Second quantization
Basics
Trond Saue
Laboratoire de Chimie et Physique Quantiques
CNRS/Université de Toulouse (Paul Sabatier)
118 route de Narbonne, 31062 Toulouse (FRANCE)
e-mail: [email protected]
Insight from Jeppe Olsen acknowledged.
Trond Saue (LCPQ, Toulouse)
Second quantization
ESQC 2015
1 / 50
Asking Nature ... and the computer
To learn about the world
the experimentalist asks Nature using his experimental apparatus
the theoretician asks the wave function Ψ
using mathematical operators Ω̂
The most important operator is the Hamiltonian
Trond Saue (LCPQ, Toulouse)
Second quantization
ESQC 2015
1 / 50
The molecular problem
The time-independent molecular Schrödinger equation
Ĥ mol Ψmol = E tot Ψmol
The molecular Hamiltonian
Ĥ mol = T̂N + T̂e + Ven + Vee + Vnn
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I
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T̂N (R) - kinetic energy of nuclei
T̂e (r) - kinetic energy of electrons
Ven (r, R) - electron-nucleus interaction
Vee (r) - electron-electron interaction
Vnn (R) - nucleus-nucleus interaction
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Second quantization
ESQC 2015
2 / 50
Simplifications
The Born-Oppenheimer approximation leads to a separation of the
I
electronic problem
H el Ψel (r; R) = E el (r)Ψel (r; R);
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H el = T̂e + Ven + Vee + Vnn
...from the nuclear problem
h
i
T̂N + E el (R) χ(R) = E tot χ(R)
... although many of us stop after the electronic part.
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Second quantization
ESQC 2015
3 / 50
Theoretical model chemistries
Electronic Hamiltonian: Ĥ =
N
X
N
ĥ(i) +
i=1
1X
ĝ (i, j) + Vnn
2
i6=j
Computational cost: xNy
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Second quantization
ESQC 2015
4 / 50
Theoretical model chemistries
Electronic Hamiltonian: Ĥ =
N
X
N
ĥ(i) +
i=1
1X
ĝ (i, j) + Vnn
2
i6=j
Computational cost: xNy
Trond Saue (LCPQ, Toulouse)
Second quantization
ESQC 2015
5 / 50
The electronic energy
The electronic wave function
Ψ(1, 2, . . . , N)
is an extraordinarily complicated mathematical beast and generally
not available in exact form.
The expectation value of the electronic Hamiltonian is
N D N
E 1X
D E X
E = Ψ Ĥ Ψ =
Ψ ĥ(i) Ψ +
hΨ |ĝ(i, j)| Ψi+hΨ |Vnn | Ψi
2
i=1
i6=j
...and can be simplified.
The constant term is
E0 = hΨ |Vnn | Ψi = Vnn hΨ|Ψi = Vnn
which follows from the normalization of the wave function.
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Second quantization
ESQC 2015
6 / 50
The electronic energy
One-electron energy
We can simplify the one-electron energy
E1
=
N D E
X
Ψ ĥ(i) Ψ
i=1
=
N ˆ
X
Ψ† (1, 2, . . . , N) ĥ(i)Ψ (1, 2, . . . , N) d1d2 . . . dN
i=1
..by noting that since electrons are indistinguishable,
all one-electron integrals have the same value
E D E
E
D D Ψ ĥ(1) Ψ = Ψ ĥ(2) Ψ = . . . = Ψ ĥ(N) Ψ
We therefore pick one and multiply with the number N of electrons
E
D E1 = N Ψ ĥ(1) Ψ
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Second quantization
ESQC 2015
7 / 50
The electronic energy
Two-electron energy
For the two-electron energy
N
E2
=
1X
hΨ |ĝ(i, j)| Ψi
2
i6=j
N
=
1X
2
ˆ
Ψ† (1, 2, . . . , N) ĝ(i, j)Ψ (1, 2, . . . , N) d1d2 . . . dN
i6=j
we can proceed in similar fashion.
