現代的な化学結合理論と計算化学 - 理論化学研究室

(1)

⌧௦ⓗ䛺໬Ꮫ⤖ྜ⌮ㄽ䛸ィ⟬໬Ꮫ

⌮Ꮫ◊✲㝔໬Ꮫ㒊㛛 ୰㔝ᬕஅ

2011ᖺ10᭶11᪥

䛿䛨䜑䛻

⌧௦䛾㔞Ꮚ໬Ꮫ⌮ㄽ

㧗⢭ᗘ䠄ᩘ㼗㼏㼍㼘㻛㼙㼛㼘㻕

㧗⬟⋡

኱つᶍ䠄ᩘ㻝㻜㻜䡚ᩘ㻝㻜㻜㻜ཎᏊ䠅

ศᏊ㌶㐨䠖ᩘ㻝㻜㻜㻜ಶ

㟁Ꮚ㓄⨨䠖ᩘ㻝㻜㻜㻌㻜㻜㻜ಶ䡚

グ㏙䛾⡆᫂䛥

䝧䞁䝊䞁

㻹㻻ᥥീ

䜿䜽䝺ᵓ㐀䚸䝕䝳䝽䞊ᵓ㐀䚸ඹ㬆

ண ⌮ㄽ

໬Ꮫ⤖ྜ䛸䛿ఱ䛛䠛

໬Ꮫ⤖ྜ

ྂ඾ⓗ

ඹ᭷⤖ྜ䠖㟁Ꮚᑐ䛾ඹ᭷

䜲䜸䞁⤖ྜ䠖㟼㟁Ẽຊ

䠄㔠ᒓ⤖ྜ䠅

ศᏊ㌶㐨⌮ㄽ

⤖ྜ䛧䛯ཎᏊ㌶㐨䛻㟁Ꮚ䠄ᑐ䠅䛜཰ᐜ䛥䜜䜛

໬Ꮫ⤖ྜ䛸䛿ఱ䛛䠛

໬Ꮫ⤖ྜ

ཎᏊ౯⤖ྜἲ

ཎᏊ㌶㐨䛻ධ䛳䛯㟁Ꮚ䛜⤖ྜ䠄䜹䝑䝥䝸䞁䜾䠅䛩䜛䠙⤖ྜ㟁Ꮚᑐ

Ꮩ❧㟁Ꮚᑐ

( )

A B

1s 1s

1 ϕ ϕ ⋅ 2αβ βα−

2s 2s

ϕ ϕ αβ⋅ ϕ ϕ αβ^{sp3 sp3}⋅

໬Ꮫ⤖ྜ䛸䛿ఱ䛛䠛

໬Ꮫ⤖ྜ

ཎᏊ౯⤖ྜἲ

౛䠊❅⣲ศᏊ㻺㻞

䠛䠛䠛

౛䠊䝣䝑໬Ỉ⣲ศᏊ㻴㻲

䠛䠛䠛

౛䠊ỈศᏊ㻴_㻞㻻

䠛䠛䠛

౛䠊䝯䝍䞁ศᏊ㻯㻴㻠

䠛䠛䠛

໬Ꮫ⤖ྜ䛸䛿ఱ䛛䠛

䡚㻝㻥㻥㻜ᖺ௦䠖 ศᏊ㌶㐨ἲ 䠚 ཎᏊ౯⤖ྜἲ

䛚䜒䛻ィ⟬ୖ䛾⌮⏤䛛䜙

➨୍㏆ఝ䛾グ㏙⬟ຊ䠄ྠ䛨䛰䛡䛾⢭ᗘ䜢ฟ䛩䛾䛻ᚲせ䛺ィ⟬

㔞䠅

ゎ㔘ୖ䛾ᡂຌ

䝣䝻䞁䝔䜱䜰㌶㐨ㄽ䠄䠤䠫䠩㻻䠉䠨䠱䠩䠫┦஫స⏝䛺䛹䠅

㻝㻥㻥㻜ᖺ௦䡚䠖 ཎᏊ౯⤖ྜἲ䛾᚟ά

⌮ㄽ䛾Ⓨᒎ䛸䝁䞁䝢䝳䞊䝍䛾Ⓨᒎ䛻䜘䜛ィ⟬㔞䛾ඞ᭹

໬Ꮫ⤖ྜᥥീ䛾⡆᫂䛥

ྂ඾ⓗ⤖ྜㄽ䛸䛾୍⮴

ඹ᭷⤖ྜ䠖 H-H

䜲䜸䞁⤖ྜ䠖 H⁺H^- H^-H⁺

໬Ꮫ཯ᛂ⌮ㄽ

ศᏊ㌶㐨ἲ

㌶㐨ᑐ⛠ᛶಖᏑ๎䠄㼃㼛㼛㼐㼣㼍㼞㼐㻙㻴㼛㼒㼒㼙㼍㼚㼚๎䠅

౛䠊䜶䝏䝺䞁䛾䠎㔞໬཯ᛂ䠖 ⇕ⓗ䛻⚗ไ

໬Ꮫ཯ᛂ⌮ㄽ

ཎᏊ౯⤖ྜἲ

どぬⓗཎᏊ౯⤖ྜἲ䠄㼂㼕㼟㼡㼍㼘㻌㼂㼍㼘㼑㼚㼏㼑㻙㻮㼛㼚㼐ἲ䠅

ᡭ㡰

䠍䠊཯ᛂ≀䛸⏕ᡂ≀䛾㼂㻮ᵓ㐀䠄䍦ᵓ㐀ᘧ䠅䜢ᥥ䛟䠎䠊཯ᛂ䛻ಀ䜛⤖ྜ䜢ᢳฟ

䠏䠊཯ᛂ≀䛸⏕ᡂ≀䛻ඹ㏻䛺ᑐ⛠ᛶ䜢⏝䛔䚸⤖ྜ䜢ᑐ⛠໬

䠐䠊཯ᛂ≀䛸⏕ᡂ≀䛾ᑐ⛠ᛶ䜢ẚ㍑

౛䠊㻴_㻞㻗㻌㻵_㻞㻌䊻㻞㻴㻵 ཯ᛂ䛿⇕ⓗ䛻⚗ไ

(2)

