First principles simulations 第一原理計算

Advanced Laser and Photon Science  レーザー・光量子科学特論 

First principles simulations   第一原理計算  

Takeshi Sato 

http://ishiken.free.fr/english/lecture.html 

[email protected] 

1.   Two electron systems 2.   Second quan6za6on

3.   Mul6configura6on 6me-dependent

Hartree-Fock method

Ac#on integral

Time-dependent varia#onal principle

S =

Z

t1

h | ( ˆ H i@

) | i

S = 0, for ! ⁰ = +

(t

) = (t

) = 0

Arbitrary = ) TDSE

Approximate = ) Variational EOMs

| (t) i = X

n

C

(t) | n i ,

Ac#on integral

Example 1: Time-dependent Configura#on Interac#on S =

Z

t1

h | ( ˆ H i@

) | i

φ2

φ3

φ4

φ5

φ_M

Ac#on integral

Example 1: Time-dependent Configura#on Interac#on S =

Z

t1

h | ( ˆ H i@

) | i

Configura#on Interac#on (CI) coefficients: Varia#onal parameters

= X

n,m

Z

t1

dt ⇣

C

C

h n | H ˆ | m i iC

C ˙

⌘ S = X

n,m

Z

t1

dt ⇣

C

C

h n | H ˆ | m i iC

h n | m i C ˙

⌘

S

C

(t) = X

m

h n | H ˆ | m i C

i C ˙

= 0

i C ˙

= X

h n | H ˆ | m i C

| (t) i = X

n

C

(t) | n i ,

Example 2: Time-dependent Hartree-Fock

φ¹ φ2

φ3

φ4

φ5

φM

Virtual

Occupied

| (t) i = | 1

1

· · · 1

000 · ··i $ det [

· · ·

]

Example 2: Time-dependent Hartree-Fock

Orbital func#ons: Varia#onal parameters

| (t) i = | 1

1

· · · 1

000 · ··i $ det [

· · ·

]

From

Homework

S =

Z

dt 2 4

X

⇣ h

i h

| ˙

i ⌘

+ 1 2

X

⇣ V

V

⌘ 3 5

S

⇤i

(t) = ˆ h

i

+

X

j=1

⇣ W ˆ

W ˆ

⌘

i

= ˆ h

+

X

j=1

⇣ W ˆ

W ˆ

⌘

W

(r

) = Z

dx

⇤i

(x

)

(x

)

| r

r

|

Mul#configura#on TD Hartree-Fock (MCTDHF)

| (t) i = X

n

C

(t) | n i ,

TD Configura#on Interac#on (CI) with given number of moving orbitals

φ2

φ³ φ⁴ φ⁵ φ_M

Occupied Virtual

General Complete- orthonormal

µ

,

, , ··

i

,

,

,

a

,

Both CI coefficients & orbital func#ons: Varia#onal parameters

Mul#configura#on TD Hartree-Fock (MCTDHF)

| (t) i = X

n

C

(t) | n i ,

Working with Slater determinant is, in general, extremely tedious:

Need techniques of second quan#za#on (1) Matrix (operator) exponen#al

exp(A) ⌘

X

A

n!

exp(A)^† = exp(A^†),

exp(A + B) = exp(A) exp(B) ( [A, B] = 0

exp(A)B exp( A) = B + [A, B] + 1

2![A,[A, B]] + 1

3![A,[A, [A, B]]] + · · ·

B ¹ exp(A)B = exp(B ¹AB)

Mul#configura#on TD Hartree-Fock (MCTDHF)

(2) Exponen#al parameteriza#on of unitary matrix (operator)

U = exp(X )

(3) Unitary transforma#on of orbitals

U : unitary U

U = U U

= 1 X : anti-Hermitian X

= X

An#-Hermi#an matrix can be parameterized more easily than unitary one

µ

(t) = X

⌫

⌫

(0)U

= X

⌫

⌫

(0) exp(X )

