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Quantum mechanics in general quantum systems (II):
perturbation theory
An Min Wang

Quantum Theory Group, Department of Modern Physics,

University of Science and Technology of China, Hefei, 230026, P.R.China
We propose an improved scheme of perturbation theory based on our exact solution [An Min
Wang, quant-ph/0602055] in general quantum systems independent of time. Our elementary start-
point is to introduce the perturbing parameter as late as possible. Our main skills are Hamiltonian
redivision so as to overcome a aw of the usual perturbation theory, and the perturbing Hamiltonian
matrix product decomposition in order to separate the contraction and anti-contraction terms. Our
calculational technology is the limit process for eliminating apparent divergences. Our central idea
is dynamical rearrangement and summation for the sake of the partial contributions from the high
order even all order approximations absorbed in our perturbed solution. Consequently, we obtain
the improved forms of the zeroth, rst, second and third order perturbed solutions absorbing the
partial contributions from the high order even all order approximations of perturbation. Then we
deduce the improved transition probability. In special, we propose the revised Fermis golden rule.
Moreover, we apply our scheme to obtain the improved forms of perturbed energy and perturbed
state. In addition, we study an easy understanding example of two-state system to illustrate our
scheme and show its advantages. All of this implies the physical reasons and evidences why our
improved scheme of perturbation theory are actually calculable, operationally ecient, conclusively
more accurate. Our improved scheme is the further development and interesting application of our
exact solution, and it has been successfully used to study on open system dynamics [An Min Wang,
quant-ph/0601051].
PACS numbers: 03.65.-w, 04.25.-g, 03.65.Ca
I. INTRODUCTION
The known perturbation theory [1, 2] is an extremely important tool for describing real quantum systems, as it
turns out to be very dicult to nd exact solutions to the Schr

dinger equation for Hamiltonians of even moderate

complexity. Recently, we see the dawn to overcome this diculty because we obtained the exact solution of the
Schroding equation [1] in general quantum systems independent of times [3]. However, this does not mean that the
perturbation theory is unnecessary because our exact solution is still an innite power series of perturbation. Our
solution is called exact one in the sense including all order approximations of perturbation. In practice, if we do
not intend to apply our exact solution to investigations of the formal theory of quantum mechanics, we often need to
cut o our exact solution series to some given order approximation in the calculations of concrete problems. Perhaps,
one argues that our exact solution will back to the usual perturbation theory, and it is, at most, an explicit form
that can bring out the eciency amelioration. Nevertheless, the case is not so. Such a view, in fact, ignores the
signicance of the general term in an innite series, and forgets the technologies to deal with an innite series in the
present mathematics and physics. From our point of view, since the general term is known, we can systematically
and reasonably absorb the partial contributions from some high order even all order approximations to the lower
order approximations just like one has done in quantum eld theory via summation over a series of dierent order
but similar feature Feynman gures. In this paper, based on such a method we develop our dynamical arrangement
and summation idea, and then propose an improved scheme of perturbation theory via introducing several useful
skills and methods.
It is very interesting that we nd a aw in the usual perturbation theory, that is, the perturbing parameter is
introduced too early so that the contributions from the high order even all order approximations of the diagonal and
o-diagonal elements of the perturbing Hamiltonian matrix are, respectively, inappropriately dropped and prematurely
cut o. For some systems, the inuences on the calculation precision from this aw can be not neglectable with the
evolution time increasing. This motivates us to set our start-point to introduce the perturbing parameter as late as
possible in order to guarantee the generality and precision. It is natural from a mathematics view if we think the

Electronic address: anmwang@ustc.edu.cn

2
perturbing parameter in a general perturbation theory is a formal multiplier. Based on this start point, we propose
Hamiltonian redivision skill and further methods so as to overcome the above aw in the usual perturbation theory,
viz. the Hamiltonian redivision makes the contributions from all order approximations of the diagonal elements of
the perturbing Hamiltonian matrix can be absorbed in our improved form of perturbed solution. Hence, this skill
advances the calculation precision in theory, extends the application range for the perturbation theory and can remove
degeneracies in some systems.
Since our exact solution series has apparent divergences, we provide the methods of perturbing Hamiltonian matrix
product decomposition in order to separate the contraction terms with apparent divergences and anti-contraction
terms without apparent divergences. Here, apparent refers to an unture thing, that is, the apparent divergences are
not real singularities and they can be eliminated by mathematical and/or physical methods, while the perturbing
Hamiltonian matrix refers to the representation matrix of the perturbing Hamiltonian in the unperturbed Hamilto-
nian representation. Then, by the limit process we can eliminate these apparent divergences in the contraction terms.
Furthermore we apply dynamical rearrangement and summation idea for the sake of the partial contributions from
the high order even all order approximations absorbed in our perturbed solution. In terms of these useful ideas, skills
and methods we build an improved scheme of perturbation theory. Without any doubt, they are given denitely
dependent on our exact solution. In fact, our exact solution inherits the distinguished feature in a c-number function
form just like the Feynman [4] path integral expression and keeps the advantage in Dyson series [5] that is a power
series of perturbation. At the same time, our exact solution is so explicit that when applying it to a concrete quantum
system, all we need to do is only the calculations of perturbing Hamiltonian matrix and the limitations of primary
functions.
As well known, a key idea of the existed perturbation theory to research the time evolution of system is to split the
Hamiltonian of system into two parts, that is
H = H
0
+ H
1
, (1)
where the eigenvalue problem of so called unperturbed Hamiltonian H
0
is solvable, and so-call perturbing Hamiltonian
H
1
is the rest part of the Hamiltonian. In other words, this splitting is chosen in such a manner that the solutions of
H
0
are known as
H
0
|