໬Ꮫ཯ᛂ⌮ㄽ

౛㻚㻌㻰㼑㼣㼑㼞㼎㼑㼚㼦㼑㼚㼑㻌䊻㻮㼑㼚㼦㼑㼚㼑㻌␗ᛶ໬཯ᛂ䛿⚗ไ

㼂㼍㼘㼑㼚㼏㼑㻌㼎㼛㼚㼐㻌⤖ྜ⾲⌧

䊻 ᑐ⛠໬

໬Ꮫ཯ᛂ⌮ㄽ

ᑐ⛠໬䡒㼍㼘㼑㼚㼏㼑㼎㼛㼚㼐⤖ྜ⾲⌧

໬Ꮫ཯ᛂ⌮ㄽ

䛭䛾௚䛾౛䠍

໬Ꮫ཯ᛂ⌮ㄽ

䛭䛾௚䛾౛䠍

໬Ꮫ཯ᛂ⌮ㄽ

䛭䛾௚䛾౛䠎

໬Ꮫ཯ᛂ⌮ㄽ

䛭䛾௚䛾౛䠎

໬Ꮫ཯ᛂ⌮ㄽ

䛭䛾௚䛾౛䠏

໬Ꮫ཯ᛂ⌮ㄽ

䛭䛾௚䛾౛䠏

(3)

໬Ꮫ཯ᛂ⌮ㄽ

䛭䛾௚䛾౛䠐

໬Ꮫ཯ᛂ⌮ㄽ

䛭䛾௚䛾౛䠐

໬Ꮫ཯ᛂ⌮ㄽ

⦎⩦ၥ㢟

ၥ䠊䜶䝏䝺䞁䛾䠎㔞໬䠄⇕ⓗ⚗ไ䠅

䠛䠛䠛

ၥ䠊䜶䝏䝺䞁䠇䝤䝍䝆䜶䞁䊻䝅䜽䝻䝦䜻䝉䞁䠄⇕ⓗチᐜ䠅

䠛䠛䠛

ཎᏊ౯⤖ྜἲ䛻䜘䜛 ཎᏊ౯⤖ྜἲ䛻䜘䜛

໬Ꮫ཯ᛂ䛾グ㏙

Ỉ⣲ศᏊ䠖 ศᏊ㌶㐨ἲ䛷䛿䠛(1)

Hartree-Fockἲ䠄MOἲ䠅

HF-SCF = σ

2

Ỉ⣲ศᏊ䠖 ཎᏊ౯⤖ྜἲ䛷䛿? (1)

ཎᏊ౯⤖ྜ(VB)ἲ

[ ]

A B

H H

A B

VB ( ) 2

H H

ϕ ϕ αβ βα

ª º

= ¬ − ¼

= −

Ỉ⣲ศᏊ䠖 ཎᏊ౯⤖ྜἲ䛷䛿? (2)

୍⯡໬ཎᏊ౯⤖ྜ(GVB)ἲ

[ ]

A B

H H

A B

GVB ( ) 2

H H

ϕ ϕ αβ βα

ª º

= ¬ − ¼

= −

Ỉ⣲ศᏊ䠖

⌧௦䛾ศᏊ㌶㐨ἲ䛷䛿䠛(2)

CAS-SCFἲ

2 2

CAS-SCF = 0.9948 σ − 0.1021 * σ

(4)

Ỉ⣲ศᏊ䠖

⌧௦䛾ཎᏊ౯⤖ྜἲ䛷䛿? (3)

CAS-VB㽤

( )

[ ] { }

A B

A A B B

H H

H H H H

A B A B A B

CAS-VB 0.9124 ( ) 2

0.0503

0.9122 H H 0.0878 H H H H ϕ ϕ αβ βα

ϕ ϕ αβ ϕ ϕ αβ

− + + −

ª º

= ¬ − ¼

ª º ª º

+ ¬ ¼ + ¬ ¼

ª º ª º

= − + ¬ ¼ ¬ + ¼

CAS-VBἲ

MO

Φi

MO

Φj

Ψ

CAS-VBἲ

VB VB

1 1 2 2

A A

Ψ = Φ + Φ

n_i: occupation number (VBᵓ㐀䛾๭ྜ) Schematic expression

CAS-VBኚ᥮䛾せㅉ

ศᏊ㌶㐨䊻 ཎᏊ㌶㐨

┤஺㟁Ꮚ㓄⨨ 䊻 ཎᏊ౯⤖ྜ(VB)㟁Ꮚ㓄⨨

䜢ἼἼື㛵ᩘ䜢ኚ䛘䛪䛻⾜䛖

Ab initio MOἲ(CASSCFἲ)䛾Ἴື㛵ᩘ䚸䜶䝛䝹䜼䞊䛿୙ኚ

ᐃ㔞ᛶ

㉁䛾పୗ䛧䛺䛔ゎ㔘䜢ಖド

H

₂

+ X 䊻 H + HX (X = F, Cl, Br, and I)

H₂+ X 䊻H + HX

H2+ F 䊻 H + HF

H₂+ Cl 䊻 H + HCl

H2+ Br 䊻 H + HBr

H₂+ I 䊻 H + HI

䝝䝰䞁䝗(Hammond)䛾௬ㄝ

Hammond௬ㄝ

George S. Hammond (1921-) USA

JACS, 77, 334 (1955)

䜒䛧䚸䠎䛴䛾≧ែ䠄౛䛘䜀䚸㑄⛣≧ែ䛸୙Ᏻᐃ䛺୰㛫య䠅䛜཯ᛂ 㐣⛬䛻䛚䛔䛶ᘬ䛝⥆䛔䛶䛔䛶⌧䜜䚸䛛䛴䚸䜋䜌ྠ䛨䜘䛖䛺䜶䝛䝹䜼䞊䜢䜒䛴䛾䛷䛒䜜䜀䚸䛭䜜䜙䛾㛫䛾┦஫ኚ᥮䛿䜋䜣䛾ᑠ䛥䛺 ศᏊᵓ㐀䛾ኚ໬䛻䜘䛳䛶䛺䛥䜜䜛䚹

