Introduction to Second-quantization II
Jeppe Olsen
Lundbeck Foundation Center for Theoretical Chemistry
Department of Chemistry, Aarhus
September 20, 2011
Jeppe Olsen (Aarhus)
Second Quantization II
September 20, 2011
1 / 31
Contents, lecture II
Spin in second quantization Form of spin-free one- and two-electron
operators, section 9, 79-83
Rotations using exponentials I: first quantization, section 8
Rotations using exponentials II: second quantization, section 8
Density matrices - bonus slides
Jeppe Olsen (Aarhus)
Second Quantization II
September 20, 2011
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Introduction to spin
Spin-functions for one electron, first quantization
Two spin-functions, α(ms ), β(ms )
ms can take the two values 12 , − 12
α( 21 ) = 1, α(− 12 ) = 0
β( 21 ) = 0, β(− 12 ) = 1
Spin properties
1
Sz α = α S 2 α =
2
1
Sz β = − β S 2 β =
2
3
α
4
3
β
4
α is ’spin-up’, β is ’spin-down’
Jeppe Olsen (Aarhus)
Second Quantization II
September 20, 2011
3 / 31
September 20, 2011
4 / 31
Introduction to spin
Spin-functions for one electron, first quantization, cont’d
Integration over spin
R
Easy: dms f (ms ) = f ( 12 ) + f (− 12 )
α, β constitute an orthonormal basis
1
2
R
R
?
dm
α
(m
)α(m
)
=
dms β ? (ms )β(ms ) = 1
s
s
s
R
dms α? (ms )β(ms ) = 0
(easily checked from definitions)
Jeppe Olsen (Aarhus)
Second Quantization II
Introduction to spin
Spin orbitals in first and second quantization
First quantization: separate functions for spatial- and spin-parts, e.g.
φp (r)α(ms )
Second quantization: Both spatial and spin-parts created by one
†
operator for example apα
†
†
a
|vaci
Ground state of H2 : a1σ
α
1σ
g
gβ
Anti-commutation relation
†
[apσ
, aqτ ]+ = δpq δστ
Jeppe Olsen (Aarhus)
Second Quantization II
September 20, 2011
5 / 31
Introduction to spin
Various types of operators
Spin-free operators
Operator that does not change spin-functions
f c α = αf c
f c β = βf c
Operators like kinetic energy and Coulomb-repulsion depend only on
spatial coordinates → spin-free
Pure spin-operators
Operators that does not change spatial functions
f c φp (r) = φp (r)f c
Operators like S+ , S− , S 2 are of this type
Mixed operators
Changes both spatial and spin functions
Jeppe Olsen (Aarhus)
Second Quantization II
September 20, 2011
6 / 31
Pure spin-operators
Example: S+
Form
S+ =
X
†
(S+ )pσqτ apσ
aqτ
pσqτ
S+ in FQ
S+ β = α
S+ α = 0
S+ φp = φp S+
(S+ is pure spin-operator)
The integrals (S+ )pσqτ
Z
Z
(S+ )pσqτ = dr dms φp (r)? σ(ms )? S+ (ms )φq (r)τ (ms )
Z
Z
= dr φp (r)? φq (r) dms σ(ms )? S+ (ms )τ (ms ) = δpq δσα δτ β
Jeppe Olsen (Aarhus)
Second Quantization II
September 20, 2011
7 / 31
September 20, 2011
8 / 31
Pure spin-operators
Example: S+
From last slide
(S+ )pσqτ = δpq δσα δτ β
So
S+ =
X
†
(S+ )pσqτ apσ
aqτ
pσqτ
=
X
†
δpq δσα δτ β apσ
aqτ
pσqτ
=
X
†
apα
apβ
p
Corresponds to the intuitive picture: changes β to α
Jeppe Olsen (Aarhus)
Second Quantization II
Spin-free operators
One-electron operators
Do you remember: general form for one-electron operator
ĥ =
X
hPQ aP† aQ
PQ
hPQ =
Z
