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L02
Wave Function Based HF Methods
武晓君 (Xiaojun Wu) [email protected]
江俊 (Jun Jiang) [email protected]
Study Subjects
Atom
Molecule
Nano-system
Surface
Interface
Solid
Information?
•
•
•
•
•
Atomic structure ?
Energy ?
Evolution/Change ?
Electron ?
Magnetism, Force, Light, Heat,……?
Challenge
• How to do simulations with Computers ?
– From atomic structure to energy
– From the variation of atomic structure to
the system energy change
Adiabatic Potential Energy Surface
• Key Issue: Define atomic position
– The location of nucleus and electrons
How to define the status of a
particle at the Microscopic scale?
Matter(de Broglie) wave
 wavefunction Ψ(x, t)
Energy, Momentum, wavelength, frequency

x

i
 ~ exp  i 2   t    exp 





xp

Et



Wavefunction:
the full description of a particle
What is wavefunction ?
Wavefunction Ψ(r, t)
describe probability amplitude
– Born's statistical interpretation
Ψ(r, t)
the squared modulus of the wave
function, | ψ(r,t ) |2,
is the probability density of measuring a
particle at a given place (or having a
given momentum), at a given time.
One-particle states in 3d space
Wavefunction is the state function, which define
all properties of the system
What is wavefunction ?
Wavefunction Ψ(r, t)
describe probability amplitude
– Born's statistical interpretation
Many particle states in 3d space
where ri is the position of the i-th particle in threedimensional space, and t is time. Altogether, this is a
complex-valued function of 3N + 1 real variables.
Wavefunction is the state function, which define
all properties of the system
Wavefunction  Information ?
Electronic wavefunction of the system
Electronic
wavefunciton
Electron
Density
Differential
Electron Density
Electronic Wavefunction for excitation
Z.Y. Gong, et.al., J. Jiang*, Y. Luo, J. Phys. Chem. A 2016, 120, 3547
Electronic Wavefunction for excitation
Metal Organic Framework（MOF): Cu3(BTC)2
Inefficient in visible light absorption
Highly efficient in visible photocatalysis (CO2→CH4)
Cu3(BTC)2
Cu3(BTC)2
0.4
Cu3(BTC)2@2H2O
Cu3(BTC)2
@2H2O
Absborance
Cu3(BTC)2@CO2+H2O
Cu3(BTC)2@2CO2
0.2
0.0
Cu3(BTC)2
@2CO2
400
500
600
Wavelength(nm)
700
Adv. Mater. 2014, 26, 4788.
CO2 adsorption—Enhance visible absorption
Electron density distribution guide charge flow
Vac.
EF
Semi1
Semi2
EF
e- e- e-
Metal
Semicond-Semicond nanoheterostructure
Type I interfacing with Metal
Vac.
e- e- e-
eEF
Semi1
Semi2
e-
Metal
h+
Semicond-(Semicond/Metal)
Nanoheterostructure Type II
T.T. Zhuang, et.al., J. Jiang*, S.H. Yu*, Angew. Chem. Int. Ed. 2015, 39, 11495
T.T. Zhuang, et.al., J. Jiang*, S.H. Yu*, Angew. Chem. Int. Ed. 2016, 55, 6396
D. Liu, et. al., J. Jiang*, Y.J. Xiong*, Nano Res. 2016, 9, 1590
Design surface/interface differential/polarization charge
Ag donate polarization charge to CuO
Lower down CO activation barrier
Vac
WorkFunction
Vac
Wf=
5.97 eV
-e
Wf=
4.45eV
Ef
Ef
CuO
Cu
O
Ag
(111)
Cu
O
Ag
Ag
Y. Bai, et. al. W.X. Huang*, J. Jiang*, Y.J.Xiong* J. Am. Chem. Soc. 2014, 136, 14650
Double-transition-atom (Co2) extract polarization chargefrom C2N electro-negativity
(2D Material) : promote O2 dsorption and activation
X.Y. Li, et. al. J. Jiang*, J. Phys. Chem. Lett. 2016, 7, 1750
Wavefunction:
Real Space .vs. Momentum space
Ψ(r,t): space distribution
•
•
 momentum probability distribution
Free particle: | ψ(r) |2 = constant, unified distribution
over space
 momentum has deterministic value: constant
Generally, ψ(r ) is a wave packet constituted by the
superposition state of plane-wave functions, a series
of free particle states with certain momentums
  x 
1
2
1
  p 
2