㑄⛣≧ែ䛾ᵓ㐀䛿䚸

྾⇕཯ᛂ䛷䛿⏕ᡂ≀䛻㏆䛟

Ⓨ⇕཯ᛂ䛷䛿཯ᛂ≀䛻㏆䛔

Energy profile: H

₂

+ X 䊻 H + HX (X = F, Cl, Br, and I)

H2+ F 䊻H + HF H2+ Cl䊻H + HCl

H2+ Br 䊻H + HBr H2+ I 䊻H + HI

Valence Bondඹ㬆ᵓ㐀

−

H X

•

H

_A _B

A B

H H X

⁺ ⁻ ^•

A B

H H X

⁻ ⁺ ^•

A B

H H

^• −

X

A B

H H X

^• ⁻ ⁺

A B

H H X

^• ⁺ ⁻

A B

H H X

⁺ ^• ⁻

A B

H H X

⁻ ^• ⁺

A B

H H

( )

X

ϕ ϕ αβ βα ϕ α

− ⋅

B B

H H X

ϕ ϕ αβ ϕ α

⋅

A A

H H X

ϕ ϕ αβ ϕ α

⋅

A B

H H X

( )

ϕ α ϕ ϕ αβ βα

⋅ −

A B B

H H H

ϕ α ϕ ϕ αβ

⋅

HA X X

ϕ α ϕ ϕ αβ

⋅

HB X X

ϕ α ϕ ϕ αβ

⋅

A A B

H H H

ϕ ϕ αβ ϕ α

⋅ HAHBX

HAHBX

Occupation number analysis (1): HF

0.0 0.2 0.4 0.6 0.8 1.0

-1.0 -0.5 0.0 0.5 1.0

Occupation Number

IRC / bohr(amu)

^{1 / 2}

H

₂

+ F −> H + HF

(5)

Occupation number analysis (1): HCl

0.0 0.2 0.4 0.6 0.8 1.0

-1.0 -0.5 0.0 0.5 1.0

Occupation Number

IRC / bohr(amu)

^{1 / 2}

H

₂

+ Cl −> H + HCl

Occupation number analysis (1): HBr

0.0 0.2 0.4 0.6 0.8 1.0

-1.0 -0.5 0.0 0.5 1.0

Occupation Number

IRC / bohr(amu)

^{1 / 2}

H

2

+ Br −> H + HBr

Occupation number analysis (1): HI

0.0 0.2 0.4 0.6 0.8 1.0

-1.0 -0.5 0.0 0.5 1.0

Occupation Number

IRC / bohr(amu)

^{1 / 2}

H

2

+ I −> H + HI

Occupation number analysis (1)

0.0 0.2 0.4 0.6 0.8 1.0

-1.0 -0.5 0.0 0.5 1.0

Occupation Number

IRC / bohr(amu)^{1 / 2} H₂+ F −> H + HF

0.0 0.2 0.4 0.6 0.8 1.0

-1.0 -0.5 0.0 0.5 1.0

IRC / bohr(amu)^{1 / 2} H₂+ Cl −> H + HCl

0.0 0.2 0.4 0.6 0.8 1.0

-1.0 -0.5 0.0 0.5 1.0

IRC / bohr(amu)^{1 / 2} H₂ + Br −> H + HBr

0.0 0.2 0.4 0.6 0.8 1.0

-1.0 -0.5 0.0 0.5 1.0

IRC / bohr(amu)^{1 / 2} H₂ + I −> H + HI

Occupation number analysis (2): HF

0.0 0.2 0.4 0.6 0.8 1.0

-1.0 -0.5 0.0 0.5 1.0

H

2

+ F −> H + HF

Occupation Number

IRC / bohr(amu)

^{1 / 2}

Occupation number analysis (2): HCl

0.0 0.2 0.4 0.6 0.8 1.0

-1.0 -0.5 0.0 0.5 1.0

H

₂

+ Cl −> H + HCl

Occupation Number

IRC / bohr(amu)

^{1 / 2}

Occupation number analysis (2): HBr

0.0 0.2 0.4 0.6 0.8 1.0

-1.0 -0.5 0.0 0.5 1.0

H

2

+ Br −> H + HBr

Occupation Number

IRC / bohr(amu)

^{1 / 2}

Occupation number analysis (2): HI

0.0 0.2 0.4 0.6 0.8 1.0

-1.0 -0.5 0.0 0.5 1.0

H

2

+ I −> H + HI

Occupation Number

IRC / bohr(amu)

^{1 / 2}

(6)

Occupation number analysis (2)

0.0 0.2 0.4 0.6 0.8 1.0

-1.0 -0.5 0.0 0.5 1.0

H₂ + F −> H + HF

IRC / bohr(amu)^{1 / 2}

0.0 0.2 0.4 0.6 0.8 1.0

-1.0 -0.5 0.0 0.5 1.0

H₂ + Cl −> H + HCl

IRC / bohr(amu)^{1 / 2}

0.0 0.2 0.4 0.6 0.8 1.0

-1.0 -0.5 0.0 0.5 1.0

H₂+ Br −> H + HBr

IRC / bohr(amu)^{1 / 2}

0.0 0.2 0.4 0.6 0.8 1.0

-1.0 -0.5 0.0 0.5 1.0

H₂+ I −> H + HI

IRC / bohr(amu)^{1 / 2}

Transition state

of chemical bond

Change of bond length (Å): HF

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

-1.0 -0.5 0.0 0.5 1.0

Bond length / ang.

IRC / bohr(amu)

^{1 / 2}

H

₂

+ F −> H + HF

Change of bond length (Å): HCl

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

-1.0 -0.5 0.0 0.5 1.0

Bond length / ang.

