Drake G.W.F. (editor) Handbook of Atomic, Molecular, and Optical Physics
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Perturbation Theory 5.3 Fermionic Many-Body Perturbation Theory (MBPT) 109
top and the bottom part by A and B, respectively, we
find for the overall contribution
1
B
·
1
A + B
·
1
B
+
1
A
·
1
A + B
·
1
B
=
1
B
+
1
A
1
(A + B)B
=
1
AB
2
. (5.67)
Thus, the contribution from these terms exactly cancels
that from the renormalization term WRW
WR
2
W
.
Generalizing (5.67), we obtain the factorization
lemma of Frantz and Mills [5.23], which implies the
cancellation of renormalization terms by the unlinked
terms originating from the principal term. This result
holds for the energy as well as for the wave function
contributions in every order of PT, as ascertained by the
linked cluster theorem, which states that
∆E = k =
∞
n=0
Φ|W|Ψ
(n)
=
∞
n=0
W(RW )
n
L
,
(5.68)
|Ψ =
∞
n=0
|Ψ
(n)
=
∞
n=0
(RW )
n
|Φ
L
, (5.69)
where the subscript L indicates that only linked dia-
grams (or terms) are to be considered. This enables
us to obtain general, explicit expressions for the nth-
order PT contributions by first constructing all possible
linked diagrams involving n vertices and by con-
verting them into the explicit algebraic expressions
using the rules of Sect. 5.3.5. Note that linked en-
ergy diagrams are always connected, but the linked
wave function diagrams are either connected or dis-
connected, each disconnected component possessing at
least one pair of particle–hole free lines extending to the
left.
To reveal a deeper structure of the result (5.69), de-
fine the cluster operator T that generates all connected
wave function diagrams,
T |Φ=
∞
n=1
(RW )
n
|Φ
C
, (5.70)
the subscript C indicating that only contributions from
connected diagrams are to be included. Since the
general component with r disconnected parts can be
shown to be represented by the term (r!)
−1
T
r
|Φ,the
general structure of the e xact wave function |Ψ is
givenbytheconnected cluster theorem, which states
that
|Ψ =e
T
|Φ . (5.71)
In other words, the wave operator W which trans-
forms the unperturbed independent particle model wave
function |Φ into the exact one according to
|Ψ =W|Φ ,
(5.72)
is given by the exponential of the cluster operator T ,
W = e
T
, (5.73)
which in turn is given by the connected wave function
diagrams. This is in fact the basis of the coupled cluster
methods [5.15,24–28] (Sect. 5.3.8).
The contributions to T may be further classified by
their excitation rank i,
T =
N
i=1
T
i
, (5.74)
where T
i
designates connected diagrams with i pairs
of free particle–hole lines, producing i-times excited
components of |Ψ when acting on |Φ.
5.3.8 Coupled Cluster Theory
Summing all HF diagrams (Sect. 5.3.6) is equivalent to
solving the HF equations. Depending on the average
electron density of the system, it may be essential to
sum certain types of PT diagrams to infinite order at
the post-HF level. A frequently used approach that is
capable of recovering a large part of the electronic cor-
relation energy is based on the connected cluster theorem
(Sect. 5.3.7), referred to in this context as the exponential
cluster Ansatz for the wave operator. Using this Ansatz,
one derives a system of energy-independent nonlinear
coupled cluster (CC) equations [5.15, 26–28] determin-
ing the cluster amplitudes of T .TheseCC equations
can be regarded as recurrence relations generating the
MBPT series [5.15], so that by solving these equations
one in fact implicitly generates all the MBPT diagrams
and sums them to infinite order. Since the solution of
the full CC equations is equivalent to the exact solution
of the Schrödinger equation, we must – in all practical
applications – introduce a suitable truncation scheme,
which implies that only diagrams of certain types are
summed.
Generally, using the cluster expansion (5.71)inthe
N-product form of the Schrödinger equation,
H
N
|Ψ ≡(H −H)|Ψ =∆E|Ψ ,
∆E = E − E
0
, (5.75)
Part A 5.3
110 Part A Mathematical Methods
premultiplying with the inverse of the wave operator,
and using the Hausdorff formula (5.11) yields
e
−T
H
N
e
T
|Φ=
∞
n=0
[ad(−T )]
n
H
N
n!
= ∆E|Φ .