Since electrons are indistinguishable,
all two-electron integrals have the same value
hΨ |ĝ(1, 2)| Ψi = hΨ |ĝ(1, 3)| Ψi = . . . = hΨ |ĝ(N − 1, N)| Ψi
We can therefore write
E2 =
1
N (N − 1) hΨ |ĝ(1, 2)| Ψi
2
where 12 N (N − 1) is the number of electron pairs.
Trond Saue (LCPQ, Toulouse)
Second quantization
ESQC 2015
8 / 50
The electronic Hamiltonian
1-electron density matrix
The one-electron Hamiltonian can be split into a free-electron part
(kinetic energy) and a term describing the electron-nucleus interaction
ĥ(1) = ĥ0 (1) + v̂en (1)
The interaction operators v̂en (i) and ĝ (i, j) are multiplicative
operators, that is, they do not contain derivatives and can be moved
around inside integrals, e.g.
D
E
V̂eN
ˆ
= N
= N
ˆ
=
Ψ† (1, 2, . . . , N) v̂eN (1)Ψ (1, 2, . . . , N) d1d2 . . . dN
N ˆ
X
v̂eN (1)Ψ† (1, 2, . . . , N) Ψ (1, 2, . . . , N) d1d2 . . . dN
i=1
v̂eN (1)n1 (1; 1)d1
where we have introduced the one-electron density matrix
ˆ
n1 (1; 10 ) = N
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Ψ† (1, 2, . . . , N) Ψ (10 , 2, . . . , N) d2 . . . dN
Second quantization
ESQC 2015
9 / 50
The electronic Hamiltonian
2-electron density matrix
The kinetic part ĥ0 is in general not multiplicative, but we can write
the expection value of kinetic energy in terms of the one-electron
density by a trick
D
T̂e
E
ˆ
= N Ψ† (1, 2, . . . , N) ĥ0 (1)Ψ (1, 2, . . . , N) d1d2 . . . dN
ˆ h
i
=
ĥ0 (10 )n(1; 10 )
d1
10 →1
The expectation value of the two-electron interaction
D E
V̂ee
=
=
ˆ
1
N(N − 1) Ψ† (1, 2, . . . , N) ĝ (1, 2)Ψ (1, 2, . . . , N) d1d2 . . . dN
2
ˆ
1
ĝ (1, 2)n2 (1, 2; 1, 2)d1d2
2
may be expressed in terms of the two-electron density matrix
ˆ
0
0
n2 (1, 2; 1 , 2 ) = N
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Ψ† (1, 2, . . . , N) Ψ (10 , 20 , . . . , N) d3 . . . dN
Second quantization
ESQC 2015
10 / 50
The electronic energy
What is needed to calculate the energy?
This exercise has shown that in order to calculate the electronic
energy we do not need the full wave function in terms of N electron
coordinates
It suffices to have the near-diagonal elements of the one-electron
density matrix and the diagonal elements of the two-electron density
matrix.
Density functional theory goes a big step further
X and proposes that
we only need the electron density ρ(r) = −e
n (1; 1)
spin
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Second quantization
ESQC 2015
11 / 50
The electronic problem
The generic form of the electronic Hamiltonian, relativistic or not, is
Ĥ =
n
X
i=1
n
1X
ĥ(i) +
ĝ (i, j) + VNN
2
i6=j
The problematic term is the two-electron interaction ĝ (i, j).
Let us for a moment drop this term, as well as VNN (a number),
and consider a two-electron system
h
i
ĥ(1) + ĥ(2) Ψ(1, 2) = E Ψ(1, 2)
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Second quantization
ESQC 2015
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The electronic problem
Separation of variables
We write the two-electron wave function as
Ψ(1, 2) = ϕa (1)ϕb (2)
Insertion into the wave equation gives
{h(1)ϕa (1)} ϕb (2) + ϕa (1) {h(2)ϕb (2)} = E ϕa (1)ϕb (2)
Division by Ψ(1, 2) gives
{h(1)ϕa (1)} {h(2)ϕb (2)}
+
=E
ϕa (1)
ϕb (2)
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Second quantization
ESQC 2015
13 / 50
The electronic problem
Separation of variables
In order for this relation to hold for any choice of electron coordinates
1 and 2, we must have
{h(1)ϕa (1)} {h(2)ϕb (2)}
=E
+
ϕa (1)
ϕb (2)
|
{z
} |
{z
}
εa
εb
A single wave equation for two electrons
h
i
ĥ(1) + ĥ(2) Ψ(1, 2) = E Ψ(1, 2)
... is thereby converted into two wave equations for single electrons
h(1)ϕa (1) = εa ϕa (1);
h(2)ϕb (2) = εb ϕb (2)
The situation is even simpler ...