IRC / bohr(amu)

^{1 / 2}

H

2

+ Cl −> H + HCl

Change of bond length (Å): HBr

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

-1.0 -0.5 0.0 0.5 1.0

Bond length / ang.

IRC / bohr(amu)

^{1 / 2}

H

2

+ Br −> H + HBr

Change of bond length (Å): HI

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

-1.0 -0.5 0.0 0.5 1.0

Bond length / ang.

IRC / bohr(amu)

^{1 / 2}

H

₂

+ I −> H + HI

Change of bond length (Å)

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

-1.0 -0.5 0.0 0.5 1.0

Bond length / ang.

IRC / bohr(amu)^{1 / 2} H₂ + F −> H + HF

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

-1.0 -0.5 0.0 0.5 1.0

IRC / bohr(amu)^{1 / 2} H₂+ Cl −> H + HCl

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

-1.0 -0.5 0.0 0.5 1.0

IRC / bohr(amu)^{1 / 2} H₂+ Br −> H + HBr

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

-1.0 -0.5 0.0 0.5 1.0

IRC / bohr(amu)^{1 / 2} H₂+ I −> H + HI

Electric dipole moment (debye): HF

0.0 0.5 1.0 1.5 2.0 2.5

-1.0 -0.5 0.0 0.5 1.0

IRC / bohr(amu)

^{1 / 2}

Dipole moment / Debye

0.0 0.5 1.0 1.5 2.0 2.5

-1.0 -0.5 0.0 0.5 1.0

IRC / bohr(amu)

^{1 / 2}

H

2

+ F −> H + HF

Electric dipole moment (debye): HCl

0.0 0.5 1.0 1.5 2.0 2.5

-1.0 -0.5 0.0 0.5 1.0

IRC / bohr(amu)

^{1 / 2}

Dipole moment / Debye

H

₂

+ Cl −> H + HCl

(7)

Electric dipole moment (debye): HBr

0.0 0.5 1.0 1.5 2.0 2.5

-1.0 -0.5 0.0 0.5 1.0

IRC / bohr(amu)

^{1 / 2}

0.0 0.5 1.0 1.5 2.0 2.5

-1.0 -0.5 0.0 0.5 1.0

IRC / bohr(amu)

^{1 / 2}

H

2

+ Br −> H + HBr

Electric dipole moment (debye): HI

0.0 0.5 1.0 1.5 2.0 2.5

-1.0 -0.5 0.0 0.5 1.0

IRC / bohr(amu)

^{1 / 2}

H

2

+ I −> H + HI

Electric dipole moment (debye)

0.0 0.5 1.0 1.5 2.0 2.5

-1.0 -0.5 0.0 0.5 1.0

IRC / bohr(amu)^{1 / 2}

Dipole moment / Debye

0.0 0.5 1.0 1.5 2.0 2.5

-1.0 -0.5 0.0 0.5 1.0

IRC / bohr(amu)^{1 / 2} H₂+ F −> H + HF

0.0 0.5 1.0 1.5 2.0 2.5

-1.0 -0.5 0.0 0.5 1.0

IRC / bohr(amu)^{1 / 2}

H₂+ Cl −> H + HCl

0.0 0.5 1.0 1.5 2.0 2.5

-1.0 -0.5 0.0 0.5 1.0

IRC / bohr(amu)^{1 / 2}

0.0 0.5 1.0 1.5 2.0 2.5

-1.0 -0.5 0.0 0.5 1.0

IRC / bohr(amu)^{1 / 2} H₂ + Br −> H + HBr

0.0 0.5 1.0 1.5 2.0 2.5

-1.0 -0.5 0.0 0.5 1.0

IRC / bohr(amu)^{1 / 2}

H₂ + I −> H + HI

䝩䝹䝮䜰䝹䝕䝠䝗䛾༢ศᏊศゎ䠖H

₂

CO 䊻 H

₂

+ CO

O C H H

H2+CO

TS䛷䛿䚸C-H⤖ྜ䛸H-H⤖ྜ䛾䛹䛱䜙䛜ඃໃ䛛䠛

Occupation number analysis

H-H

TS C-H C⁺H^-, C^-H⁺

H⁺H^-, H^-H⁺ CH

HH Others

TS

TS䛷䛿䚸

C-H⤖ྜ70%, H-H⤖ྜ24%

䋻CH⤖ྜ䛜ඃໃ䛷䛒䜛

Polyene䠖ບ㉳≧ែ䛾㟁Ꮚᵓ㐀

Polyene䛾ບ㉳≧ែ䠖retinal 䛾ග␗ᛶ໬䛾䝰䝕䝹䚸ᇶ♏ⓗ䛺⌮ゎ

1¹B1u: HOMO-LUMO 䠍㟁Ꮚບ㉳䠄チᐜ䠅

2¹A_1g: HOMO-LUMO 䠎㟁Ꮚບ㉳䠄⚗ไ䠅

1¹B_1u> 2¹A_1g

Polyene䠖ບ㉳≧ែ䛾㟁Ꮚᵓ㐀

Clear picture

2¹Ag: ⤖ྜ஺᭰䛾཯㌿䛧䛯 covalent≧ែ

1¹Bu: 2㔜⤖ྜ㒊䛜ศᴟ䛧䛯 ionic≧ែ

Butadiene

Polyene䠖ບ㉳≧ែ䛾㟁Ꮚᵓ㐀

(8)

