔ࢆ⏝࠸࡚ヲ⣽࡟᳨ウࡋࡓ࡜ࡇࢁࠊSCF ィ⟬ࢆ཰᮰ࡉࡏࡿ࠺࠼࡛㔜せ࡞ࡶࡢࡲ࡛࢝ࢵࢺࡋ࡚࠸ࡿࡇ࡜ࡀ᫂ࡽ࠿࡜࡞ࡗࡓࠋࡑࡢࡼ࠺࡞࢝ࢵࢺࢆࡋ࡚࠸ࡿㅖ᮲௳ࢆ᳨ウࡋࡓ࡜ࡇࢁࠊ㔜࡞ࡾ⾜ิࡢ್࡜㛵㐃ࡀ࠶ࡿࡇ࡜ࡀࢃ࠿ࡗࡓࠋࡑࡇ࡛ࠊSchwegler-Challacombe ἲ࡟᪂ࡓ࡟㔜࡞ࡾ⾜ิ S ࡢ್࡟ᇶ࡙ࡃ᮲௳ࢆຍ࠼ࡓಟṇἲࢆᥦ᱌~ࡋࡓࠋලయⓗ࡟ࡣࠊ K pr < e1 ࠿ࡘ S pq < e S ࠿ࡘ S rs < e S ࡢሙྜ࡟ࠊ pq | rs ࡢィ⟬ࢆ┬␎ࡍࡿࠋ㜈್e S ࢆᐃࡵࡿࡓࡵࠊ཰᮰ࡋ࡞࠿ࡗࡓ᭱ึࡢᐇ㦂࡛࢝ࢵࢺ࢜ࣇࡉࢀ࡚࠸ࡿ 4 ୰ᚰ✚ศ࡟ᑐᛂࡍࡿࠊ S ࡢศᕸࢆ⏝࠸ࡓࠋ࡞ࡿ࡭ࡃ࢝ࢵࢺ࢜ࣇᩘࢆῶࡽࡉ࡞࠸ࡼ࠺࡟ࠊ S ࡢ኱㒊ศࡼࡾࡶ㜈್ࢆࢆ኱ࡁࡃࡍࡿࡓࡵ
࡟ࠊ S ࡢᖹᆒ+ᶆ‽೫ᕪ2 ࢆ⏝࠸ࠊ e S = 0.05 ࡜ࡋࡓࠋ࠙ᐇ㦂⤖ᯝ࠾ࡼࡧ⪃ᐹࠚ
SCF ィ⟬ࡢ᭱኱⧞ࡾ㏉ࡋᅇᩘࢆ 100 ᅇ࡜ࡋࠊࡑࡢ୰࡛཰᮰ࡍࡿ࠿࡝࠺࠿ࠊࡲࡓ཰᮰ࡲ࡛࡟せࡋࡓ CPU᫬㛫ࡢィ ࢆ⾜ࡗࡓࠋ౑⏝ࡋࡓ CPU ࡣࠊIntel (R) Pentium (R) D CPU 3GHz ࡛࠶ࡿࠋ
1.

2.
3.