Trond Saue (LCPQ, Toulouse)
Second quantization
ESQC 2015
14 / 50
The electronic problem
Indistinguishability
Electrons can not be distinguished,
so it suffices to solve a single wave equation
h(1)ϕx (1) = εx ϕx (1);
x = a, b, c, . . .
However, the form
Ψ(1, 2) = ϕa (1)ϕb (2)
is not an acceptable wave function:
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electrons are identical particles
electrons are fermions:
the wave function must be antisymmetric under particle exchange
This leads to the form
1
Ψ(1, 2) = √ {ϕa (1)ϕb (2) − ϕb (1)ϕa (2)}
2
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Second quantization
ESQC 2015
15 / 50
The electronic problem
Slater determinants
This result is readily generalized:
The exact wave function for a system of N non-interacting electrons is
an antisymmetrized product of one-electron wave functions (orbitals).
1
Ψ(1, 2, . . . , N) = √ Âϕ1 (1)ϕ2 (2) . . . ϕN (N) = |ϕ1 (1)ϕ2 (2) . . . ϕN (N)|
N!
where  is the anti-symmetrization operator.
The wave function for a system of N interacting electrons is typically
expanded in an N-electron basis of Slater determinants.
The fermionic nature of electrons is not built into the electronic
Hamiltonian.
This is achieved with second quantization !
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Second quantization
ESQC 2015
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First quantization
The quantum-mechanical Hamiltonian Ĥ is obtained from its classical
counterpart, the Hamiltonian function H ≡ H(r, p), by replacing the
dynamical variables (position r and momentum p) by operators:
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in the coordinate representation:
r → r̂ = r;
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p → p̂ = −i~
∂
= −i~∇
∂r
in the momentum representation:
r → r̂ = i~
∂
;
∂p
p → p̂ = p
...in order to obey the fundamental commutator relation
[ri , pj ] = i~δij
Quantization leads to discrete values of the energy E
(as well as angular momentum etc.)
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Second quantization
ESQC 2015
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Intermission: the double slit experiment
Researchers of Hitachi has reproduced the famous double-slit experiment
using an electron microscope as electron source, an “electron biprism” as
double slit and a very sensitive electron detector.
Trond Saue (LCPQ, Toulouse)
Second quantization
ESQC 2015
18 / 50
Intermission: the double slit experiment
Researchers of Hitachi has reproduced the famous double-slit experiment
using an electron microscope as electron source, an “electron biprism” as
double slit and a very sensitive electron detector.
Trond Saue (LCPQ, Toulouse)
Second quantization
ESQC 2015
19 / 50
Intermission: the double slit experiment
Researchers of Hitachi has reproduced the famous double-slit experiment
using an electron microscope as electron source, an “electron biprism” as
double slit and a very sensitive electron detector.
Trond Saue (LCPQ, Toulouse)
Second quantization
ESQC 2015
20 / 50
Intermission: the double slit experiment
Researchers of Hitachi has reproduced the famous double-slit experiment
using an electron microscope as electron source, an “electron biprism” as
double slit and a very sensitive electron detector.
This information is contained in the wave function.
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Second quantization
ESQC 2015
21 / 50
Second quantization
The wave function is interpreted as a probability amplitude.
For a single electron
ψ † (r)ψ(r)d3 r
represents the probability of finding the electron in an infinitesimal
volume element d 3 r around position r.
This means that
ˆ
ψ † (r)ψ(r)d3 r = 1
What if ψ † (r) and ψ(r) are operators, creating and annihilating an
electron (with some amplitude) at position r ???
