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Many Electron AtomsMany Electron Atoms
Juan Ignacio Rodríguez Hernández
Escuela Superior de Física y MatemáticasInstituto Politécnico Nacional
Mexico City
August 2016
1
Many Electron AtomsMany Electron Atoms
2
−ℏ
2
2mn
∇ R2−∑
i=1
Nℏ
2
2m∇ r i
2−∑
i=1
N1
4 πε0
e2
|R− r j|+∑
i=1
N
∑j> i
N1
4 πε0
e2
|r i−r j|¿Ψ ( R , r i)=EΨ ( R , r i)
¿¿¿
H Ψ=EΨ
For gold N=79, so we have 3*79=237 independent variables !!!
Atomic Units (a.u.)Atomic Units (a.u.)
3
1h 1em 02 e=
{−ℏ
2
2 μ∇ r
2−
e2
4 πε0
1r}ψ ( r )=Eψ ( r )
{−12∇ r
2−1r}ψ ( r )=Eψ ( r )
Atomic Units (a.u.)Atomic Units (a.u.)
4
5 1 15 31 27.2 2.20 10 6.58 10 2.63 10 / .)Hartree eV cm Hz kJ mol � � �
{−12∇ r
2−1r}ψ ( r )=Eψ ( r ) E=−0 .5 a .u .≡−0.5 Hartrees
1Bohr=0 . 529 A=5 . 29×10−11m
Energy:
Lenth:
Mass:me=9 .1095×10−31kg
Charge:
e=1 .6022×10−19C
Many electron atom HamiltonianMany electron atom Hamiltonianin a.u.in a.u.
−ℏ2
2mn
∇ R2−∑
i=1
N ℏ2
2m∇ r i
2 −∑i=1
N1
4 πε0
e2
|R− r i|+∑
i=1
N
∑j>i
N1
4 πε 0
e2
|r i− r j|¿Ψ ( R , r i )=EΨ ( R , r i )
¿¿¿
−ℏ
2
2mn
∇R2−∑
i=1
N12∇ r i
2 −∑i=1
Ne2
|R− r i|+∑
i=1
N
∑j>i
Ne2
|r i− r j|¿Ψ ( R , r i )=EΨ ( R , r i )
¿¿¿
−∑i=1
N12∇ r i
2−∑
i=1
N ZN
|R− r i|+∑
i=1
N
∑j>i
N1
|r i− r j|¿Ψ ( r i)=EΨ ( r i)
¿¿¿
5
Stern-Gerlach ExperimentStern-Gerlach Experiment
Beam of Hydrogen or Silver atoms
7
s2≡ sx
2+ s y
2+ sz
2
[ sx , s y ]= iℏ sz [ s y , s z ]=i ℏ sx [ sz , sx ]=i ℏ s y
s2 f=ℏ2 s( s+1) f s=
12
s z f=ℏm s f ms=−12,12
Spin angular momentum Spin angular momentum
8
s2 f=ℏ2 s( s+1) f ; s=12
s z f=ℏm s f ; mM=−12,
12
Spin angular momentum Spin angular momentum
s z f 1/2=ℏ12f 1/ 2
s z f −1/2=−ℏ12f−1/2
s zα=ℏ12α
s z β=−ℏ12β
9
s2 f=ℏ2 s( s+1) f ; s=12
s z f=ℏm s f ; m s=−12,12
A postulate: A postulate: Spin is an intrinsic variable Spin is an intrinsic variable
s zα=ℏ12α s z β=−ℏ
12β
s=√32ℏ
ms=−12,12
10
Spin Variable Spin Variable
α (m s)=δms 1/2 β(ms )=δms−1 /2
ms=1/2,−1/2
α(m s=1/2)=1 α(m s=−1/2)=0
β(ms=1/2)=0 β(ms=−1/2)=1
11
Orthonormality ofOrthonormality of spin funcitons spin funcitons