e 1 ࢆ ኚ ࠼ ࡓ ሙ ྜ 㸸  e1 = 1.0 ´10-11, 1.0 ´10-12 , 1.0 ´10-13, 1.0 ´10-14 ࡜ ኚ ࠼ ࡓ ࡜ ࡇ ࢁ ࠊ
e1 = 1.0 ´10-14 ࡛཰᮰ࡋࡓࠋl ࢆኚ࠼ࡓሙྜ㸸 l = 1.1, 1.2, 1.5 ࡜ኚ࠼ࡓ࡜ࡇࢁࠊ l = 1.5 ࡛཰᮰ࡋࡓࠋ-10
ᥦ᱌ἲ(ィ⟬ἲࡢಟṇ)㸸 e1 = 1.0 ´10 , l = 1.0 ࡛཰᮰ࡋࡓࠋ
ୖグࡢ࠺ࡕࠊ཰᮰ࡋࡓࡶࡢࡣࡍ࡭࡚ SCF ィ⟬ 36 ᅇ࡛ィ⟬ࡀ཰᮰ࡋࡓࠋ4 ୰ᚰ✚ศࡢ⥲ᩘ 3859610 ࡢ࠺ࡕ࢝ࢵࢺ࢜ࣇࡉࢀࡓᩘ࡜ࠊSCF ィ⟬ 1 ᅇศࡢ᫬㛫ࢆࠊィ⟬᫬㛫ࡢᑠࡉ࠸㡰࡟୪࡭ࡓࡢࡀୗࡢ⾲࡛࠶ࡿࠋẚ㍑ࡢࡓࡵࠊ࢝ࢵࢺ࢜ࣇ࡞ࡋࡢሙྜࢆᇶ‽࡜ࡋ࡚ࠊᕪࢆྑഃ࡟グࡋࡓࠋࡲࡓࠊ཰᮰ࡋ࡞࠿ࡗࡓ᭱
ึࡢᐇ㦂࡟ࡘ࠸࡚ࡶࠊཧ⪃࡜ࡋ࡚♧ࡋࡓࠋ
Schwegler-Challacombe ࢝ࢵࢺ࢜ࣇ᮲௳-10

e1 = 1.0 ´10 , l = 1.0 㸦཰᮰ࡏࡎ㸧e1 = 1.0 ´10-10 , l = 1.0  ᥦ᱌ἲ
e1 = 1.0 ´10-14 , l = 1.0
e1 = 1.0 ´10-10 , l = 1.5࢝ࢵࢺ࢜ࣇᩘィ⟬᫬㛫(⛊)ᇶ‽࠿ࡽࡢᕪ(⛊)

2154210316- 101286231

319-7308298

3260

1972

326003260

Schwegler-Challacombe ࢝ࢵࢺ࢜ࣇ࡞ࡋ(ᇶ‽)ᐇ㦂⤖ᯝ࡟ࡼࢀࡤࠊ l ࡢ್ࢆ኱ࡁࡃࡍࡿࠊ࠶ࡿ࠸ࡣ e 1 ࢆᑠࡉࡃࡍࡿࡇ࡜࡛ SCF ィ⟬ࢆ཰᮰ࡉࡏࡿࡇ࡜ࡀ࡛ࡁࡓࡀࠊィ⟬᫬㛫▷⦰ࡢຠᯝࡣᑠࡉ࠸ࠋᮏᐇ㦂⣔ࡣ 3 ṧᇶ࡜ᑠࡉ࠸ࡶࡢࡢࠊࢱࣥࣃࢡ㉁࡛ࡣࠊ
ィ⟬᫬㛫▷⦰ຠᯝࡣ኱ኚ㔜せ࡛࠶ࡿࠋ࢜ࣜࢪࢼࣝἲ࡛ࡣ 4 ୰ᚰ✚ศࡢ࠺ࡕᚲせ࡞ࡶࡢ࡜ࡑ࠺࡛࡞࠸ࡶࡢࢆ㐺ษ࡟㑅ูࡍࡿࡓࡵ࡟ࡣ᮲௳ࢆࡁࡘࡃࡍࡿࡋ࠿࡞࠸ࡀࠊᥦ᱌ἲ࡛ࡣ㔜࡞ࡾ✚ศ࡟ᇶ࡙ࡃ㜈್ࡢ㏣ຍ࡟ࡼࡾࠊ㑅ูࡀຠᯝⓗ࡟⾜ࢃࢀ࡚࠸ࡿࡶࡢ࡜⪃࠼ࡽࢀࡿࠋ
ᮏ◊✲ࡣࠕ࢖ࣀ࣮࣋ࢩࣙࣥᇶ┙ࢩ࣑࣮ࣗࣞࢩࣙࣥࢯࣇࢺ࢙࢘࢔ࡢ◊✲㛤Ⓨࠖ࡟ࡼࡾ⾜ࢃࢀࡓࠋཧ⪃ᩥ⊩[1] A. D. Becke: J. Chem. Phys., 98-7, 5648-5652 (1993).
[2] E. Schwegler and M. Challacombe: J. Chem. Phys., 105-7, 2726-2734 (1996).