ཎᏊ౯⤖ྜἲ䛻䜘䜛 ཎᏊ౯⤖ྜἲ䛻䜘䜛

໬Ꮫ཯ᛂ䝎䜲䝘䝭䜽䝇䛾グ㏙

INTRODUCTION INTRODUCTION

z໬Ꮫ཯ᛂᶵᵓ䛾ゎ᫂

1. 㑄⛣≧ែ䜔䝫䝔䞁䝅䝱䝹䡡䜶䝛䝹䜼䞊᭤㠃䛾Ỵᐃ (㧗䛔⢭ᗘ䜢䜒䛳䛯Ἴື㛵ᩘ䛸䜶䝛䝹䜼䞊) 2. ཯ᛂ㐣⛬䛾⤖ྜᥥീ

(

^covalent

)

ion

(

^ionic

)

cov HL HL

HL=a Φ +a Φ

Ψ

ex)VB-HL(Heitler-London)Ἴື㛵ᩘ

zCASVBἲ

䞉䞉䞉_CASSCFἼື㛵ᩘ䜢䛭䛾ᐦᗘ䜢ኚ䛘䜛䛣䛸䛺䛟䚸VBἼື㛵ᩘ䛻ኚ᥮䛩䜛ᡭἲ

CASSCFἼື㛵ᩘ 䞉䞉䞉㓄⨨㛵ᩘ(┤஺䝇䝢䞁ᅛ᭷㛵ᩘ)䛾⥺ᙧ⤖ྜ

䞉䞉䞉VB㛵ᩘ(㠀┤஺䝇䝢䞁ᅛ᭷㛵ᩘ)䛾⥺ᙧ⤖ྜ

CASVBἼື㛵ᩘ

VBᵓ㐀䛻ኚ᥮

CASSCF CSF

i i i

Ψ =

¦

CΦ ^CASVB i ^VBi

i

Ψ =

¦

AΦ Ψ^CASSCF²= Ψ^CASVB²

䊻 ;

䠆CASSCFἲ䛾⢭ᗘ(Ἴື㛵ᩘ䚸䜶䝛䝹䜼䞊)䜢䛭䛾䜎䜎䛻䚸VBᵓ㐀䛻䜘䜛

᫂☜䛺ᥥീ䛜ᚓ䜙䜜䜛䚹

CASVB(complete

CASVB(completeactive space valence bond)active space valence bond)ἲἲ

1. CASSCFἲ䛻䜘䜛ศᏊ㌶㐨䛸┤஺㟁Ꮚ㓄⨨㛵ᩘ 䛾Ỵᐃ

{ }

¦

^Φ ^Φ ⁼^Φ

( )

= Ψ

i

i CSF i CSF i CSF i i

CASSCF C ^, λ

{ }λi

2. active ㌶㐨㛫䛾⥺ᙧ⤖ྜ䛻䜘䛳䛶ᒁᅾ໬㌶㐨(localized MO:LMO) 䜢Ỵᐃ (ཎᏊ㌶㐨ᵝ䛾┤஺LMO)

*䛣䛣䛷䛿ᒁᅾ໬䛾᪉ἲ䛸䛧䛶Boys’ localization䜢⏝䛔䛯䚹

䛣䛾᪉ἲ䛷䛿┤஺LMO䜢ᚓ䜛䛣䛸䛻䛺䜛䛜䚸௚䛻䜒㠀┤஺LMO䜢ᚓ䜛᪉ἲ䜒䛒䜛䚹㠀┤஺

LMO䛸䛩䜛䛣䛸䛷ཎᏊ㌶㐨䛻䜘䜚㏆䛔㌶㐨䜢ᚓ䜛䛣䛸䛜䛷䛝䜛䛜ィ⟬㈇Ⲵ䛜㔜䛟䛺䜛䚹 3. LMO䛻䜘䜛CAS-CI䜢⾜䛔䚸䛭䛾㐺ᙜ䛺⥺ᙧ⤖ྜ䜢䛸䜛䛣䛸䛷spin paired

function(VBᵓ㐀)䛾⤌䛻ኚ᥮䛩䜛䚹 spin paired function

*spin-paired function䛿┤஺䛿䛧䛶䛔䛺䛔䛜䚸S²䛾ᅛ᭷㛵ᩘ䛸䛺䛳䛶䛔䜛⥺ᙧ⊂❧䛺䝇䝢䞁㛵ᩘ䛷䛒䜛䚹

{ }ϕi

{ }λi 䋻{ }ϕi

iCSF

Φ

( ) ( ) ( ) ( )

( ) (( ) ( ) ( ) ( )1 2 1 2)

2 2 1 1 2 2 1 Y 1 -

X ≡ ϕX ϕY +ϕY ϕX ⋅ α β −β α

• ௨ୖ䜘䜚CASSCF㛵ᩘ䛸CASVB㛵ᩘ䛿ྠ୍㛵ᩘ䛾ู䛾⾲⌧䛸䛺䛳䛶䛔䜛䚹䛴䜎䜚䚸CASVBἲ䛿CASSCFἲ䛾⢭ᗘ䜢ኻ䛖䛣䛸䛺䛟໬Ꮫⓗᥥീ䜢ᚓ䜙䜜䜛᪉

ἲ䛷䛒䜛䚹

䛣䛾㛵ಀᘧ䜘䜚

䛷䛒䜛䚹

**********************************************************************

CASVB CASVB༨᭷ᩘ ༨᭷ᩘ

䛸䛩䜛䛸n_i䛿VBᵓ㐀䛾㔜䜏(weight)䛸䛺䜛䚹 { } ( ) { }

( ) ¦ ¦

¦

Φ

=

° Φ

¿

°¾

½ Φ

= Φ Φ

= Ψ

Φ

= Φ Φ

= Ψ

VB i i CSF i i

i i

VB i VB i VB i i CASVB

i CSF i CSF i i

CSF i i CASSCF

A A C

C

ϕ λ ,

,

CSF j VB i ij j ijCj with

Ai=

¦

Ω Ω =Φ Φ

¦

⁼^Φ ^Φ

=

j

VB j VB i ij j ij i

i A SA with S

n ^* (¦ = )

i

ni 1

༨᭷ᩘ

༨᭷ᩘnnii䛿ྛ䛿ྛVBVBᵓ㐀䛾ᐤ୚䜢ᐃ㔞ⓗ䛻⾲䛩㔞ᵓ㐀䛾ᐤ୚䜢ᐃ㔞ⓗ䛻⾲䛩㔞 (

(┤஺ᇶᗏ䛻䛚䛡䜛ಀᩘ䛾஧஌䛻┦ᙜ䛩䜛㔞┤஺ᇶᗏ䛻䛚䛡䜛ಀᩘ䛾஧஌䛻┦ᙜ䛩䜛㔞))

{ }

¦

Φ Φ =Φ ( )

= Ψ

i

i VB i VB i VB i i

CASVB A , ϕ

n_i nj

A_j A_i

VB

Φi

VB

Φ

j CASSCF

Φ

CAS-VBἲ

᭷ຠ

V

CAS-VBἲ

᭷ຠ???