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Second quantization
ESQC 2015
22 / 50
Field operators
In order to represent electrons (fermions) field operators must obey
the following anti-commutation relations
h
i
ψ̂ † (r), ψ̂ † (r0 )
= ψ̂ † (r)ψ̂ † (r0 ) + ψ̂ † (r0 )ψ̂ † (r) =
0
+
h
h
i
ψ̂(r), ψ̂(r0 )
+
i
ψ̂(r), ψ̂ † (r0 )
+
=
ψ̂(r)ψ̂(r0 ) + ψ̂(r0 )ψ̂(r)
=
ψ̂(r)ψ̂ † (r0 ) + ψ̂ † (r0 )ψ̂(r)
=
0
= δ(r − r0 )
Bosons obey corresponding commutator relations.
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Second quantization
ESQC 2015
23 / 50
Expansion of field operators
Suppose that we have some orthonormal orbital basis {ϕp (r)}M
p=1
ˆ
ϕ†p (r)ϕq (r)d3 r = hϕp |ϕq i = Spq = δpq
We now expand the field operators in this basis
X
X
ϕq (r)aq ; ψ̂ † (r) =
ϕ†q (r)aq†
ψ̂(r) =
q
q
We find the expansion coefficients ap and ap† by
ˆ
ˆ
†
3
†
ap = ϕp (r)ψ̂(r)d r; ap = ψ̂ † (r)ϕp (r)d3 r
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Second quantization
ESQC 2015
24 / 50
Expansion of field operators
This is perhaps easier seen using bracket notation, for instance
E X
X
ψ̂(r) =
ϕq (r)ap → ψ̂ =
|ϕq i ap
q
q
... such that
D
E X
X
ϕp |ψ̂ =
hϕp |ϕq i aq =
δpq aq = ap
q
q
The expansion coefficients âp and âp† are operators as well:
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âp is denoted an annihilation operator
âp† is denoted a creation operator and is the conjugate of âp
†
which also means that âp† = âp
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Second quantization
ESQC 2015
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Annihilation and creation operators
The algebra of the annihilation and creation operators follows from
the algebra of the field operators. We have
h
i
ψ̂(r), ψ̂ † (r0 ) = δ(r − r0 )
+
.. from which we deduce
ˆ
ˆ
i
h
†
3
†
† 0
0 3 0
ϕp (r)ψ̂(r)d r, ψ̂ (r )ϕq (r )d r
âp , âq =
+
+
Remembering that the integral signs are like summation signs we
obtain
ˆ ˆ
h
i
h
i
âp , âq†
=
ϕ†p (r)ϕq (r0 ) ψ̂(r), ψ̂ † (r0 ) d3 rd3 r0
+
+
ˆ ˆ
=
ϕ†p (r)ϕq (r0 )δ(r − r0 )d3 rd3 r0
ˆ
ϕ†p (r)ϕq (r)d3 r = δpq
=
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Second quantization
ESQC 2015
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Algebra of annihilation and creation operators
We just found that (using an orthonormal basis)
h
i
h
i
ψ̂(r), ψ̂ † (r0 ) = δ(r − r0 ) ⇒
âp , âq†
= δpq
+
+
In a similar manner we find that
h
i
i
h
ψ̂ † (r), ψ̂ † (r0 )
= 0 ⇒ âp† , âq†
+
h
i
ψ̂(r), ψ̂(r0 )
Trond Saue (LCPQ, Toulouse)
+
+
= 0 ⇒
Second quantization
[âp , âq ]+
= 0
= 0
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Occupation-number vectors
Let us consider a simple example:
We have 4 orbitals {ϕ1 , ϕ2 , ϕ3 , ϕ4 } (M=4).
4
With two electrons (N=2) we can build
= 6 determinants.
2
One example is
1 ϕ1 (1) ϕ3 (1) Φ (1, 2) = √ 2! ϕ1 (2) ϕ3 (2) or, in short-hand notation
Φ (1, 2) = |ϕ1 ϕ3 |
We can map this into an occupation-number vector
Φk (1, 2) = |ϕ1 ϕ3 |
→
|ki = |k1 , k2 , k3 , k4 i = |1, 0, 1, 0i
... where occupation numbers kp are either 0 or 1,
since electrons are fermions.
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Second quantization
ESQC 2015
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Occupation-number vectors
Vacuum state and annihilation
Another example is
1
Φm (1, 2, 3) = √
3!