  p  eipx / dp
  x  eipx / dx
The position-space and momentumspace wave functions are Fourier
transforms of each other
The properties of wavefunction?
•
ψ(r ) contains many free-particle states, each of which
has certain momentum, and their coefficient ψ(p )
describes the probability of a certain momentum
• State superposition principle:
If ψ1, ψ2, …, ψn are possible states of a system, their linear
combination is also a possible state.
The probability of the ψi state is
How to get wavefunction?
Simple case: Free particle
i

xp

Et




  x, t   A exp 
Wave Equation

i
 E
t
 2 p 2



2
2m x
2m
2
p2
E
2m
2

 2
i

t
2m x 2
External
2
 p2




2
 E  
 V    
 V  x, t  
Field V(x) i
2
t
 2m

 2m x

Quantum Mechanics
Classical Mechanics
(Newton)
High
Low
Velocity
Mass
Relativity
Wave Theory of Light
(Huygens)
Maxwell’s
EM Theory
Quantum Theory
Quantum Electrodyamics
Electricity and Magnetism
(Faraday, Ampere, et al.)
The most simple & realistic system：
Hydrogen or Hydrogen-like atom

2 2
2 2
Ze 2
H 
N 
e 

2M
2me
40 r
me
1

M 1836
central force field V(r)
Born-Oppenheim approximation
（BO, adiabatic approximation）
Frozen Nucleus
2
2

Ze
'
Hˆ  Tˆe  VˆeN  
 e2  
2m
r
 2 2 Ze'2


(



)

(
r
)

E

(
r
)
central force field particle
2m
r
 2 2 Ze'2


(
 
) (r )  E (r )
2m
r
 (r , ,  )  R(r )Ylm ( ,  )
 nlm  N nl r
n, l , m
n 1
e
 r
Ylm( , )
(r, , )  Rn,l (r )l , m ( )m ( )
eimφ
R


(r ) * Rn ,l (r )r dr   n , n ' l ,l '
2
n ', l '
l ', m '
m
( ) * l , m ( ) sin d   l ,l ' m , m '
( ) *  m ( )d   m , m '
Orbital wavefunction of Hydrogen
 nm  Rn (r )m ( ) m ( ) E   1
n
2
 nm (r, , )  n  m
n principal quantum number, shell 1, 2, 3, 4, …
l angular quantum number, subshell 0, 1, 2, 3, …, n-1
m magnetic quantum number,
-l, -l+1, …, l-1, l
Complex wavefunction
 nm  Rn (r )m ( ) m ( )
Real wavefunction
 nm  Rn (r ) ( )e

eimφ
m

 
Rn (r ) ( ) cos m
 
m
im
 n m  Rn (r ) ( )e   Rn (r )m ( ) sin m
m

im
n
Atomic Orbital
n principal quantum number, shell 1, 2, 3, 4, …
l angular quantum number, subshell 0, 1, 2, 3, …, n-1
m magnetic quantum number,
-l, -l+1, …, l-1, l
• principal quantum number: n
(1) n determines
orbital
energy
2
2
Z
Z
E n   R  2   2 *13.6eV
n
n
l: 0 1 2 3 4 5 6
Shape: s p d f g h i
n=1,2,3,* (only for Hydrogen-like system)
E2s = E2p = ¼ E1s
E3s
E3p
E1s
(2) n with l determine degeneracy
n 1
g   (2l  1)  n 2
l 0
Atomic Orbital
n principal quantum number, shell 1, 2, 3, 4, …
l angular quantum number, subshell 0, 1, 2, 3, …, n-1
m magnetic quantum number,
-l, -l+1, …, l-1, l
• Angular quantum number: l
(1) l determines orbital shape
l: 0 1 2 3 4 5 6
Shape: s p d f g h i
(2) l determines orbital magnetic moment
e 
 