dx φ?P (x) h(x) φQ (x)
Separation of spatial and spin parts
P → pσ, Q → qτ
ĥ =
X
†
hpσqτ apσ
aqτ
pσqτ
hpσqτ =
Z
dr
Z
dms φ?p (r)σ ? (ms ) h(r, ms ) φq (r)τ (ms )
Jeppe Olsen (Aarhus)
Second Quantization II
September 20, 2011
9 / 31
Spin-free operators
One-electron operator
The integrals hpσqτ
Z
Z
hpσqτ = dr dms φ?p (r)σ ? (ms )h(r)φq (r)τ (ms )
Z
Z
= dms σ ? (ms )τ (ms ) dr φ?p (r)h(r)φq (r) = δστ hpq
Introduce Epq =
P
†
†
†
a
a
=
a
a
+
a
pα qα
pβ aqβ
σ pσ qσ
(generator of the linear group)
The operator then become
ĥ =
P
†
pσqτ δστ hpq apσ aqτ =
P
pq
hpq Epq
hpq depends only on orbitals
Jeppe Olsen (Aarhus)
Second Quantization II
September 20, 2011
10 / 31
Spin-free operators
Two-electron operator
General two-electron operator in SQ
X
1
† †
ĝ =
gpσ,qτ,r µ,sν apσ
ar µ asν aqτ
2 pσ,qτ,r µ,sν
Simplification for spin-free two-electron operator
gpσ,qτ,r µ,sν = gpqrs δστ δµν
Z
gpqrs = drdr0 φ?p (r)φ?r (r0 )g c (r, r0 )φq (r)φs (r0 )
Gives spin-free two-electron operator
X
1X
† †
apσ
ar τ asτ aqσ )
ĝ =
gpqrs (
2 pqrs
στ
Jeppe Olsen (Aarhus)
Second Quantization II
September 20, 2011
11 / 31
Spin-free operators
Two-electron operator, cont’d
A simple rewrite to obtain E-operators
X
1X
† †
ĝ =
gpqrs (
apσ
ar τ asτ aqσ )
2 pqrs
στ
X
1X
† †
=−
gpqrs (
apσ
ar τ aqσ asτ )
2 pqrs
στ
X
1X
†
=−
gpqrs (
apσ
(−aqσ ar†τ + δqr δστ )asτ )
2 pqrs
στ
X
X
1X
†
†
†
apσ
asσ )
=
gpqrs (
apσ aqσ ar τ asτ − δqr
2 pqrs
σ
στ
1X
1X
=
gpqrs (Epq Ers − δqr Eps ) =
gpqrs epqrs
2 pqrs
2 pqrs
Definition: epqrs = Epq Ers − δqr Eps
Jeppe Olsen (Aarhus)
Second Quantization II
September 20, 2011
12 / 31
Spin-free operators
The nonrelativistic Hamiltonian
Terms
One-electron part containing kinetic energy and nuclear attraction ←
ĥ
Two-electron part containing electron-electron repulsion ← ĝ
Spin-free !
The form
Ĥ = ĥ + ĝ
X
1X
=
hpq Epq +
gpqrs (Epq Ers − δqr Eps )
2
pq
pqrs
X
1X
=
hpq Epq +
gpqrs epqrs
2
pqrs
pq
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Second Quantization II
September 20, 2011
13 / 31
Rotations of orbitals using exponential mappings
Problem
A set of orthonormal spin-orbitals is given φ
Obtain all orthonormal spin-orbitals φ̃ that can be obtained as linear
combinations of φ
φ̃ = φU
hφ̃P |φ̃Q i = δPQ
We want a simple parameterization of all orthonormal spin-orbitals φ̃
Occurs in SCF and MCSCF optimization
( Note: P, Q now refers to spin-orbitals)
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Second Quantization II
September 20, 2011
14 / 31
Rotations of orbitals using exponential mappings
May the elements of U be used as independent parameters?
No, because:
The required orthonormality of φ̃ requires that U is unitary
U† U = 1
The unitary conditions on U gives constraints on its elements
The elements of U can therefore not be used as independent
parameters
A parameterization of unitary matrices may be obtained in terms of
exponential matrices
Jeppe Olsen (Aarhus)
Second Quantization II
September 20, 2011
15 / 31
Rotations of orbitals using exponential mappings
Exponential matrices
Definition
∞
X
1 n
exp(A) =
A
n!