ϕ1 (1)
ϕ1 (2)
ϕ1 (3)
ϕ2 (1)
ϕ2 (2)
ϕ2 (3)
ϕ4 (1)
ϕ4 (2)
ϕ4 (3)
= |ϕ1 ϕ2 ϕ4 |
→
|mi = |1, 0, 1, 1i
A special occupation-number vector is the vacuum state
|vaci = |0, 0, 0, 0i
Annihilation operators reduce occupation numbers by one and
therefore all give zero when acting on |vaci
âp |vaci = 0;
∀âp
This even serves as a definition of the vacuum state.
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Second quantization
ESQC 2015
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Occupation-number vectors
Creation operators
Creation operators increase occupation numbers by one
â1† |vaci = |1, 0, 0, 0i
â2† |vaci = |0, 1, 0, 0i
â3† |vaci = |0, 0, 1, 0i
â4† |vaci = |0, 0, 0, 1i
..but, since they refer to fermions,
occupation numbers can not be greater than one
â1† |1, 0, 0, 0i = â1† â1† |vaci = 0
This follows directly from the special case
h
i
âp† , âp†
= âp† âp† + âp† âp† = 2âp† âp† = 0
+
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Second quantization
ESQC 2015
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Occupation-number vectors
More creation
We can build ONVs corresponding to N = 2
â1† |0, 1, 0, 0i = |1, 1, 0, 0i = â1† â2† |vaci
Using the algebra of creation operators we find
â2† |1, 0, 0, 0i = â2† â1† |vaci = −â1† â2† |vaci = − |1, 1, 0, 0i
..showing how the fermion antisymmetry is built into the operators.
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Second quantization
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Fock space
..or occupation-number space
Occupation number vectors (ONVs) have the general form
|ki = |k1 , k2 , . . . , kM i ;
kp = 0 or 1
and reside in a Fock space of dimension 2M .
Any ONV can be generated from the vacuum state
|ki =
M
Y
âp†
kp
|vaci
p=1
An inner-product in Fock space is defined by
hk|mi = δk,m =
M
Y
δkp ,mp
p=1
and is one if all occupation numbers are identical, zero otherwise.
A special case
hvac|vaci = 1
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Second quantization
ESQC 2015
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Fock space
..or occupation-number space
The dual vector hk| is obtained by conjugation, e.g. starting from
|ki = |1, 0, 1, 1i = â1† â3† â4† |vaci
... we have
†
hk| = h1, 0, 1, 1| = hvac| â1† â3† â4† = hvac| â4 â3 â1
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notice the change of operator order under conjugation
annihilation operators become creators when operating to the left
the dual vacuum state can therefore be defined by
hvac| âp† = 0;
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Second quantization
∀âp†
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The number operator
Notice that in Fock space there is no restriction on particle number
N, except N ≤ M.
For a one-electron system, we saw in first quantization that
ˆ
ψ † (r)ψ(r)d3 r = 1
With field operators expanded in some orthonormal orbital basis
{ϕp (r)}M
p=1 we obtain
ˆ
ψ̂ † (r)ψ̂(r)d3 r =
X ˆ
X
X †
ϕ†p (r)ϕq (r)d3 r âp† âq =
δpq âp† âq =
âp âp
pq
pq
p
... which defines the number operator N̂. For instance
N̂ |1, 0, 1, 1i = 3 |1, 0, 1, 1i
The occupation number vectors are eigenvectors of the number
operator.
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Second quantization
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Counting electrons
The field operators do not relate to specific electrons; rather, they
sample contributions to the electron quantum field in space
Quantum field theory explains why electrons are the same everywhere;
they all belong to the same field !
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Second quantization
ESQC 2015
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The number operator
and commutator algebra
Suppose that
N̂ |ki = N |ki
What about N̂ap† |ki ?