= E

, (2)
where |

is the eigenvector of H
0
and E

is the corresponding eigenvalue. Whole |

, in which takes over all

possible values, form a representation of the unperturbed Hamiltonian. It must be emphasized that the principle of
Hamiltonian split is not just the best solvability mentioned above in more general cases. If there are degeneracies,
the Hamiltonian split is also restricted by the condition that the degeneracies can be completely removed via the
usual diagonalization procedure of the degenerate subspaces and the Hamiltonian redivision proposed in this paper,
or specially, if the remained degeneracies are allowed, it requires that the o-diagonal element of the perturbing
Hamiltonian matrix between any two degenerate levels are always vanishing so as to let our improved scheme of
perturbation theory work well. As an example, it has been discussed in our serial study [6]. In addition, if the
cut-o approximation of perturbation is necessary, it requires that for our improved scheme of perturbation theory,
the o-diagonal elements of H
1
matrix is small enough compared with the diagonal element of H = H
0
+ H
1
matrix
in the unperturbed representation.
Nevertheless, there are some known shortcomings in the existed perturbation theory, for example, when H
1
is
not so small compared with H
0
that the high order approximations should be considered, and/or when the partial
contributions from the higher order approximations become relatively important to the studied problems, and/or the
evolution time is long enough, the usual perturbation theory might be dicult to calculate to an enough precision
in an eective way, even not feasible practically since the lower approximation might break the physical symmetries
and/or constraints. In order to overcome these diculties and problems, we recently study and obtain the exact
solution in general quantum systems via explicitly expressing the time evolution operator as a c-number function
and a power series of perturbation including all order approximations [3]. In this paper, our purpose is to build
an improved scheme of perturbation theory based on our exact solution [3] so that the physical problems are more
accurately and eectively calculated. For simplicity, we focus on the cases of Schrodinger dynamics [1]. It is direct to
extend our improved scheme to the cases of the von Neumann dynamics [7].
Just well-known, quantum dynamics and its perturbation theory have been suciently studied and have many
successful applications. Many famous physicists created their nice formulism and obtained some marvelous results.
An attempt to improve its part content or increase some new content as well as some new methods must be very
dicult in their realizations. However, our endeavors have obtained their returns, for examples, our exact solution
[3], perturbation theory and open system dynamics [6] in general quantum systems independent of time.
In this paper, we start from proposing our ideas, skills and methods. We expressly obtain the improved forms of the
zeroth, rst, second and third order approximations of perturbed solution absorbing partial contributions from the
3
high order even all order approximations, nding the improved transition probability, specially, the revised Fermis
golden rule, and providing an operational scheme to calculate the perturbed energy and perturbed state. Furthermore,
by studying a concrete example of two state system, we illustrate clearly that our solution is more ecient and more
accurate than the usual perturbative method. In short, our exact solution and perturbative scheme are formally
explicit, actually calculable, operationally ecient, conclusively more accurate (to the needed precision).
This paper is organized as the following: in Sec. II we nd a aw of the usual perturbation theory and introduce
Hamiltonian redivision to overcome it. Then, we propose the skill of the perturbing Hamiltonian matrix product
decomposition in order to separate the contraction terms with apparent divergences and anti-contraction terms without
apparent divergences. By the limit process we can eliminate these apparent divergences. More importantly, we use
the dynamical rearrangement and summation idea so that the partial contributions from the high order even all
order approximations are absorbed in our perturbed solution and the above aw is further overcome; in Sec. III we
obtain the improved forms of the zeroth, rst, second and third order perturbed solutions of dynamics absorbing
partial contributions from the high order even all order approximations; in Sec. IV we deduce the improved transition
probability, specially, the revised Fermis golden rule. In Sec. V we provide a scheme to calculate the perturbed
energy and the perturbed state; in Sec. VI we study an example of two state system in order to concretely illustrate
our solution to be more eective and more accurate than the usual method; in Sec. VII we summarize our conclusions
and give some discussions. Finally, we write an appendix as well as a supplementary where some expressions are
calculated in order to derive out the improved forms of perturbed solutions.
II. SKILLS AND METHODS IN THE IMPROVED SCHEME OF PERTURBATION THEORY
In our recent work [3], by splitting a Hamiltonian into two parts, using the solvability of eigenvalue problem of
one part of the Hamiltonian, proving an useful identity and deducing an expansion formula of operator binomials
power, we obtain an explicit and general form of the time evolution operator in the representation of solvable part
(unperturbed part) of the Hamiltonian. Then we nd out an exact solution of the Schrodinger equation in general
quantum systems independent of time
|(t) =

l=0
A
l
(t)|(0) =

l=0

l
(t)
_

|(0)
_
|

, (3)
where
A
l
(t) =

l
(t)|

, (4)
A

0
(t) =

e
iEt

, (5)
A

l
(t) =

1, ,
l+1
_
l+1

i=1
(1)
i1
e
iE
i
t
d
i
(E[, l])
_
_
_
l

j=1
H
jj+1
1
_
_

l+1

. (6)
and all H
jj+1
1
=
j
|H
1
|
j+1
form so-called perturbing Hamiltonian matrix, that is, the representation matrix
of the perturbing Hamiltonian in the unperturbed Hamiltonian representation. While
d
1
(E[, l]) =
l

i=1
_
E
1
E
i+1
_
, (7)
d
i
(E[, l]) =
i1

j=1
_
E
j
E
i
_
l+1

k=i+1
(E
i
E

k
) , (8)
d
l+1
(E[, l]) =
l

i=1
_
E
i
E

l+1
_
, (9)
here 2 i l.
It is clear that there are apparent divergences in the above solution. For example
A

1
(t) =
_
e
iEt
E

e
iE

t
E

_
H

1
. (10)
4
when E

= E

(which can appear in the summation or degeneracy cases), it is = iH

1
t. As pointed out
in our paper [3], we need to understand A

k
(t) in the sense of limitation. Moreover, in practice, we should present
how to calculate their limitation in order to eliminate the apparent divergences.
Now, the key problems are how and when to introduce the cut-o approximation in order to obtain the perturbed
solution. For studying and solving them, we rst need to propose some skills and methods in this section. These skills
and methods prot from the fact that the general term is clearly known and explicitly expressed in our exact solution.
By using them we can derive out the improved forms of perturbed solution absorbing the partial contributions from
the high order even all order approximations of perturbation. It will be seen that all the steps are well-regulated
and only calculational technology is to nd the limitation of primary functions. In other words, our exact solution
and perturbation theory are easily calculative and operational, and they have better precision and higher eciency.
Frankly speaking, before we know our exact solution, we are puzzled by too many irregular terms and very trouble
dependence on previous calculation steps. Moreover, we are often anxious about the precision of the results in such
some calculations because those terms proportional to t
a
e
iE
i
t
(a = 1, 2, ) in the high order approximations
might not be ignorable with time increasing. Considering the contributions of these terms can obviously improve the
precision. However, the known perturbation theory does not give the general term, considering this task to absorb
reasonably the high order approximations is impossible.
Since our exact solution has given the explicit form of any order approximation, that is a general term of an
arbitrary order perturbed solution, and their forms are simply the summations of a power series of the perturbing
Hamiltonian. Just enlightened by this general term of arbitrary order perturbed solution, we use two skills and
dynamical rearrangement and summation method to build an improved scheme of perturbation theory, which are
respectively expressed in the following two subsections.
A. Hamiltonian Redivision
The rst skill starts from the decomposition of the perturbing Hamiltonian matrix, that is the matrix elements of
H
1
in the representation of H
0
, into diagonal part and o-diagonal part:
H
jj+1
1
= h
j
1

jj+1
+ g
jj+1
1
, (11)
so as to distinguish them because the diagonal and o-diagonal elements can be dealed with in the dierent way. In
addition, it makes the concrete expression of a given order approximation can be easily calculated. Note that h
j
1
has
been chosen as its diagonal elements and then g
jj+1
1
has been set as its o-diagonal elements:
g
jj+1
1
= g
jj+1
1
(1
jj+1
). (12)
As examples, for the rst order approximation, it is easy to calculate that
A

1
(h) =

1,2
_
2

i=1
(1)
i1
e
iE
i
t
d
i
(E[, l])
_
(h
1
1

12
)
1

2
=
(ih

1
t)
1!
e
iEt

, (13)
A

1
(g) =

1,2
_
2

i=1
(1)
i1
e
iE
i
t
d
i
(E[, l])
_
g
12
1
(1
12
)
1

2
=
_
e
iEt
E

e
iE

t
E

_
g

1
. (14)
Note that here and after we use the symbol A

i
denoting the contribution from the ith order approximation, which
is dened by (6), while its argument indicates the product form of perturbing Hamiltonian matrix. However, for the
second order approximation, since
2

j=1
H
jj+1
1
= (h
1
1
)
2

12

23
+ h
1
1
g
23
1

12
+ g
12
1
h
2
1

23
+ g
12
1
g
23
1
. (15)
we need to calculate the mixed product of diagonal and o-diagonal elements of perturbing Hamiltonian matrix.
Obviously, we have
A