M
2 me


 
e
2 me
l (l  1)  l (l  1)
eh
 l (l  1)u B
4me
Atomic Orbital
n principal quantum number, shell 1, 2, 3, 4, …
l angular quantum number, subshell 0, 1, 2, 3, …, n-1
m magnetic quantum number,
-l, -l+1, …, l-1, l
• Magnetic quantum number: m
(1) m determines the direction of angular momentum
The choice of directions is 2l+1
l=2
Atomic Orbital
n principal quantum number, shell 1, 2, 3, 4, …
l angular quantum number, subshell 0, 1, 2, 3, …, n-1
m magnetic quantum number,
-l, -l+1, …, l-1, l
l: 0 1 2 3 4 5 6
Shape: s p d f g h i
Most systems contain
many particles
N-particle system is much more complicated
Practical Solutions
Semiempirical
method
Complete Active
Space
Multiconfiguration SCF
(MC-SCF)
Configuration
Interaction
theory
CI
Hartree-Fock
theory
Quantum
Chemistry
Moller-Plesset
perturbation theory
MP2
Density
Functional
Theory
(DFT)
Couple Cluster
theory CCSD(T)
Approx. I：B-O
Separate nucleus and electrons
E(R) depends on atom position  Potential Energy Surface (PES)
http://www.sciencemag.org/content/329/5995/1057/F3.expansion.html
1D PES
Multi-Dimen. PES
Simpification:
Electronic wavefunction：Ψ(r1; r2; …; rn)
 Cannot be separated
How to calculate the
wavefunction in the multidimensional space?
Approx.II
Single-electron
Simplify Hamiltonian
Single-e Schrodinger eqn:
There are no two particles with the same
quantum numbers: (n, l, m, s)
Here s = ±1/2 is spin quantum number
Pauli exclusion principle
Many-body wavefunction without interactions？
Hartree products（HP many-body wavefunciton）
Total energy
•
Approx. III：Mean-field
approximation
for the many-body wavefunction:
Hartree Approximation: the electrons do not
interact explicitly with the others, but each
electron interacts with the medium potential given
by the other electrons
Using the Lagrange’s multipliers method  Hartree equations:

M
N
N
ZA
 1  2 



  Φ j (j)
 2 i 
ri A
A 1
i 1 j 1

j i
Variation method:
where:
2

1
dτ j  Φ i (i)  ε i Φ i (i)

ri j

N
1 N N
E   Hi   J i j
2 i 1 j 1
i 1
j i
 1 2 M ZA 
H i   Φ i (i)h i Φ i (i)dτ i  Φ i (i)   i   
Φ i (i)dτ i - 1-e integral
2
r
A 1
iA 

1 2
2
J i j   Φ i (1) Φ j (2)dτ 1 dτ 2
r12
-Coulomb 2-e integral
- represents the repulsive interactions between two set of classic charge
densities described by Φi and Φj
•
Approx. III：Mean-field
approximation
for the HP (hartree product) many-body wavefunction:
Hartree Approximation: the electrons do not
interact explicitly with the others, but each
electron interacts with the medium potential given
by the other electrons
Variation method:
N
1 N N
E   Hi   J i j
2 i 1 j 1
i 1
j i
where:
 1 2 M ZA 
H i   Φ i (i)h i Φ i (i)dτ i  Φ i (i)   i   
Φ i (i)dτ i - 1-e integral
2
r
A 1
iA 

J i j   Φ 2i (1)
1 2
Φ j (2)dτ 1 dτ 2
r12
-Coulomb 2-e integral
- represents the repulsive interactions between two set of classic charge
densities described by Φi and Φj
Using the Lagrange’s multipliers method  Hartree equations:


M
N N
2 1
Z
1
  2   A 
Φ (i)  ε Φ (i)
Φ
(j)
dτ


i
j
i
i
j
 2
 i
ri A i 1 j1 
ri j
A 1


j i
N
ε i  Hi   Jij
j 1
j i
- Molecular orbital energy
N
N -1
N
i 1
i 1 j  i  1
E    i    J i j Total energy
In order to find Φi we need Φi  SCF procedure
ρ i (r)  Φ i (r)
N
2
i-th electron density
N
ρ tot (r)   ρ i (r)   Φ i (r)
i 1
2
Total electron density
i 1
Solution: Self-Consistent Field （SCF）
Wavefunction of
many electrons
Two-electron system:
Pauli exclusion principle
Another solution:
For our two electron problem, we can satisfy the
antisymmetry principle by a wavefunction like:
Slater wavefunction of N-electrons system
Fock, Slater 1930
• Satisfy the antisymmetry principle : exchanging the position of two
particles leads to a negative sign on the Determinants
• Satisfy the Pauli exclusion principle: SD=0 when two particles hold the
same status (same quantum numbers)
Hartree-Fock Solution
Fock, Slater 1930
E  Ψ HΨ
SD
SD
N
1 N N
 Hi  (J i j  K i j)
2 i1 j1
i 1
Interaction energy of electrons
 1