n=0
1
= 1 + A + A2 + · · ·
2
Straightforward extension of Taylor-expansion of exp(x), where x is a
number
Converges for any choice of finite A ( assuming finite dimension)
Jeppe Olsen (Aarhus)
Second Quantization II
September 20, 2011
16 / 31
Rotations of orbitals using exponential mappings
Exponential matrices, cont’d
Relations
exp(A) exp(−A) = 1
exp (A)† = exp(A† )
B exp(A)B−1 = exp(BAB−1 )
exp(A + B) = exp(A) exp(B) iff [A, B] = 0
The Baker-Campbell-Hausdorf (BCH) expansion
1
exp(A)B exp(−A) = B + [A, B] + [A, [A, B]]
2
1
+ · · · + [A, [A, · · · , [A, B] · · · ]] + · · ·
{z
}
n! |
n
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Second Quantization II
September 20, 2011
17 / 31
Rotations of orbitals using exponential mappings
Unitary matrices as exponentials of antihermitian matrices
Any unitary matrix may be written as the exponential of an
antihermitian matrix
U = exp(−κ)
U = exp(iκ)
κ† = −κ(κ is antihermitian, USED HERE)
κ† = κ(κ is hermitian)
Relies on two theorems
1
exp(−κ) is unitary for κ antihermitian
2
All unitary matrices can be written in this form
1: exp(−κ) is unitary
exp(−κ)† exp(−κ) = exp(κ) exp(−κ)
=1
Jeppe Olsen (Aarhus)
Second Quantization II
September 20, 2011
18 / 31
Rotations of orbitals using exponential mappings
Unitary matrices as exponentials of antihermitian matrices, cont’d
2: All unitary matrices may be written as exp(−κ)
A unitary matrix U may be diagonalized U = VV† ,
VV† = 1
is a complex diagonal matrix and |i | = 1
|i | = 1 → i = exp(iδi ), δi is real
Therefore U = V exp(iδ)V† = exp(−(−iVδV† ))
−iVδV† is antihermitian, (−iVδV† )† = iVδV†
Jeppe Olsen (Aarhus)
Second Quantization II
September 20, 2011
19 / 31
Rotations of orbitals using exponential mappings
Unitary matrices as exponentials of antihermitian matrices, cont’d
What is so great about U = exp(−κ)
It as easy to obtain a independent set of parameters for an
antihermitian matrix
1
2
Elements at or below the diagonal are independent parameters
Obtain the elements above the diagonal as κQP = −κ?PQ , P > Q
A parameterization of all sets of orthonormal orbitals
φ̃ ← U ← κ ← κPQ , P > Q
The energy as a function of orbitals
E = E (φ̃) = E (κlow ), for example the SCF energy
The energy is now a function of a set of independent parameters
Standard methods like Newton method may be used to optimize E
Of some use for HF, essential for MCSCF
Jeppe Olsen (Aarhus)
Second Quantization II
September 20, 2011
20 / 31
Rotation of creation-operators
Rotation of orbitals → rotation of creation operators
Rotated orbitals: φ̃P =
P
Q
φQ UQP , U = exp(−κ)
A creation
creating on electron in spin-orbital φ̃P :
P operator
†
†
ãP = Q aQ UQP
An ONV of rotatedPspin-orbitals
† †
ãP† 1 ãP† 2 · · · |vaci = Q1 Q2 ··· aQ
a · · · |vaciUQ1 P1 UQ2 P2 · · ·
1 Q2
Complicated form
An operator form of ã†
ãP† =
P
Q
†
aQ
(exp(−κ))QP
We will show that ãP† = exp(−κ̂)aP† exp(κ̂)
P
κ̂ = PQ κPQ aP† aQ (κ̂ is an antihermitian one-electron operator)
The exponential of an operator is defined as the exponential of matrix
and has the same properties and relations.
Jeppe Olsen (Aarhus)
Second Quantization II
September 20, 2011
21 / 31
Rotation of creation-operators
exp(−κ̂)aP† exp(κ̂) =
P
Q
†
exp(−κ)QP aQ
The BCH expansion gives
1
exp(−κ̂)aP† exp(κ̂) = aP† + [aP† , κ̂] + [[aP† , κ̂], κ̂] + · · ·
2
Calculating the commutator gives
exp(−κ̂)aP† exp(κ̂)
X
X 1
†
†
†
= aP −
κQP aQ +
(κ)2QP aQ
+ ···
2!
Q
Q
X †
1
aQ (1 − κ + (κ)2 + · · · )QP
=
2!
Q
X
†
=
exp(−κ)QP aQ
Q
Jeppe Olsen (Aarhus)
Second Quantization II
September 20, 2011
22 / 31
Rotation of creation-operators
A compact form of ONV with rotated operators
] i = ㆠㆠ· · · ㆠ|vaci
|ONV
P1 P2
PN
= exp(−κ̂)aP† 1 exp(κ̂) exp(−κ̂)aP† 2 exp(κ̂)
· · · exp(−κ̂)aP† N exp(κ̂)|vaci
= exp(−κ̂)aP† 1 aP† 2 · · · aP† N |vaci
= exp(−κ̂)|ONV i
] i = exp(−κ̂)|ONV i
|ONV
Applying a single exponential operator rotates all the spin-orbitals , used
for generating all sets of orthonormal orbitals in SCF and MCSCF.