(5.76)
In fact, this expansion terminates, so that using (5.74)
and projecting onto |Φ we obtain the energy expression
∆E =H
N
T
2
+
1
2
H
N
T
2
1
,
(5.77)
while the projection onto the manifold of excited states
{|Φ
i
} relative to |Φ≡|Φ
0
gives the system of CC
equations
Φ
i
|H
N
+[H
N
, T ]+
1
2
[[H
N
, T ], T ]+···|Φ=0 .
(5.78)
Approximating, e.g., T by the most important pair clus-
ter component T ≈ T
2
gives the so-called CCD (coupled
clusters with doubles) approximation
Φ
(2)
i
H
N
+[H
N
, T
2
]+
1
2
[[H
N
, T
2
], T
2
]|Φ=0 ,
(5.79)
the superscript (2) indicating pair excitations relative
to |Φ.
Equivalently, (5.77)and(5.78) can be written in the
form
∆E =
H
N
e
T
C
, (5.80)
Φ
i
H
N
e
T
C
Φ
= 0 ,
(5.81)
the subscript C again indicating that only connected di-
agrams are to be considered. The general form of CC
equations is
a
i
+
j
b
ij
t
j
+
j≤k
c
ijk
t
j
t
k
+···=0 , (5.82)
where a
i
=Φ
i
|H
N
|Φ
0
, b
ij
=Φ
i
|H
N
|Φ
j
C
, c
ijk
=
Φ
i
|H
N
|Φ
j
⊗ Φ
k
C
, etc. Writing the diagonal linear
term b
ii
in the form
b
ii
= ∆
i
+ b
ii
, (5.83)
this system can be solved iteratively by rewriting it in
the form
t
(n+1)
i
= ∆
−1
i
a
i
+ b
ii
t
(n)
i
+
j
b
ij
t
(n)
j
+
j≤k
c
ijk
t
(n)
j
t
(n)
k
+···
. (5.84)
Starting with the zeroth approximation t
(0)
i
= 0, the first
iteration is
t
(1)
i
= ∆
−1
i
a
i
, (5.85)
which yields the second-order PT energy when used
in (5.77). Clearly, the successive iterations generate
higher and higher orders of the PT. At any truncation
level, a size extensive result is obtained.
The CC methods belong to the most accurate
and often used tools in computations of molecu-
lar electronic structure and several general-purpose
codes are available for this purpose (for reviews
see [5.29–32]). The standard approach truncates the
cluster operator (5.74) at the singly (S) and doubly
(D) excited level (the CCSD method [5.33]) and
is often supplemented by a perturbative account
of the triply-excited (T) cluster components [the
CCSD(T) method] for greater accuracy [5.34]. To
avoid the breakdown of the latter method in quasi-
degenerate situations, one can employ one of the
renormalized versions of CCSD(T) [5.35]. The CC
ansatz (5.71) has also been exploited in the context
of the equation-of-motion (EOM)andthelinear-
response formalisms, enabling the computation of the
excitation energies and of properties other than the en-
ergy (dipole and quadrupole moments, polarizabilities,
etc., [5.29–32].
At this stage it is important to recall that the
above described MBPT and CC approaches pertain
to nondegenerate, lowest-lying closed-shell states of
a given symmetry species. Although the CC meth-
ods are often used even for open-shell states by
relying on the unrestricted HF (UHF) reference [of
the different-orbitals-for-different-spins (DODS) type],
a proper description of such states requires a multi-
reference (MR) generalization based on the effective
Hamiltonian formalism [5.6, 31, 32, 36–38]. Unfor-
tunately, such a generalization is not unambiguous.
The two existing formulations, the so-called valence
universal [5.6, 37]andstate universal [5.38]meth-
ods, are computationally demanding and often plagued
with the intruder state and other problems [5.15, 36].
For these reasons, no general-purpose codes have
yet been developed and very few actual applica-
tions have been carried out [5.31, 32] (see, however,
the recently formulated SU CC approach for gen-
eral model spaces [5.39, 40]). Nonetheless, the MR
CC formalism proved to be very useful in the for-
mulation of the so-called state selective or state
specific approaches (e.g., the reduced MR CCSD
method [5.41–46]).
Most recently, the CC approach has been used to
handle bosonic-type problems of the vibrational struc-
ture in molecular spectra and, generally, multimode
dynamics [5.47].