2
(hh) =

1,2,3
_
3

i=1
(1)
i1
e
iE
i
t
d
i
(E[, 2])
_
(h
1
1

12
h
2
1

23
)
1

3
=
(ih

1
t)
2
2!
e
iEt

, (16)
5
A

2
(hg) =

1,2,3
_
3

i=1
(1)
i1
e
iE
i
t
d
i
(E[, 2])
_
h
1
1
g
23
1

12

3
=
_

e
iEt
(E

)
2
+
e
iE

t
(E

)
2
+ (it)
e
iEt
E

_
h

1
g

1
, (17)
A

2
(gh) =

1,2,3
_
3

i=1
(1)
i1
e
iE
i
t
d
i
(E[, 2])
_
h
2
1
g
12
1

23

3
=
_
e
iEt
(E

)
2

e
iE

t
(E

)
2
(it)
e
iE

t
E

_
g

1
h

1
, (18)
A

2
(gg) =

1,2,3
_
3

i=1
(1)
i1
e
iE
i
t
d
i
(E[, 2])
_
g
12
1
g
23
1

1

3
=

1
_
e
iEt
(E

E
1
)(E

)

e
iE
1
t
(E

E
1
)(E
1
E

)
+
e
iE

t
(E

)(E
1
E

)
_
g
1
1
g
1

1
. (19)
In the usual time-dependent perturbation theory, the zeroth order approximation of time evolution of quantum state
keeps its original form

(0)
(t)
_
= e
iEt
|

, (20)
where we have set the initial state as |

for simplicity. By using our solution, we easily calculate out the contributions
of all order approximations from the product of completely diagonal elements h of the perturbing Hamiltonian matrix
to this zeroth order approximation

1, ,
l+1
_
l+1

i=1
(1)
i1
e
iE
i
t
d
i
(E[, l])
_
_
_
l

j=1
h
j
1

jj+1
_
_

l+1
=
(ih

1
t)
l
l!
e
iEt

. (21)
Therefore, we can absorb the contributions of all order approximation parts from the product of completely diagonal
elements h of the perturbing Hamiltonian matrix into this zeroth order approximation to obtain

(0)
(t)
_
= e
i(E+h

1
)t
|

. (22)
Similarly, by calculation, we can deduce that up to the second approximation, the perturbed solution has the following
form
|

(t) =

_
_
_
e
i(E+h

1
)t

+
_
_
e
i(E+h

1
)t
e
i

+h

t
(E

+ h

1
)
_
E

+ h

1
_
_
_
g

1
+

1
_
_
e
i(E+h

1
)t
[(E

+ h

1
) (E
1
+ h
1
1
)]
_
(E

+ h

1
)
_
E

+ h

1
__

e
i(E
1
+h

1
1
)t
[(E

+ h

1
) (E
1
+ h
1
1
)]
_
(E
1
+ h
1
1
)
_
E

+ h

1
__
+
e
i

+h

t
_
(E

+ h

1
)
_
E

+ h

1
__ _
(E
1
+ h
1
1
)
_
E

+ h

1
__
_
_
g
1
1
g
1

1
_
_
_
_

|(0)
_
|

+O(H
3
1
). (23)
However, for the higher order approximation, the corresponding calculation is heavy. In fact, it is unnecessary to
calculate the contributions from those terms with the diagonal elements of H
1
since introducing the following skill.
6
This is a reason why we omit the relevant calculation details. Here we mention it only for verication of the correctness
of our exact solution in this way.
The results (22) and (23) are not surprised because of the fact that the Hamiltonian is re-divisible. Actually, we
can furthermore use a trick of redivision of the Hamiltonian so that the new H
0
contains the diagonal part of H
1
,
that is,
H

0
= H
0
+

1
|

|, (24)
H

1
= H
1

1
|

| =

1
|

|. (25)
In other words, without loss of generality, we always can choose that H

1
has only the o-diagonal elements in the H

0
(or H
0
) representation and
H

0
|

= (E

+ h

1
) |

= E

. (26)
It is clear that this redivision does not change the representation of the unperturbed Hamiltonian, but can change the
corresponding eigenvalues. In spite that our skill is so simple, it seems not be suciently transpired and understood
from the fact that the recent some textbooks of quantum mechanics still remain the contributions from the diagonal
elements of the perturbing Hamiltonian matrix in the expression of the second order perturbed state. It is clear
that the directly cut-o approximation in the usual perturbation theory drops the contributions from all higher order
approximations of the diagonal element of the perturbing Hamiltonian matrix. From our point of view, the usual
perturbation theory introduces the perturbing parameter too early so that this aw is resulted in.
If there is degeneracy, our notation has to be changed as
E
i
E
ia
i
= E
i
, (27)

ij

ij

a
i
a
j
, (28)

ij

ij
+
ij

a
i
a
j
. (29)
Thus, we can nd
A
a,

1
(h) =
(ih

1
t)
1!
e
iEt

aa

, (30)
A
a,

1
(g) =
_
e
iEa
t
E
a
E

e
iE

t
E
a
E

_
g
a,

+
(ig
a,a

1
t)
1!
e
iEa
t

. (31)
This seems to bring some complications. However, we can use the trick in the usual degenerate perturbation theory,
that is, we are free to choose our base set of unperturbed kets |
a
in such a way that that H
1
is diagonalized in
the corresponding degenerate subspaces. In other words, we should nd the linear combinations of the degenerate
unperturbed kets to re-span the zero-order eigen subspace of H
0
so that

a
|H
1

b
_
= g
a,b
1
= d
a
1

ab
. (32)
(If there is still the same values among all of d
a
, this procedure can be repeated in general.) This means that
g
a,a

1
= 0. Then, we use our redivision skill again, that is
H

0
= H
0
+

/ D,
h

1
|

| +

D, a
d
a
1
|
a

a
|, (33)
H

1
= H
1

/ D,
h

1
|

D, a
d
a
1
|
a

a
|. (34)
where D is a set of all degenerate subspace-indexes. Thus, the last term in Eq. (31) vanishes,
A
a,