,

() a

(t) = X

⌫

a

(0)U

= X

⌫

a

(0) exp(X )

a

(t) = X

⌫

a

(0)U

= X

⌫

a

(0) exp(X )

Mul#configura#on TD Hartree-Fock (MCTDHF)

Proof: From BCH expansion (3) Unitary transforma#on of orbitals

µ

(t) = X

⌫

⌫

(0)U

= X

⌫

⌫

(0) exp(X )

,

() a

(t) = X

⌫

a

(0)U

= X

⌫

a

(0) exp(X )

a

(t) = X

⌫

a

(0)U

= X

⌫

a

(0) exp(X )

() a

(t) = exp( ˆ X )a

(0) exp( X ˆ ) a

(t) = exp( ˆ X )a

(0) exp( X ˆ )

X ˆ = X

µ⌫

X

a

(0)a

(0) ⌘ X

µ⌫

E ˆ

(0)

Mul#configura#on TD Hartree-Fock (MCTDHF)

(4) Unitary transforma#on of Slater determinants

µ

(t) = X

⌫

⌫

(0)U

= X

⌫

⌫

(0) exp(X )

,

() a

(t) = exp( ˆ X )a

(0) exp( X ˆ ) a

(t) = exp( ˆ X )a

(0) exp( X ˆ )

X ˆ = X

µ⌫

X

a

(0)a

(0) ⌘ X

µ⌫

E ˆ

(0)

| n(0) i = a

(0)a

(0)a

(0) · · · |i

| n(t) i = a

(t)a

(t)a

(t) · · · |i

= exp( ˆ X ) | n(0) i

Mul#configura#on TD Hartree-Fock (MCTDHF)

(5) Unitary transforma#on of total wave func#on

(6) Varia#on and Time deriva#ve of total wave func#on

| (t) i = X

n

C

(t) | n(t) i

= exp( ˆ X ) X

n

C

(t) | n(0) i

X ˙

= h

(t) | ˙

(t) i

| (t) ˙ i = X

n

C ˙

(t) | n(t) i + X

µ⌫

X ˙

E ˆ

| (t) i

| (t) i = X

n

C

(t) | n(t) i + X

µ⌫

X

E ˆ

| (t) i

( ˆ E

⌘ ˆ a

ˆ a

)

Both CI coefficients & orbital func#ons: Varia#onal parameters Insert previous results into TDVP and require

Mul#configura#on TD Hartree-Fock (MCTDHF)

| (t) i = X

n

C

(t) | n i ,

S/ C _n ^⇤ (t) = S/ X _µ⌫ (t) = 0

iC˙_n = X

m

hn| Hˆ i X

µ⌫

Eˆ_⌫^µX˙_⌫^µ

!

|miC_m

General equa#ons of mo#on

iX "

h |Eˆ_⌫^µ 1 X

n

|nihn|

!

Eˆ | i h |Eˆ 1 X

n

|nihn|

!

Eˆ_⌫^µ| i

# X˙

= h |Eˆ_⌫^µ 1 X

n

|nihn|

!

Hˆ| i h |Hˆ 1 X

n

|nihn|

!

Eˆ_⌫^µ| i

In case of complete CI expansion within the given orbitals

Mul#configura#on TD Hartree-Fock (MCTDHF)

| (t) i = X

n

C

(t) | n i ,

D

= h | E

| i , P

= h | E

| i

iC˙_n = X

m

hn| 0

@Hˆ X

ij

E_jⁱR_jⁱ 1

A |miC_m

i | ˙

i = ˆ Q 0

@ h ˆ |

i +

X

(D

)

P

W ˆ

|

i 1 A +

X

|

i R

R^j_i ⌘ ih ^j| ˙_ii : Arbitrary Hermitian matrix

One (D) and two (P) par#cle reduced density matrices Q ˆ = 1 P

j

|

ih

|

( ˆ E

= ˆ a

a ˆ

a ˆ

ˆ a

)

