1E09ཎᏊ࣭ศᏊ⣔ࡢࢩ࣮ࣗࣞࢹ࢕࣮ࣥ࢞᪉⛬ᘧࡢゎἲ I.
཯ᑐ⛠໬ࡢ㧗㏿࢔ࣝࢦࣜࢬ࣒
*(㔞໬◊ , JST-CREST**) ‫ۑ‬୰ᔱ ᾈஅ*,**, ୰㎷ ༤*,**1. ⥴ゝ: ᡃࠎࡣࠊཎᏊ࣭ศᏊ⣔ࡢࢩ࣮ࣗࣞࢹ࢕࣮ࣥ࢞᪉⛬ᘧࢆ㠀ᖖ࡟⢭ᐦ࡟ゎࡃ᪉ἲ࡜ࡋ࡚ Free
Complement (FC) ἲࢆᥦ᱌ࡋ[1]ࠊᑡᩘ㟁ᏊཎᏊ࣭ศᏊ⣔࡛ࡣ㠀ᖖ࡟Ⰻ࠸⤖ᯝࢆᚓ࡚ࡁࡓ[2]ࠋࡉࡽ
࡟ࠊ୍⯡ࡢཎᏊ࣭ศᏊ⣔࡟ᑐࡋ࡚ࡣࠊ」㞧࡞✚ศィ⟬ࢆ୍ษ⾜࠺ᚲせࡢ࡞࠸ Local Schrödinger
Equation (LSE) ἲࢆ⏝࠸ࡿࡇ࡜࡛ཎ⌮ୖࡢᅔ㞴ࡉࡣゎỴࡉࢀࡓ[3]ࠋ
ࡋ࠿ࡋࠊࡼࡾከ㟁Ꮚ⣔࡬ࡢ㐺⏝ࡢ㝿ࡣࠊ㟁Ꮚࡢ Distinguishablility ࠿ࡽ᮶ࡿ㟁Ꮚᗙᶆࡢ཯ᑐ⛠໬ࡢၥ㢟ࡀ࠶ࡿࠋ཯ᑐ⛠໬ࡣ O(N!) (N ࡣ㟁Ꮚᩘ)ࡢ⨨᥮᧯సࢆᚲせ࡜ࡍࡿࡀࠊࡼࡃ▱ࡽࢀ࡚࠸ࡿࡼ࠺
࡟ Slater ⾜ิᘧࢆ౑࠺ࡇ࡜࡛ O(N3)ࡲ࡛ῶࡌࡿࡇ࡜ࡀ࡛ࡁࡿࠋࡋ࠿ࡋࠊἼື㛵ᩘࡀ㟢ࢃ࡟┦㛵㛵ᩘࢆྵࡴሙྜࡣ┤᥋ Slater ⾜ิᘧࢆ౑࠺ࡇ࡜ࡀ࡛ࡁ࡞࠸ࠋ(Slater ⾜ิᘧ)(Jastrow 㛵ᩘ)࡜࠸࠺ヨ⾜㛵ᩘࢆ⏝࠸ࡿኚศࣔࣥࢸ࢝ࣝࣟἲ࡛ࡣࠊ┦㛵㛵ᩘࢆྵࡴ Jastrow 㒊ศࢆ᏶඲ᑐ⛠໬ࡍࡿࡇ࡜࡛ࡇࡢၥ㢟ࢆゎỴࡋ࡚࠸ࡿࡀࠊ≀⌮ⓗ࡟ࡣ㟁Ꮚ཯Ⓨࡣ㌶㐨㛫ࡈ࡜࡟␗࡞ࡿᐤ୚ࢆࡍࡿࡓࡵࡇࡢグ㏙ࡣṇࡋࡃ࡞ࡃࠊ஦ᐇࠊExact ゎ࠿ࡽࡣࡲࡔ㐲࠸⤖ᯝࡋ࠿ᚓࡽࢀ࡞࠸[4]ࠋFC ἲ࡛⏕ᡂࡉࢀࡿἼື㛵ᩘࡣࠊFC ἲࡢ࣮࢜ࢲ࣮ࡢቑຍ࡜ඹ࡟㒊ศ┦㛵㛵ᩘࢆྵࡴࠋ㒊ศ┦㛵㛵ᩘࡣ㌶㐨㛫ࡈ࡜ࡢ␗࡞ࡿ㟁Ꮚ཯Ⓨࡢᐤ୚ࢆ࠺ࡲࡃ⾲⌧࡛ࡁࡿࡓࡵἼື㛵ᩘࡣࡼࡾ⮬↛࡞ᙧࢆࡋ࡚࠸ࡿࡀࠊ㒊ศ┦㛵㛵ᩘࢆྵࡴἼື㛵ᩘࡢ཯ᑐ⛠໬ࡣᐜ࡛᫆ࡣ࡞ࡃ」㞧࡛࠶ࡿࠋᮏ◊✲࡛ࡣࡇࡢࡼ࠺࡞」㞧࡞㒊ศ┦㛵㛵ᩘࢆྵࡴ FCἼື㛵ᩘ࡟ᑐࡍࡿຠ⋡ࡢⰋ࠸཯ᑐ⛠໬࢔ࣝࢦࣜࢬ࣒࡜ࣉࣟࢢ࣒ࣛࢆ㛤Ⓨࡍࡿࠋ2. ᡭἲ: ᡃࠎࡢၥ㢟ࡣ௨ୗࡢࡼ࠺࡟┦㛵㒊ศࡢ㛵ᩘ: e
ᩘ: d( N -r )(r )(r ಶࡢ㟁Ꮚࡀ┦㛵)࡜㠀┦㛵㒊ศࡢ㛵࠿ࡽ࡞ࡿ Complement function (cf)ࢆ཯ᑐ⛠໬ࡋࡓ࠸ࠋ