1
(g) =
_
e
iEa
t
E
a
E

e
iE

t
E
a
E

_
g
a,

. (35)
7
In fact, under the preconditions of H
1
is diagonal in the degenerate subspaces, we can directly do replacement
g
ij
1
g
ia
i
,ja
j
1

ij
(36)
from the non-degenerate case to the degenerate case. For simplicity, we always assume that H
1
has been diagonalized
in the degenerate subspaces from now on.
It must be emphasized that the Hamiltonian redivision skill leads to the fact that the new perturbed solution can
be obtained by the replacement
E
i
E
i
+ h
i
1
(37)
in the non-degenerate perturbed solution and its conclusions. With degeneracy present, if our method is to work well,
the degeneracy should be completely removed in the diagonalization procedures of the degenerate subspaces and the
Hamiltonian redivision, that is, for any given degenerate subspace, d
a
1
= d
b
1
if a = b. In other words, E

a
= E

b
if a = b. This means that all of eigenvalues of H

0
are no longer the same, so we can back to the non-degenerate
cases. Or specially, if we allow the remained degeneracies, the o-diagonal element of the perturbing Hamiltonian
matrix between any two degenerate levels are always vanishing. This implies that there is no extra contribution from
the degeneracies in the any more than the zeroth order approximations. It is important to remember these facts.
However, how must we proceed if the degeneracies are not completely removed by the usual diagonalization procedure
and our Hamiltonian redivision as well as the special cases with the remained degeneracies stated above are not valid.
It is known to be a challenge in the usual perturbation theory. Although our exact solution can apply to such a
kind of cases, but the form of perturbed solution will get complicated because more apparent divergences need to
be eliminated and then some new terms proportional to the power of evolution time will appear in general. We will
study this problem in the near future. Based on the above reasons, we do not consider the degenerate case from now
on.
From the statement above, we have seen that there are two equivalent ways to obtain the same perturbed solution
and its conclusions. One of them is to redene the energy level E
i
as E

i
(or E

i
), think E

i
(or E

i
) to be explicitly
independent on the perturbing parameter from a redened view, and then use the method in the usual perturbation
theory to obtain the result from the redivided H

1
(or H

1
). The other way is to directly derive out the perturbed
solution from the original Hamiltonian by using the standard procedure, but the rearrangement and summation are
carried out just like what we have done above. From our point of view, this is because the perturbing parameter is
only a formal multiplier in mathematics and it can be introduced after redening E

i
. It is natural although this
problem seems not be noticed for a long time. The rst skill, that is, the Hamiltonian redivision skill will be again
applied to our scheme to obtain improved forms of perturbed energy and perturbed state in Sec. III.
The Hamiltonian redivision not only overcomes the aw of the usual perturbation theory, but also has three obvious
advantages. Firstly, it advances the calculation precision of perturbation theory because it makes the contributions
from all order approximations of the diagonal elements of the perturbing Hamiltonian matrix are naturally included.
Secondly, it extends the applicable range of perturbation theory based on the same reason since the diagonal elements
of the perturbing Hamiltonian no longer is needed to be smaller. Lastly, it can be used to remove the degeneracies,
which is important for the perturbation theory.
For simplicity, in the following, we omit the

(or

) in H
0
, H
1
as well as E

, and always let H

1
have only its
o-diagonal part and let H
0
have no degeneracy unless particular claiming.
B. Perturbed Hamiltonian matrix product decomposition and apparent divergence elimination
In this subsection, we present the second important skill enlightened by our exact solution, that is, the perturbing
Hamiltonian matrix product decomposition, which is a technology to separate the contraction terms with apparent
divergences and anti-contraction terms without apparent divergences, and then we can eliminate these apparent diver-
gences by the limit process. More importantly, we can propose so-called dynamical rearrangement and summation
idea in order to absorb the partial contributions from the high order even all order approximation of perturbing
Hamiltonian into the lower order terms of our perturbation theory. It is a key method in our improved scheme of
perturbation theory.
Let us start with the second order approximation. Since we have taken H
jj+1
1
only with the o-diagonal part
g
jj+1
1
, the contribution from the second order approximation of the perturbing Hamiltonian is only A

2
(gg) in
eq.(19). However, we nd that in the above expression of A

2
(gg), the apparent divergence has not been completely
eliminated or the limitation has not been completely found out because we have not excluded the case E

= E

(or
=

). This problem can be xed by introducing a perturbing Hamiltonian matrix product decomposition
g
12
1
g
23
1
= g
12
1
g
23
1

13
+ g
12
1
g
23
1

13
, (38)
8
where
13
= 1
13
. Thus, the contribution from the second order approximation is made of two terms, one
so-called contraction term with the function factor and the other so-called anti-contraction term with the function
factor. Obviously, the contraction term has the apparent divergence and anti-contraction term has no the apparent
divergence. Hence, in order to eliminate the apparent divergence in the contraction term, we only need to nd its lim-
itation. It must be emphasized that we only consider the non-degenerate case here and after for simplication. When
the degeneration happens, two indexes with the same main energy level number will not have the anti-contraction.
In terms of the above skill, we nd that the contribution from the second order approximation is made of the
corresponding contraction- and anti-contraction- terms
A

2
(gg) = A

2
(gg; c) + A

2
(gg; n), (39)
where
A

2
(gg; c) =

1,2,3
_
3

i=1
(1)
i1
e
iE
i
t
d
i
(E[, 2])
_
g
12
1
g
23
1

13

3
=

1
_

e
iEt
(E

E
1
)
2
+
e
iE
1
t
(E

E
1
)
2
+ (it)
e
iEt
E

E
1
_
|g
1
1
|
2

, (40)
A

2
(gg; n) =

1,2,3
_
3

i=1
(1)
i1
e
iE
i
t
d
i
(E[, 2])
_
g
12
1
g
23
1

13

3
=

1
_
e
iEt
(E

E
1
)(E

)

e
iE
1
t
(E

E
1
)(E
1
E

)
+
e
iE

t
(E

)(E
1
E

)
_
g
1
1
g
1

. (41)
The above method can be extended to the higher order approximation by introducing a skill of perturbing Hamil-
tonian matrix product decomposition, or simply called it g-product decomposition when the perturbing Hamiltonian
matrix is o-diagonal. For a sequential product of o-diagonal elements g with the form

m
k=1
g

k+1
1
(m 2), we
dene its (m1)th decomposition by
m

k=1
g

k+1
1
=
_
m

k=1
g

k+1
1
_

1m+1
+
_
m

k=1
g

k+1
1
_

1m+1
. (42)
When we calculate the contributions from the nth order approximation, we will rst carry out (n 1) the rst
decompositions, that is
n

k=1
g

k+1
1
=
_
n

k=1
g

k+1
1
__
n1

k=1
_

k+2
+

k+2
_
_
. (43)
Obviously, from the fact that H
1
is usually taken as Hermit one, it follows that
g
jj+1
1
g
j+1j+2
1

jj+2
=

g
jj+1
1

jj+2
. (44)
When the contribution from a given order approximation is considered, the summation over one of two subscripts will
lead to the contraction of g-production. More generally, for the contraction of even number g-production
_
_
m

j=1
g
jj+1
1
m1

k=1

k+2
_
_

m+1
= |g
2
1
|
m
_
m1

k=1

k+2
_

m+1

, (45)
and for the contraction of odd number g-production,
_
_
m

j=1
g
jj+1
1
m1

k=1

k+2
_
_

m+1
=

m1
_
m1

k=1

k+2
_

m+1
g

1
, (46)
where
1

m+1
is a factor appearing in the expression of our solution.
9
Then, we consider, in turn, all possible the second decomposition, the third decomposition, and up to the (n1)th
decomposition. It must be emphasized that after calculating the contributions from the terms of lower decompositions,
some of terms in the higher decompositions may be trivial because there are some symmetric and complementary
symmetric indexes in the corresponding results, that is, the products of these results and the given

or

are zero. In other words, such some higher decompositions do not need to be considered. As an example, let us
analyze the contribution from the third order approximation. It is clear that the rst decomposition of a sequential
production of three o-diagonal elements becomes
g
12
1
g
23
1
g
34
1
= g
12
1
g
23
1
g
34
1