V

Ref : H. Nakano, K. Nakayama, and K. Hirao, J. Mol. Struct(Theochem) 461-462, 55(1999)

1. CAS-VBἲ䛸ືຊᏛ䜢⤖䜃䛴䛡䜛䛣䛸䛷䚸ືⓗ䛺⤒㊰ୖ䛻䛚䛔䛶

᫂☜䛺⤖ྜᥥീ䛜ᚓ䜙䜜䜛䛛䠛

2. ≉䛻䚸TS䜢㏻䜙䛺䛔⤒㊰ୖ䛷䛿䛹䛾䜘䛖䛺ᥥീ䛜ᚓ䜙䜜䜛䛛䠛

TS

RAB RBC

TS

RAB RBC VAB

VBC

VAB VBC

IRC

䡐 z㟼ⓗ䛺⤒㊰ୖ(IRC)䛷䛾⤖ྜ䛾ゎᯒ

zືⓗ䛺⤒㊰ୖ䛷䛾⤖ྜ䛾ゎᯒ

z

z ཯ᛂa (཯ᛂa (Ỉ⣲஺᥮཯ᛂỈ⣲஺᥮཯ᛂ))

X䡡 + H₂ 䊻 HX + H䡡

X = F ( H , Cl ) z

z ཯ᛂ཯ᛂb (Diels-b (Diels-AlderAlder཯ᛂ཯ᛂ))

ethylene(C₂H₄)+cyclopentadiene(C₅H₆) 䊻norborene(C₇H₁₀) zz ศᏊ㌶㐨ィ⟬ศᏊ㌶㐨ィ⟬CASSCFCASSCFἲἲ

active✵㛫 ᇶᗏ㛵ᩘ(Dunning)

཯ᛂa ->CAS(3,3) H,C : cc-pVDZ H 1s , F 2p䃢, Cl 3p䃢 F,Cl : aug-cc-pVDZ

཯ᛂb - >CAS(6,6) C1-C6 2p䃟

zz ືຊᏛィ⟬ືຊᏛィ⟬VerletVerletἲἲ time step 0.5fs

CALCULATIONS CALCULATIONS

1 2 3

4 5 6 7

+ →

1 2

3 5 4 6 7

H p^䃢 X

H H

཯ᛂa

C1 C2

C6 C5

C4 C3

཯ᛂb

: 2p䃟

n[HH

n[HH]]䍵n[F䡡H-H] + n[F䡡H⁺H^-] + n[F䡡H^-H⁺] n[HH(cov

n[HH(cov)] n[HH(ionicn[HH(ionic)])]

n[HF

n[HF]]䍵n[F-H H䡡] + n[F⁺H^-H䡡] + n[F^-H⁺H䡡] n[HF(cov

n[HF(cov)])] n[HF(ionic)]n[HF(ionic)]

཯ᛂa

CASVB

CASVB༨᭷ᩘゎᯒ༨᭷ᩘゎᯒ

VBඹ㬆ᵓ㐀䠖8ಶ

n[tail]䍵n[F⁺H䡡H^-] + n[F^-H䡡H⁺]