A[f (1, 2,..., N )] = A éëe( r ) (1, 2,..., r ) × d ( N -r ) (r + 1, r + 2,..., N ) ùû ,(1)ࡇࡇ࡛ࠊ A ࡣ཯ᑐ⛠໬࣮࢜࣌ࣞࢱ࡛࠶ࡿࠋ᭱ࡶ⡆༢࡞࢔࢖ࢹ࢕࢔ࡣ┦㛵㟁Ꮚࡢ཯ᑐ⛠໬ࡣ㟢ࢃ࡟⾜࠸ࠊṧࡾࡢ㠀┦㛵㒊ศࡣ⾜ิᘧࢆ⏝࠸ࡿࡸࡾ᪉࡛࠶ࡿࠋࡇࢀࡣ O(Nr+3)ࡢィ⟬࣮࢜ࢲ࣮ࢆᚲせ࡜ࡍࡿࠋࡋ࠿ࡋࠊ⤒㦂ⓗ࡟Ⰻ࠸⢭ᗘࡀᚓࡽࢀࡿ FC ἲࡢ Order 4 ࡛ࡣ᭱㧗ḟ࡛ 8 㟁Ꮚࡀ┦㛵ࡋࠊO(N11)ࡢィ⟬࣮࢜ࢲ࣮࡛࠶ࡿࠋࡇࢀࡣ⌧ᐇⓗ࡟ N ࡢ኱ࡁ࡞⣔࡟㐺⏝࡛ࡁࡿ࡛ࣞ࣋ࣝࡣ࡞࠸ࠋᮏ◊✲࡛ࡣࠊ┦㛵㛵ᩘࡢ┦㛵ࢲ࢖࢔ࢢ࣒ࣛࢆᑟධࡍࡿࡇ࡜࡛ࠊ࡝ࡢࡼ࠺࡟㟁Ꮚࡀ┦㛵ࡋ࡚࠸ࡿ࠿ࢆ᭱ึ࡟ゎᯒࡋࠊ2 㟁Ꮚ┦㛵㡯ࡀ⾜ิᘧࡢ୰࡟ྲྀࡾ㎸ࡲࢀࡓ Generalized Slater ⾜ิᘧࢆᑟධࡍࡿࡇ࡜࡛ࠊ᭱ᑠࡢィ⟬࣮࢜ࢲ࣮࡟ῶࡌࡿࡇ࡜ࡢ࡛ࡁࡿ᪉ἲࢆ㛤Ⓨࡋࡓࠋ౛࠼ࡤࠊ6 㟁Ꮚ⣔࡛e( r ) (1, 2,..., r ) = f12 (1, 2) f13 (1,3), d ( N -r ) (r + 1, r + 2,..., N ) = Õ i6=1 oi (i) (ࡇࡇ࡛ࠊ oi ࡜ f ij ࡣࡑࢀࡒࢀ 1 㟁Ꮚ㛵ᩘཬࡧ 2 㟁Ꮚ㛵ᩘ)ࡢ࡜ࡁࡢ཯ᑐ⛠໬ࡣࠊé
f12 (1* , 2)o2 (2)êf (1* , 2)o3 (2)ê1!
N() ê(+1) × o (1* ) 13Af =o4 (2)1N! êo5 (2)êêo6 (2)
ëf12 (1* ,3)o2 (3)f12 (1* , 4)o2 (4)f12 (1* ,5)o2 (5)f12 (1* , 6)o2 (6)f12 (2* ,1)o2 (1)f12 (2* ,3)o2 (3)f12 (2* , 4)o2 (4)f12 (2* ,5)o2 (5)f12 (2* , 6)o2 (6)f13 (1* ,3)o3 (3)f13 (1* , 4)o3 (4)f13 (1* ,5)o3 (5)f13 (1* , 6)o3 (6)f13 (2* ,1)o3 (1)f13 (2* ,3)o3 (3)f13 (2* , 4)o3 (4)f13 (2* ,5)o3 (5)f13 (2* , 6)o3 (6)o4 (3)o4 (4)o4 (5)o4 (6)o4 (1)o4 (3)o4 (4)o4 (5)o4 (6)o5 (3)o5 (4)o5 (5)o5 (6)o5 (1)o5 (3)o5 (4)o5 (5)o5 (6)o6 (3)o6 (4)o6 (5)o6 (6)o6 (1)o6 (3)o6 (4)o6 (5)o6 (6)