13

24
+ g
12
1
g
23
1
g
34
1

13

24
+g
12
1
g
23
1
g
34
1

13

24
+ g
12
1
g
23
1
g
34
1

13

24
. (47)
Thus, the related contribution is just divided into 4 terms
A

3
(ggg) = A

3
(ggg; cc) + A

3
(ggg; cn) + A

3
(ggg; nc) + A

3
(ggg, nn). (48)
In fact, by calculating we know that the second decomposition of the former three terms do not need to be considered,
only the second decomposition of the last term is nontrivial. This means that
A

3
(ggg; nn) = A

3
(ggg; nn, c) + A

3
(ggg; nn, n), (49)
where we have added
13
in the denition of A

3
(ggg; nn, c), and
13
in the denition of A

3
(ggg; nn, n). Ob-
viously, in the practical process, this feature largely simplies the calculations. It is easy to see that the number of
all of terms with contractions and anti-contractions is 5. For convenience and clearness, we call the contributions
from the dierent terms in the decomposition of g-product as the contractions and anti-contractions of g-product. Of
course, the contraction and anti-contraction refer to the meaning after summation(s) over the subscript(s) in general.
Moreover, here and after, we drop the argument gg g in the ith order approximation A
i
since its meaning has
been indicated by i after the Hamiltonian is redivided. For example, the explicit expressions of all contraction- and
anti-contraction terms in the third order approximation A
3
can be calculated as follows:
A

3
(cc) =

1, ,4
_
4

i=1
(1)
i1
e
iE
i
t
d
i
(E[, 3])
_
_
_
3

j=1
g
jj+1
1
_
_
_
2

k=1

k+2
_

=
_

2e
iEt
(E

)
3
+
2e
iE

t
(E

)
3
+ (it)
e
iEt
(E

)
2
+ (it)
e
iE

t
(E

)
2
_

2
g

1
, (50)
A

3
(cn) =

1, ,4
_
4

i=1
(1)
i1
e
iE
i
t
d
i
(E[, 3])
_
_
_
3

j=1
g
jj+1
1
_
_

13

24

l+1

1
_

e
iEt
(E

E
1
) (E

)
2

e
iEt
(E

E
1
)
2
(E

)
+
e
iE
1
t
(E

E
1
)
2
(E
1
E

e
iE

t
(E

)
2
(E
1
E

)
+ (it)
e
iEt
(E

E
1
) (E

)
_
|g
1
1
|
2
g

1

1
, (51)
A

3
(nc) =

1, ,4
_
4

i=1
(1)
i1
e
iE
i
t
d
i
(E[, 3])
_
_
_
3

j=1
g
jj+1
1
_
_

13

24

l+1

1
_
e
iEt
(E

E
1
) (E

)
2

e
iE
1
t
(E

E
1
) (E
1
E

)
2

e
iE

t
(E

E
1
) (E

)
2
+
e
iE

t
(E

E
1
) (E
1
E

)
2
+ (it)
e
iE

t
(E

) (E
1
E

)
_
g

g
1

1
, (52)
10
A

3
(nn, c) =

1, ,4
_
4

i=1
(1)
i1
e
iE
i
t
d
i
(E[, 3])
_
_
_
3

j=1
g
jj+1
1
_
_

4

13

24

12
_

e
iEt
(E

E
1
)
2
(E
1
E
2
)
+
e
iEt
(E

E
2
)
2
(E
1
E
2
)
+
e
iE
1
t
(E

E
1
)
2
(E
1
E
2
)

e
iE
2
t
(E

E
2
)
2
(E
1
E
2
)
+ (it)
e
iEt
(E

E
1
) (E
1
E
2
)
_
g
1
1
g
12
1
g
2

1

2

, (53)
A

3
(nn, n) =

1, ,4
_
4

i=1
(1)
i1
e
iE
i
t
d
i
(E[, 3])
_
_
_
3

j=1
g
jj+1
1
_
_

4

13

24

12
_
e
iEt
(E

E
1
) (E

E
2
) (E

)

e
iE
1
t
(E

E
1
) (E
1
E
2
) (E
1
E

)
+
e
iE
2
t
(E

E
2
) (E
1
E
2
) (E
2
E

)

e
iE

t
(E

) (E
1
E

) (E
2
E

)
_
g
1
1
g
12
1
g
2

1

2

. (54)
In the above calculations, the uesed technologies mainly to nd the limitation, dummy index changing and summation,
as well as the replacement g
ij
1

ij
= g
ij
1
since g
ij
1
has been o-diagonal.
It must be emphasized that, in our notation, A

i
represents the contributions from the ith order approximation.
The other independent variables are divided into i 1 groups and are arranged sequentially corresponding to the
order of g-product decomposition. That is, the rst variable group represents the rst decomposition, the second
variable group represents the second decomposition, and so on. Every variable group is a bit-string made of three
possible element c, n, k and its length is equal to the number of the related order of g-product decomposition, that is,
for the jth decompositions in the ith order approximation its length is i j. In each variable group, c corresponds
to a function, n corresponds to a function and k corresponds to 1 (non-decomposition). Their sequence in the
bit-string corresponds to the sequence of contraction and/or anti-contraction index string. From the above analysis
and statement, the index string of the jth decompositions in the i order approximation is:
ij

k=1
(
k
,
k+1+j
) . (55)
For example, for A
5
, the rst variable group is cccn, which refers to the rst decomposition in ve order approxi-
mation and the terms to include the factor
13

24

35

46
in the denition of A
5
(cccn). Similarly, cncc means
to insert the factor
13

24

35

46
into the denition of A
5
(cncc). When there are nontrivial second contrac-
tions, for instance, two variable groups (ccnn, kkc) represent that the denition of A
5
(ccnn, kkc) has the factor
(
13

24

35

46
)
36
. Since there are fully trivial contraction (the bit-string is made of only k), we omit their
related variable group for simplicity.
Furthermore, we pack up all the contraction- and non-contraction terms in the following way so that we can obtain
conveniently the improved forms of perturbed solution of dynamics absorbing the partial contributions from the high
order even all order approximations. We rst decompose A

3
, which is a summation of all above terms, into the
three parts according to e
iE
i
t
, (it)e
iE
i
t
and (it)
2
e
iE
i
t
/2:
A

3
= A

3
(e) + A

3
(te) + A

3
(t
2
e). (56)
Secondly, we decompose its every term into three parts according to e
iEt
, e
iE
1
t
(

1
e
iE
1
t
) and e
iE

t
:
A

3
(e) = A

3
(e
iEt
) + A

3
(e
iE
1
t
) + A

3
(e
iE

t
), (57)
A

3
(te) = A

3
(te
iEt
) + A

3
(te
iE
1
t
) + A

3
(te
iE

t
), (58)
A

4
(t
2
e) = A

3
(t
2
e
iEt
) + A

3
(t
2
e
iE
1
t
) + A

3
(t
2
e
iE

t
). (59)
11
Finally, we again decompose every term in the above equations into the diagonal and o-diagonal parts about and