Part A 5.3
Perturbation Theory 5.4 Time-Dependent Perturbation Theory 111
5.4 Time-Dependent Perturbation Theory
5.4.1 Evolution Operator PT Expansion
By introducing the evolution operator U(t, t
0
)
|Ψ(t)=U(t, t
0
)|Ψ(t
0
) , (5.86)
time-dependent Schrödinger equation
i
∂
∂t
|Ψ(t)=H|Ψ(t)
(5.87)
becomes
i
∂
∂t
U(t, t
0
) = HU(t, t
0
). (5.88)
Clearly,
U(t
0
, t
0
) = 1 ,
U(t, t
0
) = U(t, t
)U(t
, t
0
),
U(t, t
0
)
−1
= U(t
0
, t) = U
†
(t, t
0
). (5.89)
If the Hamiltonian is time independent then
U(t, t
0
) = exp
−
i
H(t − t
0
)
. (5.90)
In the interaction picture (subscript I)
|Ψ(t)
I
= exp
i
H
0
t
|Ψ(t) , (5.91)
where now
H = H
0
+ V , (5.92)
the Schrödinger equation becomes
i
∂
∂t
|Ψ(t)
I
= V(t)
I
|Ψ(t)
I
, (5.93)
known as Tomonaga-Schwinger equation [5.48]. Analo-
gously, the evolution operator in this picture (we drop
the subscript I from now on) satisfies
i
∂
∂t
U(t, t
0
) = V(t)U(t, t
0
), (5.94)
with the initial condition U(t
0
, t
0
) = 1. This differential
equation is equivalent to an integral equation
U(t, t
0
) = 1 −
i
t
t
0
V(t
1
)U(t
1
, t
0
)dt
1
. (5.95)
Iterating we get [5.49,50]
U(t, t
0
)
=
∞
n=0
−
i
n
×
t
t
0
dt
1
t
1
t
0
dt
2
···
t
n−1
t
0
dt
n
V(t
1
)V(t
2
) ···V(t
n
)
=
∞
n=0
(−i/ )
n
n!
×
t
t
0
dt
1
···
t
t
0
dt
n
T [V(t
1
) ···V(t
n
)] , (5.96)
where T [···] designates the time-ordering or chrono-
logical operator.
5.4.2 Gell–Mann and Low Formula
For a time-independent perturbation, one introduces the
so-called adiabatic switching by writing
H
α
(t) = H
0
+ λe
−α|t|
V,α>0 (5.97)
so that H
α
(t →±∞) = H
0
and H
α
(t → 0) = H =
H
0
+ λV .Then
|Ψ(t)
I
= U
α
(t, −∞|λ)|Φ
0
, (5.98)
with U
α
(t, −∞|λ) obtained with V
α
(t) = λe
−α|t|
V (all
in the interaction picture). The desired energy is then
given by the Gell–Mann and Low formula [5.51]
∆E = lim
α→0+
i αλ
∂
∂λ
lnΦ
0
|U
α
(0, −∞|λ)|Φ
0
,
(5.99a)
or
∆E =
1
2
lim
α→0+
i αλ
∂
∂λ
lnΦ
0
|U
α
(∞, −∞|λ)|Φ
0
,
(5.99b)
which result from the asymmetric energy formula (5.27).
One can similarly obtain the perturbation expansion for
the one- or two-particle Green functions, e.g.,
G
µν
t, t
= lim
α→0+
T
a
µ
(t)a
†
ν
(t
)U
α
(∞, −∞|λ)
U
α
(∞, −∞|λ)
,
(5.100)
Part A 5.4
112 Part A Mathematical Methods
with the operators in the interaction representation and
the expectation values in the noninteracting ground
state |Φ
0
. Analogous expressions result for G(rt, r
t
),
etc., when the creation and annihilation operators are
replaced by the corresponding field operators.
5.4.3 Potential Scattering and Quantum
Dynamics
The Schrödinger equation for a free particle of energy
E =
2
k
2
/2m, moving in the potential V(r),
∇
2
+ k
2
ψ(k, r) = v(r)ψ(k, r),
v(r) =
2m/
2
V(r),
(5.101)
has the formal solution
ψ(k, r) = Φ(k, r) +
G
0
k, r, r
v
r
ψ
k, r
dr
,
(5.102)
where Φ(k, r) is a solution of the homogeneous equation
[v(r) ≡ 0] and G
0
(k, r, r
) is a classical Green function
∇
2
+ k
2
G
0
k, r, r
= δ
r −r
.