f12 (3* , 2)o2 (2)f12 (3* ,1)o2 (1)f12 (3* , 4)o2 (4)f12 (3* ,5)o2 (5)f12 (3* , 6)o2 (6)f12 (4* , 2)o2 (2)f13 (3* , 2)o3 (2)f13 (3* ,1)o3 (1)f13 (3* , 4)o3 (4)f13 (3* ,5)o3 (5)f13 (3* , 6)o3 (6)f13 (4* , 2)o3 (2)f12 (4* ,3)o2 (3)f13 (4* , 3)o3 (3)f12 (4* ,1)o2 (1)f13 (4* ,1)o3 (1)f12 (4* ,5)o2 (5)f13 (4* ,5)o3 (5)f12 (4* , 6)o2 (6)f13 (4* , 6)o3 (6)o4 (2)o4 (1)o4 (4)o4 (5)o4 (6)o4 (2)o4 (3)o4 (1)o4 (5)o4 (6)o5 (2)o5 (1)o5 (4)o5 (5)o5 (6)o5 (2)o5 (3)o5 (1)o5 (5)o5 (6)o6 (2)o6 (1)o6 (4)o6 (5)o6 (6)o6 (2)o6 (3)o6 (1)o6 (5)o6 (6)

f12 (5* , 2)o2 (2)f12 (5* ,3)o2 (3)f12 (5* , 4)o2 (4)f12 (5* ,5)o2 (1)f12 (5* , 6)o2 (6)f12 (6* , 2)o2 (2)f12 (6* ,3)o2 (3)f12 (6* , 4) o2 (4)f12 (6* ,5)o2 (5)f13 (5* , 2)o3 (2)f13 (5* ,3)o3 (3)f13 (5* , 4)o3 (4)f13 (5* ,5)o3 (1)f13 (5* , 6)o3 (6)f13 (6* , 2)o3 (2)f13 (6* ,3)o3 (3)f13 (6* , 4)o3 (4)f13 (6* ,5)o3 (5)o4 (2)o4 (3)o4 (4)o4 (1)o4 (6)o4 (2)o4 (3)o4 (4)o4 (5)o5 (2)o5 (3)o5 (4)o5 (1)o5 (6)o5 (2)o5 (3)o5 (4)o5 (5)o6 (2)o6 (3)o6 (4)o6 (1)o6 (6)o6 (2)o6 (3)o6 (4)o6 (5)+ (-1) × o1 (3* )+ (-1) × o1 (5* )+ (-1) × o1 (2* )+ (-1) × o1 (4* )+ (-1) × o1 (6* )f12 (6* ,1)o2 (1) ùú
f13 (6* ,1)o3 (1) úúo4 (1)
úo5 (1)úúo6 (1)
û,(2)ࡢࡼ࠺࡟⾲ࢃࡉࢀࡿࠋࡇࡢሙྜࡣ 3 㟁ᏊTable I. FC Ἴື㛵ᩘࡢ཯ᑐ⛠໬ィ⟬࣮࢜ࢲ࣮4ࡀ┦㛵ࡋ࡚࠸ࡿ࡟ࡶ㛵ࢃࡽࡎ O(N )࡛῭
ࡴࠋࡉࡽ࡟(2)ᘧࢆࡼࡃぢࡿ࡜ࠊ஫࠸ࡢ⾜FC order᭱㧗ḟࡢࢣ࣮ࢫ