We can rewrite this as
h
i
N̂âp† |ki = âp† N̂ + N̂, âp† |ki
We need to solve the commutator
i
h
i Xh
N̂, âp† =
âq† âq , âp†
q
We may use a commutator rule such as
h
i
h
i h
i
ÂB̂, Ĉ = ÂB̂Ĉ − ĈÂB̂ = ÂB̂Ĉ−ÂĈB̂ + ÂĈB̂ − ĈÂB̂ = Â B̂, Ĉ + Â, Ĉ B̂
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Second quantization
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The number operator
and commutator algebra
The algebra of creation and annihilation operators is, however,
expressed in terms of anti-commutators
† †
âp , âq + = 0;
[âp , âq ]+ = 0;
âp , âq† + = δpq
We therefore rather form the rule
h
i
h
i h
i
ÂB̂, Ĉ = ÂB̂Ĉ−ĈÂB̂ = ÂB̂Ĉ+ÂĈB̂ − ÂĈB̂−ĈÂB̂ = Â B̂, Ĉ − Â, Ĉ B̂
+
+
... which gives

h

i X

X † âq âq , âp† − âq† , âp† âq  = âp†
N̂, âp† =
âq† âq , âp† =


+
+
| {z } | {z }
q
q
=δpq
=0
Our final result is thereby
h
i
N̂âp† |ki = âp† N̂ + N̂, âp† |ki = âp† N̂ + 1 |ki = (N + 1) âp† |ki
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Second quantization
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The number operator
and commutator algebra
What about N̂ap |ki ?
We can write this as
h
i
N̂âp |ki = âp N̂ + N̂, âp |ki
We can proceed as before, but instead we note that
I
I
h
i† †
h † †i
† †
† †
Â, B̂ = ÂB̂ − B̂Â = B̂ Â − Â B̂ = − Â , B̂
X
†
N̂ † =
âp† âp = N̂ (hermitian operator)
p
... so that
h
i† †
N̂âp |ki = âp N̂ − N̂, âp
|ki = âp N̂ − 1 |ki = (N − 1) âp |ki
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Second quantization
ESQC 2015
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Counting electron pairs
Let us have a look at the second quantization operator
N̂pair
=
=
=
=
ˆ
1
ψ̂ † (1)ψ̂ † (2)ψ̂(2)ψ̂(1)d1d2
2
ˆ
1X
ϕ†p (1)ϕ†q (2)ϕr (2)ϕs (1)d1d2 âp† âq† âr âs
2 pqrs
ˆ
ˆ
1X
ϕ†p (1)ϕs (1)d1 ϕ†q (2)ϕr (2)d2 âp† âq† âr âs
2 pqrs
1X
1X † †
{δps δqr } âp† âq† âr âs =
âp âq âq âp
2 pqrs
2 pq
Operator algebra
âp† âq† âq âp = −âp† âq† âp âq = −âp† δpq − âp âq† âq = âp† âp âq† âq − δpq âp† âq
...shows that it counts electron pairs
N̂pair =
Trond Saue (LCPQ, Toulouse)
1 N̂ N̂ + 1
2
Second quantization
ESQC 2015
39 / 50
What does the second-quantized electronic
Hamiltonian look like ?
The first-quantized form
Ĥ =
N
X
i=1
N
1X
ĥ(i) +
ĝ (i, j) + Vnn
2
i6=j
The second-quantized form
ˆ
Ĥ =
ψ̂ † (1)h(1)ψ̂(1)d1 +
1
2
ˆ
ψ̂ † (1)ψ̂ † (2)ĝ (1, 2)ψ̂(2)ψ̂(1)d1d2 + Vnn
(notice the order of electron coordinates in the two-electron operator)
This gives a formula for finding the second-quantized form of any
one- and two-electron operator.
Trond Saue (LCPQ, Toulouse)
Second quantization
ESQC 2015
40 / 50
What does the second-quantized electronic
Hamiltonian look like ?