Hi   Φ *i (r)hi Φ i (r)dr   Φ *i (r)  2i  v i  Φ i (r)dr
 2

J i j   Φ2i (1)
1 2
Φ j (2)dr1 dr2
r12
K i j   Φ*i (r1 )Φ*j (r1 )
Coulomb interaction energy
1
Φ i (r2 )Φ j (r2 )dr1 dr2 Exchange energy of electrons with same spi
r1  r2
Slater determinant is better than HP (hartree product)
exchange operator:


1
K j (1)Φi (1)   Φ j (2) Φi (2)dτ 2 Φ j (1)
r12


Minimizing the energy by varying the spin orbitals leads to the Hartree-Fock equations:
N
N
2
1
1
 1 2

*



v
Φ
(r)

Φ
(r'
)
Φ
(r)dτ

Φ
(r'
)Φ
(r'
)
Φ j (r)dτ j   i  i (r)

i i
i
j 
j
i
 j
 2 i
r - r'
r - r'

j1
j 1
Fock operator:
N
molecular orbital energies:
f i  h i   (J j  K j )
j 1
N
ε i  Φ i f i Φ i  H i   (J i j  K i j )
j 1
Restricted and Unrestricted HF
Restricted HF
restricted wave-function
RHF  1s1s 2 s
Φ1(x)=φ1(r)α(ω) Φ2(x)=φ1(r)β(ω)
Restricted wave-function for Li atom
But: K1s()2s( )≠0 and K1s()2s()=0
1s() and 1s() electrons will experience different potentials so that it will be more
convenient to describe the two kind of electrons by different wave-functions
Unrestricted HF
No restriction on spatial wavefunction for spin orbit
Φ1(x)=φα1(r)α(ω) Φ2(x)=φβ1(r)β(ω)
UHF  1s 1s 2s
Unrestricted wave-function for Li atom
 iα  jα  δ ij
 iβ  βj  δ i j
 iα  βj  S αβ
ij
UHF solution: possible spin
containment
Closed Shell System RHF is good
RHF and UHF present same results
Open Shell System
UHF
Advantages: efficient with two sets of spatial function
Dis-advantage：mixing high-order spin states
 causing spin contamination
How to check and solve ?
OK if errors <10%
Gaussian software, use iop(5/14)=2 输出 <S2>
Recommended Solution
Restricted Open-Shell HF (ROHF)
Wavefunction is composed of many determinants
1. Good for energy and wavefunction
2. Bad for spin-dependent properties
Compare
Unrestricted HF (UHF)
Wavefunction is composed ofa single determinants
1. Energy: EUHF ≤ ERHF or EROHF
2. Good for spin-dependent properties
Hartree-Fock-Roothann Equation
Ψ 0  Ψ HF
Φ1 (x 1 )
Φ (x )
 Ψ SD (r1 , r2 ,..., rN )  (N! )1 /2 1 2

Φ2 (x 1 ) ...
Φ2 (x 2 ) ...


Φ N (x 1 )
Φ N (x 2 )

Φ1 (x N ) Φ2 (x N ) ... Φ N (x N )
Roothann
Linear Combination of Atomic Orbital – Molecular Orbital (LCAO-MO)
K
 i   c i 
 1
i=1,2,...,K
{μ} – a set of known functions :
atomic basis or plane wave basis set
 c11

c
C   21
...

c
 K1
FC=SC
 1 0

 0 2
 
... ...

0 0

where





...  K 
...
...
...
0
0
...
c12
c 22
...
... c1K
... c 2 K
... ...
cK 2
... c KK







Turn HF problem to Matrix problem
Density matrix element
N /2
P  2  ca c*a
a
Single-electron
Ignore ele-ele
interactions
Mean Field
Koopman’s Theorem
The first ionization energy = Energy of HOMO
The electron affinity energy = Energy of LUMO
HF method
Good at predicting geometry
Bad at predicting binding energy
What will be the next?