Operation count0

No correlation

O(N3)

1

2 electrons correlated

O(N3)

2

4 electrons correlated

O(N4)

3

6 electrons correlated

O(N5)

4

8 electrons correlatedO(N6)ิᘧྠኈࡣࡓ࠿ࡔ࠿ᩘ⾜(࠶ࡿ࠸ࡣᩘิ)ࡋ࠿㐪ࡗ࡚࠸࡞࠸ࡓࡵࠊ⾜ิᘧࡢ Laplaceᒎ㛤ࢆ⏝࠸ࡓ Determinant update ἲࢆ౑࠺ࡇ࡜ࡀ࡛ࡁࡿࠋ⤖ᯝ࡜ࡋ࡚ࠊࡉࡽ࡟ 13

࣮࢜ࢲ࣮㏿ࡃ࡞ࡾࠊO(N )ࡢィ⟬࣮࢜ࢲ࣮࡛῭ࡴࠋTable I ࡟ࠊFC ἲࡢ࣮࢜ࢲ࣮࡟ᑐࡍࡿᮏ◊✲ࡢ࢔ࣝࢦࣜࢬ࣒ࡢィ⟬࣮࢜ࢲ࣮ࢆ♧ࡍࠋὀ┠ࡍ࡭ࡁ࡜ࡇࢁࡣࠊ(N ࡀ኱ࡁ࠸࡜ࡁ)1 ✀ࡢࡳ┦㛵㛵ᩘࡀྵࡲࢀࡿሙྜࡣ┦㛵㛵ᩘࡀఱࡶྵࡲࢀ࡞࠸
ሙྜ࡜ྠࡌ O(N3)ࡢィ⟬࣮࢜ࢲ࣮࡛῭ࡴࡇ࡜࡛࠶ࡿࠋ᭱㧗ḟࡢ 8 㟁Ꮚࡀ┦㛵㡯࡟ྵࡲࢀࡿሙྜ࡛ࡶࡇࡢ཯ᑐ⛠໬㒊఩ࡣ O(N6)ࡢィ⟬࡛῭ࡴࠋTable II ࡛ࡣࠊࡶࡋ N!ࡢ⨨᥮᧯సࢆ⾜ࡗࡓሙྜ࡜௒ᅇࡢ
᪉ἲ࡛ࡢࠊFC ἲࡢ Order 1 ࡜ 2 ࡛⌧ࢀࡿ᭱ࡶィ⟬࣮࢜ࢲ࣮ࡢ኱ࡁ࠸ cf ࡢ཯ᑐ⛠໬࡟࠿࠿ࡿィ⟬᫬㛫ࢆẚ㍑ࡋࡓࡶࡢࢆ♧ࡋ࡚࠸ࡿࠋConventional ࡞ࡸࡾ᪉࡛ࡢከ㟁Ꮚ⣔ࡢィ⟬ࡣ⌧ᐇⓗ࡟୙ྍ⬟࡞ィ⟬᫬㛫ࢆせࡍࡿࡀࠊᮏ᪉ἲࡢᑟධ࡟ࡼࡾⴭࡋࡃィ⟬ࡀຍ㏿࡛ࡁ࡚࠸ࡿࡇ࡜ࡀศ࠿ࡿࠋࡇࡢࡼ࠺࡟ࠊ
ᮏ◊✲࡟ࡼࡾ FC LSE ἲࡢከ㟁ᏊཎᏊ࣭ศᏊ⣔ࡢᐇ⏝ィ⟬ࡀ༑ศ࡟ྍ⬟࡟࡞ࡗࡓࠋ
Table II. FC Ἴື㛵ᩘࡢ཯ᑐ⛠໬ࡢ Timing ࢸࢫࢺ(༢఩: s) (in Maple)
N (㟁Ꮚᩘ)FC order 1