(5.103)
For an in-going plane wave Φ(k, r) ≡ Φ
k
i
(r) =
(2π)
3/2
exp(ik
i
· r) with the initial wave vector k
i
and appropriate asymptotic boundary conditions
(outgoing spherical wave with positive phase vel-
ocity), when G
0
(k, r, r
) ≡ G
(+)
0
(|r− r
|) =−(4π|r −
r
|)
−1
e
ik|r−r
|
,(5.102)isreferredtoastheLippmann–
Schwinger equation [5.52]. It can be equivalently
transformed into the integral equation for the Green
function
G
(+)
r, r
= G
(+)
0
r, r
+
G
(+)
0
r, r
v
r
× G
(+)
r
, r
dr
, (5.104)
representing a special case of the Dyson equation.
In the time-dependent case, considering the scatter-
ing of a spinless massive particle by a time-dependent
potential V(r, t), we get similarly
ψ(r, t) =
Φ(r, t)+
G
0
r, r
; t, t
V
r
, t
ψ
r
, t
dr
dt
,
(5.105)
where the zero-order time-dependent Green function
now satisfies the equation
i
∂
∂t
− H
0
G
0
r, r
; t, t
= δ
r −r
δ
t − t
.
(5.106)
Again, for causal propagation one chooses the time-
retarded or causal Green function or propagator
G
(+)
0
(r, r
; t, t
).
5.4.4 Born Series
Iteration of (5.105)givestheBorn sequence
ψ
0
(r, t) = Φ(r, t),
ψ
1
(r, t) = Φ(r, t) +
G
(+)
0
r, r
; t, t
× V
r
, t
Φ
r
, t
dr
dt
, (5.107a)
ψ
2
(r, t) = Φ(r, t) +
G
(+)
0
r, r
; t, t
× V
r
, t
ψ
1
r
, t
dr
dt
, (5.107b)
and, generally
ψ
n
(r, t) = Φ(r, t) +
G
(+)
0
r, r
; t, t
× V
r
, t
ψ
n−1
r
, t
dr
dt
. (5.108)
Summing individual contributions gives the Born series
for ψ(r, t) ≡ ψ
(+)
(r, t),
ψ(r, t) =
∞
n=0
χ
n
(r, t), (5.109)
where
χ
0
(r, t) = Φ(r, t),
χ
n
(r, t) =
G
n
r, r
; t, t
Φ
r
, t
dr
dt
,
(5.110)
with
G
n
r, r
; t, t
=
G
1
r, r
; t, t
× G
n−1
r
, r
; t
, t
dr
dt
,(n > 1)
G
1
r, r
; t, t
= G
(+)
0
r, r
; t, t
V
r
, t
.
(5.111)
In a similar way we obtain the Born series for the
scattering amplitudes or transition matrix elements.
5.4.5 Variation of Constants Method
An alternative way of formulating the time-dependent
PT is the method of variation of the constants [5.53,54].
Start again with the time-dependent Schrödinger equa-
tion (5.87) with H = H
0
+ V , and assume that H
0
is
time-independent, while V is a time-dependent pertur-
bation. Designating the eigenvalues and eigenstates of
Part A 5.4
Perturbation Theory References 113
H
0
by ε
i
and |Φ
i
, respectively [cf. (5.48)], the general
solution of the unperturbed time-dependent Schrödinger
equation
i
∂
∂t
|Ψ
0
=H
0
|Ψ
0
(5.112)
has the form
|Ψ
0
=
j
c
j
|Φ
j
exp
−
i
ε
j
t
, (5.113)
with c
j
representing arbitrary constants, and the sum
indicating both the summation over the discrete part and
the integration over the continuum part of the spectrum
of H
0
.