s zα=12α
s z β=−12β
∑ms=−1/ 2
1/2
α*(ms )α (ms )=1
∑ms=−1/ 2
1/2
β* (ms ) β (ms )=1
∑ms=−1/ 2
1/2
α* (ms ) β (ms )=0
12
Spin orbitals Spin orbitals
{−12∇ 2−
1r}ψ ( r )=Eψ ( r )
ψ ( r )=ψ ( x , y , z ) ψ ( r ,ms )=ψ ( x , y , z ,ms )
The Hamiltonian is not dependent of the spin operator (to first approximation):
ψ ( r ,ms )=φ( r )gms(ms )
ψ ( r ,1 /2)=φ ( r )α
ψ ( r ,−1 /2)=φ ( r ) β
13
Orthomalized Spin orbitals Orthomalized Spin orbitals
⟨ψ i|ψ j⟩=∫ψ i¿( r ,ms )ψ j ' ( r ,ms )dτ≡∑
m s
∫φ i¿( r ) gm si
¿(m s)φ j¿( r )gmsj¿ (ms )d r
=[∫φi¿( r )φ j ( r )d r ]∑
m s
gmsi
¿(ms )gmsj(m s)=δij δm
simsj
If φi form an orthonormalized set then so do the spin orbitals ψi ‘s
Variational TheoremVariational TheoremGiven a system whose Hamiltonian operator is time independent and whose lowest-energy eigenvalues is E0, if Ф is any normalized-well behaved function of the coordinates of the system’s particles that satisfies the boundary condiition of the problem, then
⟨φ|H|φ ⟩≥E0
14
Variational TheoremVariational Theorem
. .. . . .
{ 1}
ˆ ˆming sg s g sE H H
Ψ g. s The ground state function is the eigenfunction with the lowest eigenvalue (ground state energy)
is the function that minimaze the (energy) functional:
1
Ψ g. s
Minimization Constrain:
15
Hartree-Fock ApproximationHartree-Fock Approximationfor the many-electron atom for the many-electron atom
−∑i=1
N12∇ r i
2−∑
i=1
N Z N
ri+∑
i=1
N
∑j>i
N1
|r i− r j|¿Ψ ( r i )=EΨ ( r i)
¿¿¿
Instead of solving:
One minimizes:
F [Ψ ]=⟨Ψ|H|Ψ ⟩
H=−∑i=1
N12∇ r i
2−∑
i=1
N Z N
ri+∑
i=1
N
∑j>i
N1
|r i− r j|
16
HF wave function: Slater HF wave function: Slater Determinant Determinant
Spin Orbital concept:
α (m s1 ) if s=1/2
β (ms 1 ) if s=−1/2¿
g (m s1 )=¿
¿
ui ( x j )≡ψ i( x j )≡φi( r j )g (m s1 )
17
Closed shell restricted Closed shell restricted Hartree-Fock Approximation Hartree-Fock Approximation
N is even and there is always a α and β spin orbitals for each spacial function :
1 1 1 1 / 2 1
1 2 1 2 / 2 2
1 1 / 2
( ) (1) ( ) (1) ... ( ) (1)
( ) (2) ( ) (2) ... ( ) (2)1
!