:
A

3
(e
iE
i
t
) = A

3
(e
iE
i
t
; D) + A

3
(e
iE
i
t
; N), (60)
A

3
(te
iE
i
t
) = A

3
(te
iE
i
t
; D) + A

3
(te
iE
i
t
; N), (61)
A

3
(t
2
e
iE
i
t
) = A

3
(t
2
e
iE
i
t
; D) + A

3
(t
2
e
iE
i
t
; N), (62)
where E
i
takes E

, E
1
and E

.
According to the above way, it is easy to obtain
A

3
(e
iEt
; D) =

1,2
e
iEt
_
1
(E

E
1
) (E

E
2
)
2
+
1
(E

E
1
)
2
(E

E
2
)
_
g
1
1
g
12
1
g
2
1

, (63)
A

3
(e
iEt
; N) =

1
e
iEt
_
1
(E

E
1
) (E

)
2
+
1
(E

E
1
)
2
(E

)
_
g
1
1
g
1
1
g

1
+

1,2
e
iEt
g
1
1
g
12
1
g
2

1

2

(E

E
1
) (E

E
2
) (E

)
, (64)
A

3
(e
iE
1
t
; D) =

1,2
e
iE
1
t
g
1
1
g
12
1
g
2
1

(E

E
1
)
2
(E
1
E
2
)
, (65)
A

3
(e
iE
1
t
; N) =

1,2
e
iE
1
t
g
1
1
g
12
1
g
2

1

1

(E

E
1
) (E
1
E
2
) (E
1
E

)
, (66)
A

3
(e
iE
2
t
; D) =

1,2
e
iE
2
t
g
1
1
g
12
1
g
2
1

(E

E
2
)
2
(E
1
E
2
)
, (67)
A

3
(e
iE
2
t
; N) =

1,2
e
iE
2
t
g
1
1
g
12
1
g
2

1

2

(E

E
2
) (E
1
E
2
) (E
2
E

)
, (68)
A

3
(e
iE

t
; D) = 0, (69)
A

3
(e
iE

t
; N) =

1
e
iE

t
_
1
(E

) (E
1
E

)
2
+
1
(E

)
2
(E
1
E

)
_
g

1
1
g
1

1
g

1,2
e
iE

t
g
1
1
g
12
1
g
2

1

1

(E

) (E
1
E

) (E
2
E

)
. (70)
In the end of this subsection, we would like to point out that the main purpose introducing the g-product decom-
position and calculating the contractions and anti-contractions of g-product is to eliminate the apparent divergences
and nd out all the limitations from the contributions of g-product contraction terms. This is important to express
the results with the physical signicance.
III. IMPROVED FORMS OF PERTURBED SOLUTION OF DYNAMICS
In fact, the nal aim using the g-product decomposition and then calculating the limitation of the contraction terms
is to absorb the partial contributions from the high order approximations of o-diagonal elements of the perturbing
Hamiltonian matrix into the lower order approximations in our improved scheme of perturbation theory. In this
section, making use of the skills and methods stated in previous section, we can obtain the zeroth, rst, second and
third order improved forms of perturbed solutions with the above features.
In mathematics, the process to obtain the improved forms of perturbed solutions is a kind of technology to deal
with an innite series, that is, according to some principles and the general term form to rearrange those terms with
12
the same features together forming a group, then sum all of the terms in such a particular group that they become
a compact function at a given precision, nally this innite series is transformed into a new series form that directly
relates to the studied problem. More concretely speaking, since we concern the system evolution with time t, we take
those terms with (iy
i
t)e
ixit
, (iy
i
t)
2
e
ixit
/2! and (iy
i
t)
3
e
ixit
/3!, with the same factor function f together
forming a group, then sum them to obtain an exponential function f exp [i (x
i
+ y
i
) t]. The physical reason to do
this is that such an exponential function represents the system evolution in theory and it has the obvious physical
signicance in the calculation of transition probability and perturbed energy. Through rearranging and summing,
those terms with factors t
a
e
iE
i
t
, (a = 1, 2, ) in the higher order approximations are absorbed into the improved
lower approximations, we thus can advance the precision, particularly, when the evolution time t is long enough. We
can call it dynamical rearrangement and summation method.
A. Improved form of the zeroth order perturbed solution of dynamics
Let us start with the zeroth order perturbed solution of dynamics. In the usual perturbation theory, it is well-known

(0)
(t)
_
=

e
iEt

|(0)|

e
iEt

, (71)
where a

|(0). Now, we would like to improve it so that it can absorb the partial contributions from higher
order approximations. Actually, we can nd that A
2
(c) and A
3
(nn, c) have the terms proportional to (it)
(it)e
iEt
_

1
1
E

E
1
|g
1
1
|
2
_

, (72)
(it)e
iEt
_

1,2
1
(E

E
1
)(E

E
2
)
g
1
1
g
12
1
g
2

1
_

. (73)
Introduce the notation
G
(2)

1
1
E

E
1
|g
1
1
|
2
, (74)
G
(3)

1,2
1
(E

E
1
)(E

E
2
)
g
1
1
g
12
1
g
2
1
. (75)
It is clear that G
(a)

has the energy dimension, and we will see that it can be called the ath revised energy. Let
us add the terms (72), (73) and the related terms in A
4
(te
iEt
, D), A
4
(t
2
e
iEt
, D), A
5
(te
iEt
, D), A
5
(t
2
e
iEt
, D),
A
6
(t
2
e
iEt
, D) and A
6
(t
3
e
iEt
) given in Appendix A together, that is,
A

I0
(t) = e
iEt
_
1 + (it)
_
G
(2)

+ G
(3)

+ G
(4)

+ G
(5)

_
+
(it)
2
2!
_
G
(2)

+ G
(3)

_
2
+
(it)
2
2!
2G
(2)

G
(4)

+
_

, (76)
Although we have not nished more calculations, from the mathematical symmetry and physical concept, we can
think
A

I0
= e
iEt
_
1 + (it)
_
G
(2)

+ G
(3)

+ G
(4)

+ G
(5)

_
+
(it)
2
2!
_
G
(2)

+ G
(3)

+ G
(4)

+ G
(5)

_
2
+
_

, (77)
New terms added to the above equation ought to, we think, appear at A
7
(t), A
8
(t), A
9
(t) and A
10
(t), or come from
the point of view introducing the higher approximations. So we have
A

I0
(t) = e
i(E+G
(2)

+G
(3)

+G
(4)

+G
(5)

)t

(78)
and then obtain the improved form of the zeroth order perturbed solution of dynamics

(0)
ET
(t)
_
I
=

A
I0
a

(t)|

. (79)
13
It is clear that G
(2)

is real. In fact, G
(3)

is also real. In order to prove it, we exchange the dummy indexes

1
and

2
and take the complex conjugate of G
(3)

, that is
G
(3)

1,2
1
(E

E
1
)(E

E
2
)
(g
2
1
)

(g
21
1
)

(g
1
1
)

1,2
1
(E

E
1
)(E

E
2
)
g
1
1
g
12
1
g
2
1
= G
(3)