The one-electron part
ˆ
Ĥ1
ψ̂ † (1)h(1)ψ̂(1)d1
X X ˆ
X
†
=
ϕp (1)h(1)ϕq (1)d1 âp† âq =
hpq âp† âq
=
p
q
p,q
Proceeding in the same way with the two-electron part we obtain
Ĥ2
=
=
ˆ
1
ψ̂ † (1)ψ̂ † (2)ĝ (1, 2)ψ̂(2)ψ̂(1)d1d2
2
1X
Vpq,rs âp† âq† âs âr
2 pq,rs
ˆ
ϕ†p (1)ϕ†q (2)ĝ (1, 2)ϕr (1)ϕs (2)d1d2
ˆ
= (ϕp ϕq |ϕr ϕs ) = ϕ†p (1)ϕ†r (2)ĝ (1, 2)ϕq (1)ϕs (2)d1d2
Dirac notation: Vpq,rs = hϕp ϕq |ϕr ϕs i =
Mulliken notation:gpq,rs
Trond Saue (LCPQ, Toulouse)
Second quantization
ESQC 2015
41 / 50
What does the second-quantized electronic
Hamiltonian look like ?
The final form is
Ĥ =
X
pq
hpq âp† âq +
1X
Vpq,rs âp† âq† âs âr + Vnn
2 pq,rs
This is a very convenient operator form:
I
I
I
The fermion antisymmetry is built into the operator
The operator is expressed in terms of one- and two-electron integrals,
which are the basic ingredients of quantum chemistry codes
The form is universal; there is no reference to the number of electrons !
..but note that it is a projected operator:
it “lives” in the space defined by the orbital set {ϕp }M
p=1 .
Trond Saue (LCPQ, Toulouse)
Second quantization
ESQC 2015
42 / 50
The electronic energy in second quantization
The electronic Hamiltonian is
Ĥ =
X
pq
hpq âp† âq +
1X
Vpq,rs âp† âq† âs âr + Vnn
2 pq,rs
The wave function is now expressed as a linear combination of
occupation number vectors (limited to occupation N)
|0i =
X
Ck |ki
k
The energy is given as the expectation value
D E X
1X
hpq Dpq +
E = 0 Ĥ 0 =
Vpq,rs dpq,rs + Vnn
2 pq,rs
pq
I
I
Matrix elements hpq and Vpq,rs depends on the operator, but are
independent of wave function
Orbital density matrices Dpq and dpq,rs are independent of operator,
but depend on wave function
Trond Saue (LCPQ, Toulouse)
Second quantization
ESQC 2015
43 / 50
Orbital density matrices
One-electron orbital density matrix
E
D Dpq = 0 âp† âq 0
I
I
I
dimension: M 2
contains all information needed to calculate expectation values of
one-electron operators
diagonalization gives natural orbitals
Two-electron orbital density matrix
E
D dpq,rs = 0 âp† âq† âs âr 0
dimension: M 4
contains all information needed to calculate expectation values of
two-electron operators
M
Data reduction: C :
→ D/d : M 2 /M 4 !!!
N
I
I
Trond Saue (LCPQ, Toulouse)
Second quantization
ESQC 2015
44 / 50
What about spin ?
By convention, the z-axis is chosen as spin-axis such that the electron
spin functions |s, ms i are eigenfunctions of ŝ 2 and ŝz
ŝ 2 |s, ms i = s (s + 1) |s, ms i ;
ŝz |s, ms i = ms |s, ms i
It is also convenient to introduce step operators
ŝ+ = ŝx + iŝy and ŝ− = ŝx − iŝy
p
ŝ± |s, ms i = s (s + 1) − ms (ms ± 1) |s, ms ± 1i
1 1
1
,
particles
with
spin
functions
denoted
|αi
=
Electrons are
spin2 2
1 12
and |βi = 2 , − 2 . The action of the spin operators is summarized by
|αi
|βi
Trond Saue (LCPQ, Toulouse)
3
4
3
4
ŝ 2
|αi
|βi
ŝz
|αi
1
− 2 |βi
1
2
Second quantization
ŝ+
0
|αi
ŝ−
|βi
0
ESQC 2015
45 / 50
Spin in second quantization
We may separate out spin from spatial parts of the creationand annihilation operators, giving
h
i
†
†
âpσ
, âqσ
= 0;
0
+
[âpσ , âqσ0 ]+ = 0;
h
†
âpσ , âqσ
0
i
+
= δpq δσσ0 ;
σ, σ 0 = α or β
We may also separate out spin in the electronic Hamiltonian.