FC order 2

N!ࡢ⨨᥮᧯స

Present: O(N )N!ࡢ⨨᥮᧯స

Present: O(N4)10411.6

0.031

438.4

1.016

114914.5

0.046

5173.31.376

1262863.2

0.062

66101.1

1.719

10 *

15(1.710 )20

16 *(3.210 )40

46 *(1.110 )30.09310 *(1.810 )

3.189

0.171

16 *(3.410 )

7.703

1.173

46 *

80.845(1.110 )*ぢ✚ࡶࡾ್

3. ⤖ㄽ: ᮏ◊✲࡛ࡣࠊ」㞧࡞㒊ศ┦㛵㛵ᩘ࡟ᑐࡋ࡚࡛ࡶຠ⋡ࡢⰋ࠸཯ᑐ⛠໬ࡢ㧗㏿࢔ࣝࢦࣜࢬ࣒
ࢆᥦ᱌ࡋࠊࡑࡢࣉࣟࢢ࣑ࣛࣥࢢࡶ᏶஢ࡋࡓࠋLSE ἲࡣ Sampling ἲࢆᇶ♏࡜ࡍࡿࡓࡵࠊGeneralized
Slater ⾜ิᘧࡢྛせ⣲ࡣ༢࡞ࡿᩘᏐ࡛࠶ࡿࡓࡵࠊVB ᆺࡢἼື㛵ᩘࡢ฼⏝࡛⏕ࡌࡿ N!ࡢィ⟬࣮࢜ࢲ࣮ࡢၥ㢟ࡶ࡞ࡃィ⟬࡛ࡁࡿࡇ࡜ࡀศ࠿ࡗ࡚࠸ࡿࠋᐇ㝿ࡢཎᏊ࣭ศᏊ⣔࡬ࡢ㐺⏝ィ⟬࡟ࡘ࠸࡚ࡣࠊ
⥆ࡃㅮ₇(1E10, 1E11)࡛ࠕFrom Atoms to MoleculeࠖࡸࠕLocal Sampling ἲࠖ࡞࡝ࡢ᪂ࡋ࠸࢔࢖ࢹ࢕࢔࡜⤖ྜࡋࡓ⤖ᯝ࡜ࡋ࡚Ⓨ⾲ࡉࢀࡿࠋ
ࡲࡓࠊFC ἲࡀᇶ♏࡜ࡍࡿ Iterative Complement (IC) ἲࡸ Piecewise Complement (PC) ἲ࡛ࡣࠊ඲ᑐ⛠ࡢࣁ࣑ࣝࢺࢽ࢔ࣥࡸ g 㛵ᩘࡢᙧ≧ࢆ⥔ᣢࡋ࡚࠸ࡿࡓࡵࠊࡑࡢ≉ᛶࢆ⏕࠿ࡋࡓࡼࡾ㧗㏿࡞ィ⟬ࡀྍ⬟࡛࠶ࡿ࡜⪃࠼ࡽࢀࡿࠋࡇࢀࡽࡢ࢔ࣝࢦࣜࢬ࣒࡟ࡘ࠸࡚ࡣᙜ᪥Ⓨ⾲ࡍࡿࠋ
References: [1] H. Nakatsuji, J. Chem. Phys. 113, 2949 (2000). H. Nakatsuji, Phys. Rev. Lett. 93, 030403(2004). H. Nakatsuji, Phys. Rev. A 72, 062110 (2005). [2] H. Nakashima and H. Nakatsuji, J. Chem. Phys.
127, 224104 (2007). Y. Kurokawa, H. Nakashima, and H. Nakatsuji, Phys. Rev. A 72, 062502 (2005). [3] H.
Nakatsuji, H. Nakashima, Y. Kurokawa, and A. Ishikawa, Phys. Rev. Lett. 99, 240402 (2007). [4] B. L.
Hammond, W. A. Lester, Jr., and P. J. Reynolds, Monte Carlo Methods in Ab Initio Quantum Chemistry(World Scientific, Singapore, 1994).