, (80)
where we have used the relations
_
g
12
1
_

= g
21
1
for any
1
and
2
since H
1
is Hermit. Similar analyses can be
applied to G
(4)

and G
(5)

. These mean that e

i(G
(2)

+G
(3)

+G
(4)

+G
(5)

)t
is still an oscillatory factor.
B. Improved form of the rst order perturbed solution of dynamics
Furthermore, in order to absorb the partial contributions from the approximation higher than zeroth order, we need
to consider the contributions from o-diagonal elements in the higher order approximations.
Well-known usual rst order perturbing part of dynamics is

(1)
(t)
_
=

_
e
iEt
E

e
iE

t
E

_
H

1
|

__
e
iEt
E

e
iE

t
E

_
g

1
_
|

. (81)
It must be emphasized that H
1
is taken as only with the o-diagonal part for simplicity. That is, we have used the
Hamiltonian redivision skill if the perturbing Hamiltonian matrix has the diagonal elements.
Thus, from A
3
(te
iEt
, N) and A
4
(te
iEt
, N), A
4
(t
2
e
iEt
, D), A
5
(t
2
e
iEt
, N), A
6
(t
2
e
iEt
, N), which are dened
and calculated in the Appendix A, it follows that
A

I1
(t) =
e
iEt
(E

)
_
1 + (it)
_
G
(2)

+ G
(3)

+ G
(4)

_
+
(it)
2
2!
_
G
(2)

_
2
+
(it)
2
2!
2G
(2)

G
(3)

+
_
g

e
iE

t
(E

)
_
1 + (it)
_
G
(2)

+ G
(3)

+ G
(4)

_
+
(it)
2
2!
_
G
(2)

_
2
+
(it)
2
2!
2G
(2)

G
(3)

+
_
g

1
. (82)
Therefore, by rewriting
A

I1
(t) =
_
_
e
i(E+G
(2)

+G
(3)

+G
(4)

)t
E

e
i

+G
(2)

+G
(3)

+G
(4)

t
E

_
_
g

1
, (83)
we obtain the improved form of the rst order perturbed solution of dynamics

(1)
(t)
_
I
=

I1
(t)a

. (84)
C. Improved second order- and third order perturbed solution
Likewise, it is not dicult to obtain
A
,

I2
(t) =

1
_

_
e
i(E+G
(2)

+G
(3)

)t
e
i(E
1
+G
(2)

1
+G
(3)

1
)t
(E

E
1
)
2
_
g
1
1
g
1
1

+
_
e
i(E+G
(2)

+G
(3)

)t
(E

E
1
) (E

)

e
i(E
1
+G
(2)

1
+G
(3)

1
)t
(E

E
1
) (E
1
E

)
+
e
i

+G
(2)

+G
(3)

t
(E

) (E
1
E

)
_
_
g
1
1
g
1

_
_
_
, (85)
14
A
,

I3
(t) =

1,2
_

e
i(E+G
(2)

)t
(E

E
1
) (E

E
2
)
2

e
i(E+G
(2)

)t
(E

E
1
)
2
(E

E
2
)
+
e
i(E
1
+G
(2)

1
)t
(E

E
1
)
2
(E
1
E
2
)

e
i(E
2
+G
(2)

2
)t
(E

E
2
)
2
(E
1
E
2
)
_
g
1
1
g
12
1
g
2
1

1
_
e
i(E+G
(2)

)t
(E

E
1
) (E

)
2
+
e
i(E+G
(2)

)t
(E

E
1
)
2
(E

)
_
g
1
1
g
1
1
g

1
+

1,2
_
e
i(E+G
(2)

)t

2
(E

E
1
) (E

E
2
) (E

)

e
i(E
1
+G
(2)

1
)t

(E

E
1
) (E
1
E
2
) (E
1
E

)
+
e
i(E
2
+G
(2)

2
)t

2
(E

E
2
) (E
1
E
2
) (E
2
E

)
_
g
1
1
g
12
1
g
2

. (86)
Therefore, the improved forms of the second- and third order perturbed solutions are, respectively,

(2)
(t)
_
I
=

A
,

I2
(t)a

, (87)

(3)
(t)
_
I
=

A
,

I3
(t)a

. (88)
D. Summary
Obviously, our improved form of perturbed solution of dynamics up to the third order approximation is
|(t) =
3

i=0

(i)
(t)
_
I
+O(H
4
1
). (89)
However, this solution absorbs the contributions from the whole A

l
(te), A

l
(t
2
e) parts up to the fth order
approximation and the whole A

l
(t
2
e), A

l
(t
3
e) parts in the sixth order approximation. After considering the
contractions and anti-contractions, we get the result corresponds to the replacement
e
iE
i
t
e
i
e
E
i
t
, (90)
in the A

l
(e) part, where

E
i
= E
i
+ h
i
+

a=2
G
(a)
i
, (91)
i = 0, 1, 2, , and
0
= . Here, we have absorbed the possible contributions from the diagonal elements of the
perturbing Hamiltonian matrix. Although the upper bound of summation index a is dierent from the approximation
order in the nished calculations, we can conjecture that it may be taken to at least 5 based on the consideration
from the physical concept and mathematical symmetry. For a 5, their forms should be similar. From our point of
view, such form is so delicate that its form happens impossibly by accident. Perhaps, there is a fundamental formula
within it. Nevertheless, we have no idea of how to prove it strictly and generally at present.
Actually, as soon as we carry out further calculations, we can absorb the contributions from higher order approxi-
mations. Moreover, these calculations are not dicult and are programmable because we only need to calculate the
limitation and summation. Therefore, the advantages of our solution have been made clear in our improved forms
of perturbed solution of dynamics. In other words, they oer clear evidences to show our improved scheme is better
than the existing method in the precision and eciency. In the following several sections, we will clearly demonstrate
these problems.
IV. IMPROVED TRANSITION PROBABILITY AND REVISED FERMIS GOLDEN RULE
One of the interesting applications of our perturbed solution is the calculation of transition probability in general
quantum systems independent of time. It ameliorates the well-known conclusion because our solution absorbs the
15
contributions from the high order approximations of the perturbing Hamiltonian. Moreover, in terms of our improved
forms of perturbed solution, it is easy to obtain the high order transition probability. In addition, for the case of
sudden perturbation, our scheme is also suitable.
Let us start with the following perturbing expansion of state evolution with time t,
|(t) =

(t)|

n=0

c
(n)

(t)|

. (92)
When we take the initial state as

_
, from our improved rst order perturbed solution, we immediately obtain
c
(1)
,I
=
e
i
e
Et
e
i
e
E

t
E

1
, (93)
where

E
i
= E
i
+ h
i
1
+ G
(2)
i
+ G
(3)
i
+ G
(4)
i
. (94)
Here, we use the subscript I for distinguishing it from the usual result. Omitting a unimportant phase factor e
i
e
Et
,
we can rewrite it as
c
(1)
,I
=
g

1
E

_
1 e
ie

t
_
, (95)
where

=

E

. Obviously it is dierent from the well known conclusion

c
(1)

=
g

1
E

_
1 e
i

t
_
, (96)
where

= E

. Therefore, our result contains the partial contributions from the high order approximations.
Considering the transition probability from