୺䛺⤖ྜᥥീ

䊠䊡䊢䊣䊤

཯ᛂb VBඹ㬆ᵓ㐀䠖175ಶ C1 C2

C5 C4 C6 C3

C1 C2 C5 C4 C3 C6

C1 C2 C5 C4 C3

C6 C3

C1 C2 C5 C4

C6 C6

C1 C2 C5 C4 C3

n[䊠]䍵 ^{C1 C2} C5 C4 C6 C3

n[ ] ^{C1 C2} C5 C4 C3 n[ ]C6

+ -

+ ^{C1 C2}

C5 C4 C3 n[ ]C6

- +

+ ^{C1 C2}

C5 C4 C3 n[ ]+C6

-

+ ^{C1 C2}

C5 C4 C3 n[ ]-C6

+ +

C1 C2 C5 C4 C3 n[ ]C6 +

-

+ ^{C1 C2}

C5 C4 C3 n[ ]C6 -

+

+ ^{C1 C2}

C5 C4 C3 n[ ]C6⁺ ^-

+ +

-

C1 C2 C5 C4 C3 n[ ]-C6

+

+ ⁺ ^- + 䞉䞉䞉

ex)䊠䛾weight

ඖ䛸䛺䜛VBᵓ㐀䛜୍䛴䛷䛺䛔VBᵓ㐀䛾༨᭷ᩘ䛿䚸ඖ䛸䛺䜛VBᵓ㐀䛾䛭䜜䛮䜜䛾༨᭷

ᩘ䛰䛡ẚ㔜䜢䛴䛡䛶ศ㓄䛩䜛䚹

౛䛘䜀䛾ඖ䛸䛺䜛VBᵓ㐀䛿

䛾4䛴䛷䛒䜛䛾䛷䚸ྛVBᵓ㐀䛾༨᭷ᩘ䛛䜙ẚ㔜䜢䛴䛡䛶䛾༨᭷ᩘ䜢ศ㓄䛧䛯䚹ẚ㔜䛾䛴䛡᪉䛾౛䜢ୗ䛻♧䛩䚹

C1 C2 C5 C4 C3 -C6

+

+ - C1 C2

C5 C4 C3 -C6

+

C1 C2 C5 C4 C3 C6

+ - C1 C2

C5 C4 C3 -C6

+ C1 C2

C5 C4 C3 C6

+ -

n[ ]^{C1 C2} C5 C4 C6 C3 -

+ C1 C2

C5 C4 C3 n[ ]-C6

+ 㽢n[ ]^{C1 C2} 䋻

C5 C4 C6 C3 -

+ + -

䠈䠈䠈

n[ ] n[ ] n[ ] n[ ]^{C1 C2}

C5 C4 C3 -C6

+

+ ^{C1 C2} + +

C5 C4 C3 C6

+ - C1 C2

C5 C4 C3 -C6

+ C1 C2

C5 C4 C3 C6

+ -

௚䛾3䛴䛾VBᵓ㐀䛻䛴䛔䛶䜒䚸ྠᵝ䛻䛧䛶 ^{C1 C2} 䛾༨᭷ᩘ䜢ศ㓄䛩䜛䚹 C5 C4

C6 C3 -

+ + -

C1 C2 C5 C4 C3 -C6

+ + -

(9)

ᶞᙧᅗ䛾౛䠖䝤䝍䝆䜶䞁

2pz

X H H

౛1䠊Ỉ⣲஺᥮཯ᛂ䠖 F+H

₂

䊻HF+H

• CASSCF/CASVBィ⟬

– Active✵㛫CAS(3,3)

• Active㌶㐨䠖F(2p_z), H(1s), H(1s)

• ᇶᗏ㛵ᩘ䠖 F: aug-cc-pVDZ, H: cc-pVDZ

• BOືຊᏛ – Verletἲ

• Time step: 0.5 fs

– ึᮇ᮲௳

• 䠍䠊ඹ⥺ⓗ䚸E_K(F) = 20 kcal/mol

• 䠎䠊ඹ⥺ⓗ䚸E_K(F) = 12.7 kcal/mol

• 䠏䠊䌴30㼻䚸E_K(F) = 30 kcal/mol

౛1䠊Ỉ⣲஺᥮཯ᛂ䠖 F+H

₂

䊻HF+H

䈄 ᶞᶞᙧᅗ

H_AH_B⤖ྜ H_BF⤖ྜ H_AF⤖ྜ

䜿䞊䝇䠍䠖 ඹ⥺ⓗ䠈E

_K

(F)=20 kcal/mol

Distance

Occupation Number

Potential Energy

Weight of Bond Structure

䜿䞊䝇䠍䠖 ඹ⥺ⓗ䠈E

_K

(F)=20 kcal/mol

Distance

Occupation Number

Potential Energy

䜿䞊䝇䠎䠖 ඹ⥺ⓗ䠈E

_K

(F)=12.7 kcal/mol

Distance

Occupation Number

Potential Energy

䜿䞊䝇3䠖䌴30 r 䠈E

_K

(F)=30 kcal/mol 䜿䞊䝇 3 䠖䌴 30 㼻䠈 E

_K

(F)=30 kcal/mol

Distance

Occupation Number

Potential Energy

(10)

౛䠎䠊㻰㼕㼑㼘㼟㻙㻭㼘㼐㼑㼞཯ᛂ

• CASSCF/CASVBィ⟬

–Active✵㛫CAS(6,6)

• Active㌶㐨

–䜶䝏䝺䞁䃟㌶㐨䚸䝅䜽䝻䝨䞁䝍䝆䜶䞁䃟㌶㐨 –䝜䝹䝪䝹䝛䞁䠖䃟㌶㐨䚸䃢㌶㐨㽢2

• ᇶᗏ㛵ᩘ䠖cc-pVDZᇶᗏ

• BOືຊᏛ

–ึᮇ᮲௳

• 䠍䠊ᑐ⛠⤒㊰

• 䠎䠊㠀ᑐ⛠⤒㊰䚸䌴30㼻

• 䠏䠊㠀ᑐ⛠⤒㊰䚸䌴60㼻

䈄䝹䝹䞊䝖ඹ㬆ᵓ㐀

I II III IV V

– I: 䜶䝏䝺䞁䠇䝅䜽䝻䝨䞁䝍䝆䜶䞁䠄཯ᛂ≀䠅

– II: 䝜䝹䝪䝹䝛䞁䠄⏕ᡂ≀䠅 – III, IV: ୍᪉䛜⤖ྜ

– V: 䝅䜽䝻䝨䞁䝍䝆䜶䞁䛾⤖ྜ஺᭰䛜཯㌿䠄཯ᛂ≀䛾๪ᵓ㐀䠅

౛䠎䠊㻰㼕㼑㼘㼟㻙㻭㼘㼐㼑㼞཯ᛂ

C1 C₂ C3 C4 C₅ C6

C1 C₂ C3 C4 C₅ C6 C1 C₂

C3 C4 C₅ C6

C1 C₂ C3 C4 C₅ C6 C1 C₂

C3 C4 C₅ C6

䜿䞊䝇䠍䠖 ᑐ⛠⤒㊰

Distance

䜿䞊䝇䠍䠖 ᑐ⛠⤒㊰

Potential Energy

㑄⛣≧ែᵓ㐀䊠

䊡䊢䊣䊤

C1 C2 C5 C4 C6 C3

䜿䞊䝇䠎䠖㠀ᑐ⛠⤒㊰䠈䌴30r

Distance

䜿䞊䝇䠎䠖㠀ᑐ⛠⤒㊰䠈䌴30r

Potential Energy

䊠䊡䊢䊣䊤

C1 C2 C5 C4 C6 C3

䜿䞊䝇䠏䠖㠀ᑐ⛠⤒㊰䠈䌴60 r

Distance

䜿䞊䝇䠏䠖㠀ᑐ⛠⤒㊰䠈䌴60 r

(11)