Importance of non-complete CI expansions

φ1

φ2

φ3

φ⁴ φ⁵ φ_M

Occupied Virtual

Ac/ve

Dynamical-core Frozen-core

φ1

φ2

φ³ φ4

φ⁵ φ_M

Occupied Virtual

TD-CASSCF (complete-ac#ve-space self-

consistent-field): core and ac#ve subspaces N = 36

MCTDHF

N

= 784

H = X

i=1

<

: 1 2

@²

@x²_i

X

a=1

Z_a q

(x_i X_a)² + c

x_iE(t)=

; + X

i>j

q 1

(x_i x_j)² + d

-4 -3 -2 -1 0

-15 -10 -5 0 5 10 15

Orbital energy / Hartree

x / bohr

orbital 1 orbital 2 orbital 3 orbital 4 nuclear

Core Valence

-0.12 -0.08 -0.04 0.00 0.04 0.08 0.12

0.0 0.5 1.0 1.5 2.0 2.5 3.0

field amplitude

time / optical cycle

0.4 PW/cm² 750 nm

Ground-state

7.5 fs

Field

-80 -60 -40 -20 0 20 40 60

0.0 0.5 1.0 1.5 2.0 2.5 3.0

dipole moment / au

time / optical cycle CAS(8e)

CAS(4e) CAS(2e) HF

h (t) | x | (t) i

1

49 784

44100

-0.8 -0.4 0.0 0.4 0.8

0.0 0.5 1.0 1.5 2.0 2.5 3.0

dipole acceleration / au

time / optical cycle 8e DC+4e FC+4e

h (t) | x ¨ | (t) i

“Dynamical”

“Frozen” or

10^-12 10^-10 10^-8 10^-6 10^-4 10^-2

0 20 40 60 80 100

intensity (a.u.)

harmonic order 8e DC+4e FC+4e

“Dynamical”

“Frozen” or

Cutoff

3-step model (Koopmans)

h (t) | x ¨ | (t) i

FT of

10^-12 10^-10 10^-8 10^-6 10^-4 10^-2

0 20 40 60 80 100

intensity (a.u.)

harmonic order Core Valence DC+4e: Net

h (t) | x ¨ | (t) i

FT of

Submission due: July 31.  

Place to submit: the office of the Nuclear Engineering & Management, 2nd  floor of the Bldg. 3. Language: English or Japanese.  

(1) 

MCTDHF includes single determinant TDHF as a special case. 

Derive the TDHF equations of motion (given in p. 7) starting from the 

MCTDHF equations (p. 15) by ignoring CI equations and inserting HF wave  function,  

in the definition of one and two particle reduced density matrices. Here N is  the number of electrons. The resultant equations will still look different 

from those in p. 7. Choose the appropriate Hermitian matrix R in order to  obtain exactly the same equations as those in p. 7. 

| i = | 1

1

1

· · · 1

0000 i

GVB(r₁,r₂) = 1

p2 [ ₁(r₁) ₂(r₂) + ₂(r₁) ₁(r₂)]

= A1 1(r1) 1(r2) + A2 2(r1) 2(r2)

A₁ = 1 + |S₁₂| p2(1 + |S₁₂|²)

S₁₂^⇤

|S₁₂|, A₂ = 1 |S₁₂|

p2(1 + |S₁₂|²)

S₁₂

|S₁₂|,

1 = 1

p2(1 + |S₁₂|)

⇢ S₁₂

|S₁₂| ¹ + ₂

2 = 1

p2(1 + |S |)

⇢ S₁₂^⇤

|S | ^{2 1}

Derive the transformation (Expressions for      below) from GVB  wave function to the MCTDHF wave function for the two-electron singlet  system, and explicitly show that MCTDHF orbitals are orthonormal. Assume  that GVB orbitals are normalized. See J. Phys. B: At. Mol. Opt. Phys. 47 , 

204031 (2014). 

A₁, A₂, ₁, ₂

+ 

A

+  A

1 2

+

= C

+ C

{

+

}

C

+ C

A

, h

| +

A i = 0