In the spirit of the general variation of constants
procedure, write the unknown perturbed wave function
|Ψ(t),(5.87), in the form
|Ψ(t)=
j
C
j
(t)|Φ
j
exp
−
i
ε
j
t
, (5.114)
where the C
j
(t) are now functions of time. Substitut-
ing this Ansatz into the time-dependent Schrödinger
equation (5.87)gives
˙
C
j
(t) = (i )
−1
k
C
k
(t)V
jk
exp[(i/ )∆
jk
t] ,
(5.115)
where
∆
jk
= ε
j
− ε
k
, V
jk
=Φ
j
|V |Φ
k
. (5.116)
Introducing again the ‘small’ parameter λ by writing
the Hamiltonian H in the form
H = H
0
+ λV(t), (5.117)
and expanding the ‘coefficients’ C
j
(t) in powers of λ,
C
j
≡ C
j
(t) =
∞
k=0
C
(k)
j
(t)λ
k
, (5.118)
gives the system of first order differential equations
˙
C
(n+1)
j
= (i )
−1
k
C
(n)
k
V
jk
exp
(i/ )∆
jk
t
,
n = 0, 1, 2,... ,
(5.119)
with the initial condition
˙
C
(0)
j
= 0, which implies that
C
(0)
j
are time independent, so that C
(0)
j
= c
j
, obtain-
ing (5.113) in the zeroth order. The system (5.119) can
be integrated to any prescribed order. For example, if the
system is initially in a stationary state |Φ
i
,thenset
C
(0)
j
=
δ
ji
for discrete states ,
δ( j − i) for continuous states ,
(5.120)
so that
C
(1)
j(i)
(t) = (i )
−1
t
−∞
V
ji
exp
(i/ )∆
ji
t
dt
,
(5.121)
assuming C
(1)
j(i)
(−∞) = 0. Clearly, |C
(1)
j(i)
(t)|
2
gives the
first order transition probability for the transition from
the initial state |Φ
i
to a particular state |Φ
j
.These
in turn will yield the first order differential cross sec-
tions [5.55].
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ˇ
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ˇ
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zek: J. Chem. Phys. 45, 4256 (1966)
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ˇ
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ˇ
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Part A 5
115
Second Quanti
6. Second Quantization
In second quantization, the characteristic
properties of eigenfunctions are transferred
to operators. This approach has the advan-
tage of treating the atomic shell as the basic
unit, as opposed to the electron configura-
tion. The creation and annihilation operators
allow one to move from configuration to con-
figuration, exposing an intrinsic shell structure.
The introduction of coefficients of fractional
parentage (cfp) then allows the calculation
of the matrix elements of an operator in one
configuration to be expressed in terms of
those of the same operator in another con-
figuration; hence the matrix elements of an
operator in all configurations may be de-
termined from the knowledge of its matrix
elements in but one. This can be viewed as
an extension of the usual Wigner-Eckart the-
orem. The basic concepts of quasispin and
quasiparticle are also introduced within this
context.
6.1 Basic Properties................................... 115
6.1.1 Definitions ............................... 115
6.1.2 Representation of States ............ 115
6.1.3 Representation of Operators ....... 116
6.2 Tensors............................................... 116
6.2.1 Construction ............................. 116
6.2.2 Coupled Forms .......................... 116
6.2.3 Coefficients of Fractional
Parentage ................................ 117
6.3 Quasispin............................................ 117
6.3.1 Fermions.................................. 117
6.3.2 Bosons..................................... 118
6.3.3 Triple Tensors ........................... 118
6.3.4 Conjugation.............................. 118
6.3.5 Dependence on Electron Number 119
6.3.6 The Half-filled Shell .................. 119
6.4 Complementarity ................................. 119
6.4.1 Spin–Quasispin Interchange ....... 119
6.4.2 Matrix Elements ........................ 119
6.5 Quasiparticles ..................................... 120
References .................................................. 121
6.1 Basic Properties
6.1.1 Definitions
The creation operator a
†
ξ
creates the quantum state ξ.
The annihilation (or destruction) operator a
η
annihilates
the quantum state η. The vacuum (or reference) state |0
satisfies the equation
a
η
|0=0 . (6.1)
Bosons satisfy the commutation relations
a
†
ξ
, a
†
η
= 0 ,
(6.2)
[a
ξ
, a
η
]=0 , (6.3)
a
ξ
, a
†
η
= δ(ξ, η) ,
(6.4)
where [A, B]≡ AB − BA. Fermions satisfy the anti-
commutation relations
a
†
ξ
, a
†
η
+
= 0 , (6.5)
[a
ξ
, a
η
]
+
= 0 , (6.6)
a
ξ
, a
†
η
+
= δ(ξ, η) , (6.7)
where [A , B]
+
≡ AB+ BA.