( ) ( ) ( ) ( ) ... ( ) ( )
N
N
N N N N
r r r
r r r
N
r N r N r N
�
r r r
r r r
M M Mr r r
Function Vector Space
HF (Slater determinant) space 18
HF Aproximation HF Aproximation
−∑i=1
N12∇ r i
2−∑
i=1
N Zn
r i
+∑i=1
N
∑j>i
N1
|r i− r j|¿Ψ ( r i)=EΨ ( r i)
¿¿¿
Instead of solving:
One minimizes:
F [Ψ ]=⟨Ψ|H|Ψ ⟩
H=−∑i=1
N12∇ r i
2−∑
i=1
N Z N
r i+∑
i=1
N
∑j>i
Ne2
|r i− r j|
19
HF ApproximationHF Approximation
F [Ψ ]=⟨Ψ|H|Ψ ⟩=⟨Ψ|−∑i=1
N12∇ r i
2−∑
i=1
N ZN
r i
+∑i=1
N
∑j>i
N1
|r i− r j||Ψ ⟩
20
F [Ψ ]=⟨Ψ|H|Ψ ⟩=⟨Ψ|∑i=1
N
f i+∑i=1
N
∑j>i
N
g ij|Ψ ⟩
f i≡−12∇ i
2−Z N
rigij≡
1|r i− r j|
F [Ψ ]=⟨Ψ|H|Ψ ⟩=∑i=1
N
⟨u i(1)|f 1|ui(1 )⟩
+∑i∑j>i
[ ⟨ui(1 )u j(2 )|g12|ui(1 )u j(2)⟩−⟨ui (1)u j(2 )|g12|ui (2)u j(1 )⟩ ]
HF ApproximationHF Approximation
21
⟨Ψ|∑i
f i|Ψ ⟩=∑i=1
N
⟨u i(1)|f 1|ui(1 )⟩
⟨Ψ|∑i∑j>i
g ij|Ψ ⟩=∑i
N
∑j>i
N
[ ⟨u i(1)u j (2)|g12|u i(1)u j (2)⟩−⟨ui(1 )u j(2)|g12|u i(2 )u j(1)⟩ ]
⟨Ψ|∑i
f i|Ψ ⟩=2∑i=1
N /2
⟨φ i(1)|f 1|φi (1)⟩
⟨Ψ|∑i∑j>i
g ij|Ψ ⟩=∑i
N /2
∑j>i
N /2
[ 2J ij−K ij ]
J ij≡⟨φi (1)φ j (2)|g12|φi(1 )φ j(2 )⟩ J ij≡⟨φi (1)φ j (2)|g12|φi(2 )φ j(1 )⟩
Minimizing the HF functionalMinimizing the HF functional
22
F [Ψ ]=⟨Ψ|H|Ψ ⟩=2∑i=1
N /2
⟨φ i(1)|f 1|φi (1)⟩+∑i=1
N /2
∑j=1
N /2
[2 J ij−K ij ]=F [φi ]
A necessary condition for the φi ‘s that minimize F[φi] is
δFδφi
=0 ; i=1, .. . ,N /2
Variational derivatives
Restricted-close shell HF Restricted-close shell HF Equations Equations
/ 2 *( ') ( ') 'ˆ( ) 2'
Nj j
j i
r r drJ r
r r
�
��
r r rr
r r/ 2 ( ) ( ') 'ˆ ( ) ( )
'
Nj i
ij i
r r drK r r
r r
�
���
r r rr r
r r
23
(−12∇2−
1r+ J (r )−K (r ))φ i( r )=ε iφi ( r ) ; i=1, .. . ,N /2
f HF φ i( r )=ε iφi ( r ) ; i=1, .. . ,N /2
f HF≡−12∇ 2−
1r+ J (r )−K (r )≡−
12∇ 2−
1r+V HF (r )
V HF (r )≡ J (r )− K (r )
Fock operator
Hartree-Fock potenatial
ε i Orbital HF energies
Restricted-close shell HF Restricted-close shell HF Equations Equations
24
(−12∇2−
1r+ J (r )−K (r ))φ i( r )=ε iφi ( r ) ; i=1, .. . ,N /2
● ONE-ELECTRON equations!!!!
● Non linear integral-differential equations
● Coupled equations
(−12∇
2−
1r+∑
i=1
N /2
2∫φ j( r ' )φ j ( r ' )d r '
|r− r '|−K (r ))φi ( r )=εi φi ( r ) ; i=1, .. . ,N /2
We have separated the n-body problem!!!
Restricted-close shell HF Restricted-close shell HF Equations Equations
25
(−12∇2−
1r+ J (r )−K (r ))φ i( r )=ε iφi ( r ) ; i=1, .. . ,N /2
J (r )=∑j≠i
N /2
2∫φ j( r ' )φ j( r ' )d r '
|r− r '|=∫
ρ' ( r ' )d r|r− r '|
ρ '( r )=∑j≠i
φ j( r )φ j( r )
Coulomb potential due to the other electrons !!!!