䜿䞊䝇䠏䠖㠀ᑐ⛠⤒㊰䠈䌴60 r

Potential Energy

䊠䊡䊢䊣䊤

C1 C2 C5 C4 C3 C6

཯ᛂ≀䚸㑄⛣≧ែ䚸⏕ᡂ≀䛛䜙䛾㟁Ꮚ≧ែⓗ㊥㞳

␗䛺䜛ᵓ㐀䛻䛚䛡䜛㟁Ꮚ≧ែⓗ䛺㊥㞳 d_X–Y= max( |W_I(X) 䌦W_I(Y)|,

|W_II(X) 䌦W_II(Y)|,

|W_III(X) 䌦W_III(Y)|,

|W_IV(X) 䌦W_IV(Y)|,

|WV(X) 䌦WV(Y)|,

|WCBs(X) 䌦WCBs(Y)| ) Case 2

Case 1

Case 3

現代的な化学結合理論と計算化学 - 理論化学研究室

⌧௦ⓗ䛺໬Ꮫ⤖ྜ⌮ㄽ䛸ィ⟬໬Ꮫ

⌮Ꮫ◊✲㝔໬Ꮫ㒊㛛 ୰㔝ᬕஅ

䛿䛨䜑䛻

໬Ꮫ⤖ྜ䛸䛿ఱ䛛䠛

໬Ꮫ⤖ྜ䛸䛿ఱ䛛䠛

㔞䠅

䠍䠊཯ᛂ≀䛸⏕ᡂ≀䛾㼂㻮ᵓ㐀䠄䍦ᵓ㐀ᘧ䠅䜢ᥥ䛟 䠎䠊཯ᛂ䛻ಀ䜛⤖ྜ䜢ᢳฟ

㼂㼍㼘㼑㼚㼏㼑㻌㼎㼛㼚㼐㻌⤖ྜ⾲⌧

ཎᏊ౯⤖ྜἲ䛻䜘䜛 ཎᏊ౯⤖ྜἲ䛻䜘䜛

CAS-VBἲ

䜢ἼἼື㛵ᩘ䜢ኚ䛘䛪䛻⾜䛖

+ X 䊻 H + HX (X = F, Cl, Br, and I)

Energy profile: H

Occupation Number

Occupation number analysis (2): HF

of chemical bond

Bond length / ang.

Dipole moment / Debye

Electric dipole moment (debye)

TS䛷䛿䚸C-H⤖ྜ䛸H-H⤖ྜ䛾 䛹䛱䜙䛜ඃໃ䛛䠛

Occupation number analysis

TS䛷䛿䚸

Polyene䠖ບ㉳≧ែ䛾㟁Ꮚᵓ㐀

Butadiene

Polyene䠖ບ㉳≧ែ䛾㟁Ꮚᵓ㐀

1. 㑄⛣≧ែ䜔䝫䝔䞁䝅䝱䝹䡡䜶䝛䝹䜼䞊᭤㠃䛾Ỵᐃ (㧗䛔⢭ᗘ䜢䜒䛳䛯Ἴື㛵ᩘ䛸䜶䝛䝹䜼䞊) 2. ཯ᛂ㐣⛬䛾⤖ྜᥥീ

᫂☜䛺ᥥീ䛜ᚓ䜙䜜䜛䚹

䛣䛾㛵ಀᘧ䜘䜚

CASVB CASVB༨᭷ᩘ ༨᭷ᩘ

᫂☜䛺⤖ྜᥥീ䛜ᚓ䜙䜜䜛䛛䠛

CALCULATIONS CALCULATIONS

CASVB

౛䛘䜀 䛾ඖ䛸䛺䜛VBᵓ㐀䛿

ᶞᙧᅗ䛾౛䠖䝤䝍䝆䜶䞁

• CASSCF/CASVBィ⟬

౛1䠊Ỉ⣲஺᥮཯ᛂ䠖 F+H

䈄 ᶞᶞᙧᅗ

䜿䞊䝇䠍䠖 ඹ⥺ⓗ䠈E

(F)=12.7 kcal/mol

౛䠎䠊㻰㼕㼑㼘㼟㻙㻭㼘㼐㼑㼞཯ᛂ

• CASSCF/CASVBィ⟬

• 䠎䠊㠀ᑐ⛠⤒㊰䚸䌴30㼻

I II III IV V

– I: 䜶䝏䝺䞁䠇䝅䜽䝻䝨䞁䝍䝆䜶䞁䠄཯ᛂ≀䠅

౛䠎䠊㻰㼕㼑㼘㼟㻙㻭㼘㼐㼑㼞཯ᛂ

㑄⛣≧ែᵓ㐀 䊠

䜿䞊䝇䠎䠖 㠀ᑐ⛠⤒㊰䠈䌴30r

䊠 䊡 䊢 䊣 䊤

䜿䞊䝇䠏䠖 㠀ᑐ⛠⤒㊰䠈䌴60 r

䊠 䊡 䊢 䊣 䊤

཯ᛂ≀䚸㑄⛣≧ែ䚸⏕ᡂ≀䛛䜙䛾 㟁Ꮚ≧ែⓗ㊥㞳

䠍䠊཯ᛂ≀䛸⏕ᡂ≀䛾㼂㻮ᵓ㐀䠄䍦ᵓ㐀ᘧ䠅䜢ᥥ䛟䠎䠊཯ᛂ䛻ಀ䜛⤖ྜ䜢ᢳฟ

TS䛷䛿䚸C-H⤖ྜ䛸H-H⤖ྜ䛾䛹䛱䜙䛜ඃໃ䛛䠛

౛䛘䜀䛾ඖ䛸䛺䜛VBᵓ㐀䛿

㑄⛣≧ែᵓ㐀䊠

䜿䞊䝇䠎䠖㠀ᑐ⛠⤒㊰䠈䌴30r

䊠䊡䊢䊣䊤

䜿䞊䝇䠏䠖㠀ᑐ⛠⤒㊰䠈䌴60 r

䊠䊡䊢䊣䊤

཯ᛂ≀䚸㑄⛣≧ែ䚸⏕ᡂ≀䛛䜙䛾㟁Ꮚ≧ែⓗ㊥㞳