6.1.2 Representation of States
For an electron in an atom, characterized by the
quantum number quartet (n m
s
m
), the identification
ξ ≡ (n m
s
m
) for fermions is made. For normalized
Slater determinants {αβ...ν} characterized by the elec-
tron states α,β,... ,ν, the equivalences
a
†
α
a
†
β
...a
†
ν
|0≡{αβ...ν} , (6.8)
0|a
ν
...a
β
a
α
≡{αβ...ν}
∗
(6.9)
are valid, where the asterisk denotes the complex conju-
gate.
Part A 6
116 Part A Mathematical Methods
For a normalized boson state {···}in which the label
ξ appears N
ξ
times, the additional factor
[N
α
!N
β
! ...N
ν
!]
−
1
2
(6.10)
must be included on the left-hand sides of the equiva-
lences (6.8)and(6.9).
6.1.3 Representation of Operators
For an operator F, consisting of the sum of operators f
i
acting on the single electron i,
F ≡
ξ,η
a
†
ξ
ξ| f|ηa
η
. (6.11)
For an operator G, consisting of the sum of operators g
ij
acting on the pair of electrons i and j,
G ≡
1
2
ξ,η,ζ,λ
a
†
ξ
a
†
η
ξ
1
η
2
|g
12
|ζ
1
λ
2
a
λ
a
ζ
. (6.12)
For an N-particle system |Ψ ,
ξ
a
†
ξ
a
ξ
|Ψ =N|Ψ . (6.13)
The representations of single-particle and two-particle
operators for bosons are identical to those given above
for fermions [6.1].
6.2 Tensors
6.2.1 Construction
If the description ξ for a single fermion or boson state
includes an angular momentum quantum number t and
the corresponding magnetic quantum number m
t
,then
the 2t + 1 components of a creation operator a
†
σ
,where
σ ≡ (t, m
t
) and −t ≤ m
t
≤ t, satisfy the commutation
relations of Racah [6.2] for an irreducible spherical ten-
sor of rank t with respect to the total angular momentum
T,givenby
T =
a
†
ξ
ξ|t|ηa
η
. (6.14)
That is, with the phase conventions of Condon and
Shortley [6.3],
T
z
, a
†
σ
= m
t
a
†
σ
, (6.15)
T
x
± iT
y
, a
†
σ
=
[
t(t + 1) − m
t
(m
t
± 1)
]
1
2
a
†
τ
,
(6.16)
where τ ≡ (t, m
t
± 1).
A spherical tensor a constructed from annihilation
operators possesses the components
˜
a
σ
, which satisfy
˜
a
σ
= (−1)
p
a
ζ
, (6.17)
with p = t − m
t
and ζ ≡ (t, −m
t
).
The 4 + 2 components of the creation operator for
an electron in the atomic shell form a double tensor of
rank
1
2
with respect to the total spin S, and rank with
respect to the total angular momentum L.
6.2.2 Coupled Forms
Tensors formed from annihilation and creation operators
can be coupled by means of the usual rules of angular
momentum theory [6.4]. The double tensor defined for
electrons in the shell by
W
(κk)
=−
a
†
a
(κk)
, (6.18)
possesses a rank κ with respect to S, and rank k with
respect to L. Its reduced matrix element, defined here as
in (5.4.1) of Edmonds [6.4], for a single electron in both
the spin and orbital spaces, is given by
s || W
(κk)
|| s
=[(2κ + 1)(2k+ 1)]
1
2
. (6.19)
The connections to tensors whose matrix elements have
been tabulated [6.5, 6]are
W
(0k)
=[(2k+ 1)/2]
1
2
U
(k)
, (6.20)
W
(1k)
=[2(2k+ 1)]
1
2
V
(k1)
. (6.21)
For terms with common spin S,sayψ and ψ
,
ψ||W
(0k)
||ψ
=
[(2S+ 1)(2k + 1)/2]
1
2
ψ||U
(k)
||ψ
.
(6.22)
This result is obtained because the ranks assigned to the
tensors imply that W
(0k)
is to be reduced with respect
to both the spin S and the orbit L, while U
(k)
is to be
reduced only with respect to L.