/ 2 ( ) ( ') 'ˆ ( ) ( )'
Nj i
ij i
r r drK r r
r r
�
���
r r rr r
r rDoes not have classical analogous
Restricted-close shell HF Restricted-close shell HF Equations Equations
26
(−12∇2−
1r+ J (r )−K (r ))φ i( r )=ε iφi ( r ) ; i=1, .. . ,N /2
J (r )=∑j≠i
N /2
2∫φ j( r ' )φ j( r ' )d r '
|r− r '|=∫
ρ' ( r ' )d r|r− r '|
ρ '( r )=∑j≠i
φ j( r )φ j( r )
Coulomb operator
/ 2 ( ) ( ') 'ˆ ( ) ( )'
Nj i
ij i
r r drK r r
r r
�
���
r r rr r
r rExchange operator
Solving the HF equation:Solving the HF equation:Self-Consistent Field (SCF) Self-Consistent Field (SCF)
method method
27
(−12∇2−
1r+ J (r )−K (r ))φ i( r )=ε iφi ( r ) ; i=1, .. . ,N /2
I. Guess the φi ‘sII. Construct J and K operators III. Solve the HF equationsIV. If the new set of orbitals thus obtained are the same than the
previous ones under certain criterion, the process is said to converge. If not:
V. the HF equations are solved again using the new orbitals to calculate J and K and repeating the process until convergence.
The Hartree-Fock-Roothaan The Hartree-Fock-Roothaan equationsequations
28
(−12∇2−
1r+ J (r )−K (r ))φ i( r )=ε iφi ( r ) ; i=1, .. . ,N /2
( ) ( )K
i ir C r
�r r
Expanding the space orbitals in a basis set:
β={χ 1 , .. . , χK } Basis set of known and well-behaved functions
K > N
The HF-Roothaan equationsThe HF-Roothaan equations
29
(−12∇2−
1r+ J (r )−K (r ))φ i( r )=ε iφi ( r ) ; i=1, .. . ,N /2
HFF C CS
HF: A set of DIFFERENTIAL non-linear equations:
HF-Roothann: A set of ALGEBRAIC non-linear equations:
The HF-Roothaan equationsThe HF-Roothaan equations
30
1 1 1 2 1
2 1 2 2 2
1 2
ˆ ˆ ˆ
ˆ ˆ ˆ
ˆ ˆ ˆ
HF HF HFK
HF HF HFKKS
HF HF HFK K K K
F F F
F F FH
F F F
� �� �� �� ��� �� �� �� �� �
K
M M M
K
1 1 1 2 1
2 1 2 2 2
1 2
K
K
K K K K
S
� �� �� �
�� �� �� �� �
K
M M M
K
Fock operator matrix representation in the basis set:
Overlap matrix:
The “unknown” matricesThe “unknown” matrices
31
Coefficients matrix:
Orbital energy matrix:
11 12 1
21 22 1
1 2 1
K
K
K K K
C C C
C C CC
C C C
� �� �� �� �� �� �
K
M M M
K
1
2
0 0
0 0
0 0 k
� �� �� �� �� �� �
K
M M M
K
The HF-Roothaan equationsThe HF-Roothaan equations
It represents a system of nonlinear algebraic equations
In general, the basis functions are not orthogonal. So S is not always the identity matrix.
represents a generalized eigenvalue problem. The matrices C and represent the eigenvectors and eigenvalues, respectively
The better the quality of the basis set, the better the solution of HFR equaitons
32
HFF C CS
Solving HF-Roothaan equationsSolving HF-Roothaan equations
33
HFF C CS
1 1D SD Step 1: Find matrix D so that
1'C D C
Step 2: Define matrices C’ and F’HF
Step 2:
F 'HF≡D−1 FHF D
F 'HFC '=C ' ε C '−1 F 'HFC '=ε
Solving HF-Roothaan equationsSolving HF-Roothaan equations
34
HFF C CS
1 1D SD Step 1: Find matrix D so that
1'C D C
Step 2: Define matrices C’ and F’HF
Step 2:
F 'HF≡D−1 FHF D
F 'HFC '=C ' ε C '−1 F 'HFC '=ε
Properties: EnergyProperties: Energy
35
E≠∑i=1
N
εi
E=⟨Ψ|H|Ψ ⟩=2∑i=1
N /2
⟨φ i(1)|f 1|φi (1)⟩+∑i=1
N /2
∑j=1
N /2
[2J ij−K ij ]
E=⟨Ψ|H|Ψ ⟩=∑i=1
N /2
εa+∑i=1
N /2
⟨φi (1)|f 1|φi(1 )⟩
Solution of HF-Roothaan Solution of HF-Roothaan equttions equttions
36
HFF C CS K > N/2
K atomic orbitals:
11 12 1
21 22 1
1 2 1
K
K
K K K
C C C
C C CC
C C C
� �� �� �� �� �� �
K
M M M
K
( ) ( )K
i ir C r
�r r
Solution of HF-Roothaan Solution of HF-Roothaan equttions equttions
37
We have K orbitals and we need only N/2 atomic orbitals, which ones should we choose????