Jeppe Olsen (Aarhus)
Second Quantization II
September 20, 2011
23 / 31
Rotation of creation-operators
Restrictions on κ̂
Hitherto: unrestricted κ̂
P
κ̂ = PQ κPQ aP† aQ with κ general antihermitian matrix → complete
general spin-orbital transformation
1
2
3
Allows the spin-orbitals to become complex (elements of κ may be
complex)
α- and β-spin-orbitals are allowed to have different spatial parts
α- and β-spin-orbitals are allowed to mix
Two types of restrictions
1
Restrict rotations to allow only real orbitals
2
Restrict rotations to ensure that spatial parts of α− and
β-spin-orbitals are identical
Jeppe Olsen (Aarhus)
Second Quantization II
September 20, 2011
24 / 31
Rotation of creation-operators
Restrictions on κ̂
Restriction to real orbitals
Restrict κ to R κ containing only real elements
Rκ
is an antisymmetric matrix, R κQP = −R κPQ , R κPP = 0
X
X
X
†
†
R
R
R
κ̂ =
κPQ aP aQ =
κPQ aP aQ +
κPQ aP† aQ
PQ
=
X
P>Q
R
κPQ aP† aQ
P>Q
+
X
P<Q
R
†
κQP aQ
aP
=
P>Q
X
P>Q
R
†
κPQ (aP† aQ − aQ
aP )
Restrictions on spin-orbitals
Rκ
pαqβ
Rκ
pαqα
= R κpβqα = 0 (no mixing of α,β-orbitals)
= R κpβqβ = R κpq (identical rotations of α,β-orbitals)
X
†
†
R
†
†
aqβ − aqα
apα − aqβ
apβ )
κ̂ =
κpq (apα
aqα + apβ
p>q
=
X
Jeppe Olsen (Aarhus)
p>q
R
κpq (Epq − Eqp )
Second Quantization II
September 20, 2011
25 / 31
Rotation of creation-operators
Restrictions on κ̂
Real and spin-conserving rotations of ONV
] i = exp(−κ̂)|ONV i
|ONV
X
R
= exp(−
κpq (Epq − Eqp ))|ONV i
p>q
This is the form that will be used in the MCSCF lectures -with small
notational differences.
Jeppe Olsen (Aarhus)
Second Quantization II
September 20, 2011
26 / 31
Density matrices
One-electron orbital density matrix
A wave function and a spinfree 1-e-operator
|0i =
X
k
Ck |ki Ω =
X
Ωpq Epq
pq
Expectation value of spin-free 1-e operator
h0|Ω̂|0i =
X
pq
Ωpq h0|Epq |0i
Consists of two parts
Integrals Ωpq : Depends on actual operator, independent of C
Matrix-elements h0|Epq |0i: Independent of operator, depends on C
Jeppe Olsen (Aarhus)
Second Quantization II
September 20, 2011
27 / 31
Density matrices
One-electron orbital density matrix, cont’d
From last slide
h0|Ω̂|0i =
P
pq
Ωpq h0|Epq |0i
The one-electron orbital density matrix, D
Dpq = h0|Epq |0i
Matrix with dimensions M by M
All the information about C needed to calculate expectation values of
spin-free one-electron operators
Data reduction: million/billion of parameters in C → M by M matrix
Diagonalization of D → natural orbitals
Jeppe Olsen (Aarhus)
Second Quantization II
September 20, 2011
28 / 31
Density matrices
Natural orbitals
Deinition
The density matrix D is Hermitian
And may therefore be diagonalized
The eigenvalues of D are the natural occupation nummbers
The eigenvectors defines the natural orbitals
Natural occupation numbers
Gives occupation of the natural orbitals in the wave function
Are between 0 and 2
Assist in the analysis of wave functions
Jeppe Olsen (Aarhus)
Second Quantization II
September 20, 2011
29 / 31
Density matrices
The density in real space
Expectation value of one-electron multiplicative operator
h0|Ω̂|0i =
=
XZ
dr φ?p (r)Ωc (r)φq (r) Dpq
pq
Z
dr Ωc (r)
X
Dpq φ?p (r)φq (r)
pq
P
The density ρ = pq Dpq φ?p (r)φq (r)
!
=
Z
dr Ω̂c (r)ρ(r)
Function in real space, function of coordinates of one electron
Sufficient to evaluate expectation values of multiplicative
1-electron-operators
Of importance for DFT...
(The density matrix D allows the calculation of expectation values of
all one-electron operators)
Jeppe Olsen (Aarhus)
Second Quantization II
September 20, 2011
30 / 31
Density matrices
Two-electron orbital density matrix
A wave function and a spinfree 2-e-operator
|0i =
X
k
Ck |ki
1X
Ω=
Ωpqrs epqrs
2 pqrs
Expectation value of spin-free 2-e-operator
1X
1X
h0|Ω̂|0i =
Ωpqrs h0|epqrs |0i =
Ωpqrs dpqrs
2 pqrs
2 pqrs
d is the 2-e orbital density matrix
Contains all information about C needed to calculate expectation
values of spin-free 2-electron operators
Dimension M 4
Cannot be explicitly constructed and stored for large molecules
Is therefore usually constructed on the flight when needed
Jeppe Olsen (Aarhus)
Second Quantization II
September 20, 2011
31 / 31