For the (non-relativistic) one-electron part we obtain
Ĥ1
=
XXD
=
XXD
=
XXD
=
XD
pq σ,σ 0
pq σ,σ 0
pq σ,σ 0
E
†
âqσ0
ϕp σ ĥ ϕq σ 0 âpσ
E
†
ϕp ĥ ϕq σ|σ 0 âpσ
âqσ0
E
†
âqσ0
ϕp ĥ ϕq δσσ0 âpσ
E
ϕp ĥ ϕq Epq ;
pq
Trond Saue (LCPQ, Toulouse)
Epq =
X
†
âqσ0
âpσ
σ
Second quantization
ESQC 2015
46 / 50
Spin in second quantization
For the (non-relativistic) two-electron part we obtain
Ĥ2
=
† †
1X X ϕp σϕq τ |ĝ | ϕr σ 0 ϕs τ 0 apσ
aqτ asτ 0 ar σ0
2 pqrs
0 0
=
† †
1X X
hϕp ϕq |ĝ | ϕr ϕs i σ|σ 0 τ |τ 0 apσ
aqτ asτ 0 ar σ0
2 pqrs
0 0
=
1X X
†
†
hϕp ϕq |ĝ | ϕr ϕs i δσσ0 δτ τ 0 apσ
aqτ
asτ 0 ar σ0
2 pqrs
0 0
=
1X X
hϕp ϕq |ĝ | ϕr ϕs i epq,rs ;
2 pqrs
0 0
στ σ τ
στ σ τ
στ σ τ
epq,rs =
στ σ τ
X
†
†
apσ
aqτ
asτ ar σ
στ
Operator algebra
†
†
†
†
†
†
†
apσ
aqτ
asτ 0 ar σ0 = −apσ
aqτ
ar σ asτ = apσ
ar σ aqτ
asτ − δqr δστ apσ
asτ
.. shows that
epq,rs = Epr Eqs − δqr Eps
Trond Saue (LCPQ, Toulouse)
Second quantization
ESQC 2015
47 / 50
Summary
Second quantization starts from field operators ψ † (1), ψ(1) sampling
the electron field in space. It provides a very convenient language for
the formulation and implementation of quantum chemical methods.
Occupation number vectors (ONVs) are defined with respect to
some (orthonormal) orbital set {ϕp (r)}M
p=1
Their occupation numbers are manipulated using creation- and
annihilation operators, âp† and âp ,
which are conjugates of each other.
The algebra of these operators is summarized
by anti-commutator relations
h
i
h
i
âp† , âq†
= 0; [âp , âq ]+ = 0;
âp , âq†
= δpq
+
+
and reflects the fermionic nature of electrons.
Trond Saue (LCPQ, Toulouse)
Second quantization
ESQC 2015
48 / 50
Summary
One-electron operators are translated
into their second quantized form by
N
X
ˆ
fˆ(i)
ψ † (1)f (1)ψ(1)d1 =
→
X D E †
ϕp fˆ ϕq âp âq
pq
i=1
Two-electron operators are translated
into their second quantized form by
N
1X
g (i, j)
2 i=1
1
2
→
=
ˆ
ψ̂ † (1)ψ̂ † (2)ĝ (1, 2)ψ̂(2)ψ̂(1)d1d2
1X
hϕp ϕq |ĝ | ϕr ϕs i âp† âq† âs âr
2 pq,rs
Nice features is that:
I
I
Antisymmetry is automatically built into the operators
They are expressed in terms of integrals,
building blocks of quantum chemistry codes
Trond Saue (LCPQ, Toulouse)
Second quantization
ESQC 2015
49 / 50
Summary
The second-quantized electronic Hamiltonian is expressed as
Ĥ =
X
hpq âp† âq +
pq
1X
Vpq,rs âp† âq† âs âr + Vnn
2 pq,rs
The electronic energy becomes
D E X
1X
E = 0 Ĥ 0 =
Vpq,rs dpq,rs + Vnn
hpq Dpq +
2 pq,rs
pq
which nicely separates
I
I
operator content,
in terms of integrals hpq and Vpq,rs , and
wave function content,
in terms of orbital density matrices Dpq and dpq,rs
(data compression)
Trond Saue (LCPQ, Toulouse)
Second quantization
ESQC 2015
50 / 50