φ1( r )=∑ν=1
K
Cv1 χ ν ( r ) → ε 1
φ2( r )=∑ν=1
K
Cv 2 χ ν( r ) → ε2
⋮
φN /2 ( r )=∑ν=1
K
C vN /2 χν ( r ) → εN /2
⋮
φK ( r )=∑ν=1
K
C vK χ ν( r ) → εK
If
ε1<ε 2<. . .<εN /2< .. .<εK
φK ( r )=∑ν=1
K
Cv 1 χK ( r ) → εK
φK−1 ( r )=∑ν=1
K
C vK−1 χ ν ( r ) → εK−1
⋮
φN /2 ( r )=∑ν=1
K
C vN /2 χν ( r ) → εN /2
⋮
φ1( r )=∑ν=1
K
Cv1 χ ν ( r ) → ε 1
Solution of HF-Roothaan Solution of HF-Roothaan equttions equttions
38
φK ( r )=∑ν=1
K
Cv 1 χK ( r ) → εK
φK−1 ( r )=∑ν=1
K
C vK−1 χ ν ( r ) → εK−1
⋮
φN /2 ( r )=∑ν=1
K
C vN /2 χν ( r ) → εN /2
⋮
φ1( r )=∑ν=1
K
Cv1 χ ν ( r ) → ε 1
ε1<ε 2<. . .<εN /2< .. .<εK
Virtual orbitals
Occupied orbitals
Properties: EnergyProperties: Energy
39
E=⟨Ψ|H|Ψ ⟩=∑i=1
N /2
εa+∑i=1
N /2
⟨φi (1)|f 1|φi(1 )⟩
E=⟨Ψ|H|Ψ ⟩=2∑i=1
occ
⟨φ i(1)|f 1|φi (1)⟩+∑i=1
occ
∑j=1
occ
[2J ij−K ij ]
E=⟨Ψ|H|Ψ ⟩=2∑i=1
N /2
⟨φ i(1)|f 1|φi (1)⟩+∑i=1
N /2
∑j=1
N /2
[2J ij−K ij ]
E=⟨Ψ|H|Ψ ⟩=∑i=1
occ
εa+∑i=1
occ
⟨φi (1)|f 1|φi(1 )⟩
Koopman’s TheoremKoopman’s Theorem
40
ε a=⟨φa|f|φa ⟩+∑b=1
N /2
(2J ab−K ab)
N E−N−1 E=⟨ NΨ|H|N Ψ ⟩−⟨N−1 Ψ a|H|
N−1 Ψ a ⟩=−ε a
N−1Ψ a is the N-1 Slater determinant obtained by eliminating
the orbitalφa
Koopman’s TheoremKoopman’s Theorem
41
N E−N−1 E=⟨ NΨ|H|N Ψ ⟩−⟨N−1 Ψ a|H|
N−1 Ψ a ⟩=−ε a
Given an N-electron Hartree-Fock single determinant with
occupied energies and virtual energies , then the
ionization potential to produce an (N-1) electron single determinant
is equal to
N Ψ
ε a ε rN−1Ψ a
ε a
? Ionization Potential!!
IP=N E−N−1 E
Using Koopman’s TheoremUsing Koopman’s Theorem
42
Atom HF-631p ExperimentalN 0.469 0.189
Mg 0.253 0.281Al 0.214 0.219Si 0.240 0.299P 0.323 0.403
HF calculations using gaussian program: HF/6-31*
Ionization energies (a.u.)