Part A 6.2
Second Quantization 6.3 Quasispin 117
The following relations hold for electrons with az-
imuthal quantum numbers [6.7]:
S=[(2 + 1)/2]
1
2
W
(10)
, (6.23)
L =[2( + 1)(2 + 1)/3]
1
2
W
(01)
,
(6.24)
i
s
i
C
(2)
i
(1)
=−
( + 1)(2 + 1)
10(2 − 1)(2 + 3)
1
2
W
(12)1
,
(6.25)
i
(s
i
·
i
) =−[( + 1)(2 + 1)/2]W
(11)0
,
(6.26)
where the tensor C
k
of Racah [6.2] is related to the
spherical harmonics by
C
(k)
q
=[4π/(2k+ 1)]
1
2
Y
kq
, (6.27)
and where the tensors of the type W
(κk)K
indicate that
the spin and orbital ranks are coupled to a resultant K.
6.2.3 Coefficients of Fractional Parentage
Let ψ and
¯
ψ denote terms of
N
and
N−1
character-
ized by (S, L) and (
¯
S,
¯
L). The coefficients of fractional
parentage (cfp) (ψ{|
¯
ψ) of Racah [6.8] allow one to
calculate an antisymmetrized function ψ by vector-
coupling
¯
ψ to the spin and orbit of the Nth electron:
|ψ=
¯
ψ
¯
ψ,
2
, SL
¯
ψ|}ψ
, (6.28)
where the sum over
¯
ψ includes
¯
S,
¯
L, and any other
quantum numbers necessary to define the spectroscopic
terms of
N−1
. The cfp’s are given by
ψ||a
†
||
¯
ψ
= (−1)
N
[N(2S+ 1)(2L + 1)]
1
2
ψ{|
¯
ψ
,
(6.29)
¯
ψ||a||ψ
= (−1)
g
[N(2S+ 1)(2L + 1)]
1
2
¯
ψ|}ψ
,
(6.30)
where g = N +
¯
S+
¯
L − s− S− − L. A tabulation for
the p, d, and f shells has been given by Nielson and
Koster [6.5].
Two-electron cfp are given by
ψ||
a
†
a
†
(κk)
||
˜
ψ
=[N(N − 1)(2S+ 1)(2L + 1)]
1
2
×
ψ{|
˜
ψ,
2
(κk)
, (6.31)
where
˜
ψ denotes a term of
N−2
, and the symbols κ and
k stand for the S and the L of a term of
2
. A tabulation
for the p, d, and f shells has been given by Donlan [6.9].
An extension to all multielectron cfp has been carried
out by Velkov [6.10].
If, through successive applications of the two-
particle operators (aa)
(00)
,astateof
N
can be reduced
to
v
, but no further, then v is the seniority number of
Racah [6.8].
If the ranks s and of a
†
are coupled to
¯
S and
¯
L of
¯
ψ,theterm
a
†
¯
ψ
(SL )
(6.32)
either vanishes, or is a term of
N
characterized by
S and L. Such a term is said to possess the godparent
¯
ψ. Redmond [6.11] has used the notion of godparents to
generate an explicit formula for the single particle cfp
[6.7].
6.3 Quasispin
6.3.1 Fermions
For electrons, the components Q
±
(≡ Q
x
± iQ
y
) and Q
z
of the quasispin Q are defined by [6.7, 12]
Q
+
=[(2 + 1)/2]
1
2
a
†
a
†
(00)
, (6.33)
Q
−
=−[(2 + 1)/2]
1
2
(aa)
(00)
, (6.34)
Q
z
=−[(2 + 1)/8]
1
2
a
†
a
(00)
+
aa
†
(00)
.
(6.35)
The term quasispin comes from the fact that the com-
ponents of Q satisfy the commutation relations of an
angular momentum vector. The eigenvalues M
Q
of Q
z
,
for a state of
N
,aregivenby
M
Q
=−(2 + 1 − N)/2 . (6.36)
The shift operators Q
+
and Q
−
connect states of the
shell possessing the same value of the seniority v of
Racah [6.8]. A string of such connected states defines
the extrema of M
Q
, from which it follows that
Q = (2 + 1 − v)/2 .
(6.37)
Part A 6.3
118 Part A Mathematical Methods
Rudzikas has placed special emphasis on quasispin in
his reworking of atomic shell theory, and he has also in-
troduced isospin to embrace electrons differing in their
principal quantum numbers n [6.13]. Concise tables of
one-electron cfp with their quasispin dependence fac-
tored out have been given [6.14], as have the algebraic
dependences on ν and S of two-electron cfp [6.15].