Using Koopman’s TheoremUsing Koopman’s Theorem
43
HF calculations using gaussian program: HF/6-31*
Orbital energies for nitrogen:
IP=ε
Using Koopman’s TheoremUsing Koopman’s Theorem
44
HF calculations using gaussian program: HF/6-31*
Orbital energies for silicon:
IP=ε
Koopman’s TheoremKoopman’s Theorem
45
EA=N E−N +1 E=−εr
Given an N-electron Hartree-Fock single determinant with
virtual energy , then :
The Electron affinity potential to produce an (N+1) electron single
Determinant is equal to
N Ψ
ε rN +1 Ψ a
ε r
? Ionization Potential!!
Hartree-Fock ApproximationHartree-Fock Approximationfor molecules for molecules
−∑i=1
N12∇ r i
2−∑
A=1
M12∇ R
A
2−∑
A=1
M
∑i=1
N Z A
|R A− r i|+∑
i=1
N
∑j>i
N1
|r i−r j|¿Ψ ( r i )=EΨ ( r i )
¿¿¿
Schrodinger equation for a molecule:
46
−∑i=1
N12∇ r i
2−∑
i=1
N Z N
ri+∑
i=1
N
∑j>i
N1
|r i− r j|¿Ψ ( r i )=EΨ ( r i)
¿¿¿
ATOMS
Born-Oppenheimer approximation:
−∑i=1
N12∇ r i
2−∑
A=1
M
∑i=1
N Z A
|R A−r i|+∑
i=1
N
∑j> i
N1
|r i−r j|¿Ψ ( r i)=EΨ ( r i )
¿¿¿
MOLECULES
Restricted-close shell HF Restricted-close shell HF Equations Equations
47
(−12∇2−
1r+ J (r )−K (r ))φ i( r )=ε iφi ( r ) ; i=1, .. . ,N /2
(−12∇2−∑
A=1
M
∑i=1
N Z A
|R A− r i|+ J (r )−K (r ))φi ( r )=εi φi( r ) ; i=1,. . . ,N /2
Restricted-close shell HF Restricted-close shell HF Equations for moleculesEquations for molecules
48
(−12∇2−∑
A=1
M
∑i=1
N Z A
|R A− r i|+ J (r )−K (r ))φi ( r )=εi φi( r ) ; i=1,. . . ,N /2
HFF C CS
Solution of HF-Roothaan Solution of HF-Roothaan equttions equttions
49
φK ( r )=∑ν=1
K
Cv 1 χK ( r ) → εK
φK−1 ( r )=∑ν=1
K
C vK−1 χ ν ( r ) → εK−1
⋮
φN /2 ( r )=∑ν=1
K
C vN /2 χν ( r ) → εN /2
⋮
φ1( r )=∑ν=1
K
Cv1 χ ν ( r ) → ε 1
ε1<ε 2<. . .<εN /2< .. .<εK
Virtual orbitals
Occupied orbitals
Properties: Electronic EnergyProperties: Electronic Energy
50
Eelectronic≠∑i=1
N
εi
Eelectronic≡⟨Ψ|H|Ψ ⟩=2∑i=1
N /2
⟨φ i(1)|f 1|φi (1)⟩+∑i=1
N /2
∑j=1
N /2
[2 J ij−K ij ]
Eelectronic=⟨Ψ|H|Ψ ⟩=∑i=1
N /2
εa+∑i=1
N / 2
⟨φ i(1)|f 1|φi (1)⟩
Total Energy (Molecular energy)Total Energy (Molecular energy)
51
ETot=Eelectronic+∑A=1
M
∑B>A
M Z A ZB
|RA−RB|
ETot ( R1 ,. . , RM )=Eelectronic( R1 , .. , RM )+∑A=1
M
∑B>A
M Z AZ B
|R A−RB|
The energy of the molecule depends on the nuclear coordinates as paremeters!!!
Molecular ground state energyMolecular ground state energy
52
Eg . s .=min
{R1 , . . , RM }0
ETot( R1 , . ., RM )
{R1 , .. , RM }→ ground state structure
Geometry optimization
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