6.3.2 Bosons
For real vibrational modes created by a
†
ν
(ν = 1, 2,
... ,d), the analogs of (6.33–6.35)are
P
+
=−
1
2
ν
a
†
ν
a
†
ν
, (6.38)
P
−
=
1
2
ν
a
ν
a
ν
, (6.39)
P
z
=
1
4
ν
a
†
ν
a
ν
+ a
ν
a
†
ν
,
(6.40)
and P is an angular momentum vector [6.16]. The eigen-
values M
P
for an n-boson state are given by
M
P
= (2n+ d)/4 , (6.41)
and can therefore be quarter-integral. Successive appli-
cation of the operator P
+
to a state |n
0
,forwhich
P
−
|n
0
=0, generates an infinite ladder of states char-
acterized by
P = (2n+ d− 4)/4 .
(6.42)
6.3.3 Triple Tensors
The creation and annihilation operators a
†
ξ
and a
ξ
for
a given state ξ can be regarded as the two components
of a tensor of rank
1
2
with respect to quasispin (either
Q or P). For electrons, this leads to triple tensors a
(qs)
(for which q = s =
1
2
) satisfying
a
(qs)
λ
a
(qs)
µ
+ a
(qs)
µ
a
(qs)
λ
=
(−1)
x+1
δ
m
q
, −m
q
δ
m
s
, −m
s
δ
m
, −m
,
(6.43)
where λ ≡ (m
q
m
s
m
), µ≡ (m
q
m
s
m
), and x = q +
s+ + m
q
+ m
s
+ m
. In terms of the coupled tensor
X
(Kκk)
= (a
(qs)
a
(qs)
)
(Kκk)
, (6.44)
the angular momenta Q, S,andL are given by
Q =−[(2 + 1)/4]
1
2
X
(100)
, (6.45)
S=−[(2 + 1)/4]
1
2
X
(010)
, (6.46)
L =−[( + 1)(2 + 1)/3]
1
2
X
(001)
. (6.47)
Furthermore, the components of X
(Kκk)
for which
M
K
= 0 are identical to the corresponding components
of 2
1
2
(a
†
a)
(κk)
when K + κ + k is odd; and
X
(Kκk)
=−(2 + 1)
1
2
δ(K, 0)δ(κ, 0)δ(k, 0) (6.48)
when K + κ + k is even.
6.3.4 Conjugation
Creation and annihilation operators can be interchanged
by the operation of the conjugation operator C [6.7, 17].
For electrons in the atomic shell,
Ca
(qs)
ξ
C
−1
= (−1)
q−m
q
a
(qs)
η
, (6.49)
where ξ ≡ (m
q
m
s
m
) and η ≡ ((−m
q
)m
s
m
).Interms
of the tensors a
†
and a,
Ca
†
C
−1
= a , CaC
−1
=−a
†
. (6.50)
Furthermore,
CX
(Kκk)
λ
C
−1
= (−1)
K−M
K
X
(Kκk)
µ
, (6.51)
where λ ≡ (M
K
M
κ
M
k
) and µ ≡[(−M
K
)M
κ
M
k
],and
C
QM
Q
= (−1)
Q−M
Q
Q − M
Q
.
(6.52)
Thus, from (6.36), the action of C takes N into 4+
2 − N;thatis,C interchanges electrons and holes. When
the case κ = k = 0 is excluded, application of (6.51)and
(6.52) yields
N
ψ || W
(κk)
||
N
ψ
=
(−1)
y
4+2−N
ψ||W
(κk)
||
4+2−N
ψ
,
(6.53)
where y = κ + k +
1
2
(v
− v) + 1, and where the senior-
ities v and v
are implied by ψ and ψ
. A similar
application to reduced matrix elements of a
†
and a gives
the following relation between cfp:
N+1
ψ{|
N
ψ
= (−1)
z
4+1−N
ψ|}
4+2−N
ψ
×
(4 + 2 − N)(2S
+ 1)(2L
+ 1)
(N + 1)(2S+ 1)(2L + 1)
1
2
,
(6.54)
where z = S+ S
− s+ L + L
− +
1
2
(v + v
− 1).The
phases y and z stem from the conventions of angular
momentum theory, which enter via quasispin. Racah
[6.2, 8] did not use this concept, and his phase choices
are slightly different from the ones above.
Part A 6.3