_
to |

after time T, we have

P

I
(t) =

1 e
ie

2
=

2
sin
2
(

T/2)
(

/2)
2
. (97)
In terms of the relation
sin
2
x sin
2
y =
1
2
[cos(2y) cos(2x)] , (98)
we have the revision part of transition probability
P

I
(t) = 2

2
cos (

T) cos (

T)
(

)
2
. (99)
If plotting
sin
2
(

T/2)
(

/2)
2
=
_

_
2
sin
2
(

T/2)
(

/2)
2
, (100)
we can see that it has a well-dened peak centered at

= 0. Just as what has been done in the usual case, we can

extend the integral range as . Thus, the revised Fermis golden rule
w = w
F
+ w, (101)
where the usual Fermis golden rule is [8]
w
F
= 2(E

2
, (102)
16
in which w means the transition velocity, (E

) is the density of nal state and we have used the integral formula
_

sin
2
x
x
2
= . (103)
while the revision part is
w = 2
_

dE

(E

2
cos (

T) cos (

T)
T (

)
2
. (104)
It is clear that

is a function of E

, and then a function of

. For simplicity, we only consider

to its second
order approximation, that is

= (

) =

1
_

_
|g
1
1
|
2

g
1
1

1
_

_ +O(H
3
1
). (105)
Again based on dE

= d

, we have
w = 2
_

+ E

2
cos [

T] cos [

() T]
T (

)
2
. (106)
It seems to not to be easy to deduce the general form of this integral. In order to simplify it, we can use the fact that

is a smaller quantity since

=
4

i=2
_
G
(i)

G
(i)

_
. (107)
For example, we can approximatively take
cos (

T) cos (

T) T (

) sin (

T) , (108)
then calculate the integral. We will study it in our other manuscript (in preparing).
Obviously, the revision comes from the contributions of high order approximations. The physical eect resulted
from our solution, whether is important or unimportant, should be investigated in some concrete quantum systems.
Recently, we reconsider the transition probability and perturbed energy for a Hydrogen atom in a constant magnetic
eld [9]. We nd the results obtained by using our improved scheme are indeed more satisfying in the calculation
precision and eciency. We will discuss more examples in our future manuscripts (in preparing).
It is clear that the relevant results can be obtained from the usual results via replacing

in the exponential
power by using

. Thus, one thing is true with the time t evolving, e

i(e

t/2)
term in the improved transition
probability can be very dierent from e
i(

t/2)
in the traditional one, which might lead to totally dierent results.
To save the space, we do not intend to discuss more here.
In fact, there is no any diculty to obtain the second- and three order transition probability in terms of our
improved forms of perturbed solution in the previous section. More higher order transition probability can be given
eectively and accurately by our scheme.
V. IMPROVED FORMS OF PERTURBED ENERGY AND PERTURBED STATE
Now we study how to calculate the improved forms of perturbed energy and perturbed state. For simplicity, we
only study them concerning the improved second order approximation. Based on the experience from the skill one
in Sec. VI, we can, in fact, set a new

E and then use the technology in the usual perturbative theory. That is, we
denote

E
i
= E
i
+ G
(2)
i
+ G
(3)
i
. (109)
17
|(t) =

_
e
i
e
Et

+
_
e
i
e
Et
e
i
e
E

t
E

_
g

1
e
i
e
Et
e
i
e
E
1
t
(E

E
1
)
2
g
1
1
g
1
1

1
_
e
i
e
Et
(E

E
1
) (E

)

e
i
e
E
1
t
(E

E
1
) (E
1
E

)
+
e
i
e
E

t
(E

) (E
1
E

)
_
g
1
1
g
1
1

_
a

+O(H
3
1
). (110)
Note that E
i
can contain the diagonal element h
i
1
of the original H
1
, and we do not obviously write h
i
1
and take
new H
1
matrix as o-diagonal in the H
0
representation.
Because that
|(t) =

e
iET t

, (111)
we have
E
T
a

=

E

_

E

E
1
(E

E
1
)
2
g
1
1
g
1
1

1
_

E

(E

E
1
) (E

E
1
(E

E
1
) (E
1
E

)
+

(E

) (E
1
E

)
_
g
1
1
g
1
1

_
a

. (112)
In the usual perturbation theory, H
1
is taken as a perturbing part with the form
H
1
= v, (113)
where is a real number that is called the perturbing parameter. It must be emphasized that

E
i
can be taken as
explicitly independent perturbing parameter , because we introduce as a formal multiplier after redenition. In
other words,

E
i
has absorbed those terms adding to it and formed a new quantity. This way has been seen in our
Hamiltonian redivision skill. Without loss of generality, we further take H
1
only with the o-diagonal form, that is
H
12
1
= g
12
1
= v
12
. (114)
Then, we expand both the desired expansion coecients a

and the energy eigenvalues E

T
in a power series of
perturbation parameter :
E
T
=

l=0

l
E
(l)
T,I
, a

l=0

l
a
(l)
;I
. (115)
1. Improved 0th approximation
If we set = 0, eq.(112) yields
E
(0)
T,I
a
(0)
;I
=

E

a
(0)
;I
, (116)
where runs over all levels. Actually, let us focus on the level = , then
E
(0)
T,I
=

E

. (117)
When the initial state is taken as

_
,
a
(0)
;I
=

. (118)
Obviously, the improved form of perturbed energy is dierent from the results in the usual perturbative theory because
it absorbs the contributions from the higher order approximations. However, the so-call improved form of perturbed
state is the same as the usual result.
18
2. Improved 1st approximation
Again from eq.(112) it follows that
E
(0)
T,I
a
(1)
;I
+ E
(1)
T,I
a
(0)
;I
=

E

a
(1)
;I
+

a
(0)

;I
. (119)
When = , it is easy to obtain
E
(1)
T,I
= 0. (120)
If = , then
a
(1)
;I
=
1
(E

)
v

. (121)
It is clear that the rst order results are the same as the one in the usual perturbative theory.
3. Improved 2nd approximation
Likewise, the following equation
E
(2)
T,I
a
(0)
;I
+ E
(1)
T,I
a
(1)
;I
+ E
(0)
T,I
a
(2)
;I
=

E

a
(2)
;I
+

a
(1)

;I

1,

E
1
(E

E
1
)
2
v
1
v
1

a
(0)

;I
+

1,

_

E

(E

E
1
) (E

E
1
(E

E
1
) (E
1
E

)
+

(E

) (E
1
E

)
_
v
1
v
1

a
(0)

;I
. (122)
is obtained and it yields
E
(2)
T;I
= 0, (123)
if we take = . When = , we have
a
(2)
;I
=

1
1
(E

) (E
1
E

)
v
1
v
1

. (124)
It is consistent with the o-diagonal part of usual result. In fact, since we have taken H

1
to be o-diagonal, it
does not have a diagonal part. However, we think its form is more appropriate. In addition, we do not consider the
revision part introduced by normalization. While E
(2)
T;I
= 0 is a new result.
4. Summary
Now we can see, up to the improved second order approximation:
E
T,


E

= E

+ G
(2)

+ G
(3)

. (125)
Compared with the usual one, they are consistent at the former two orders. It is not strange since the physical law is
the same. However, our improved form of perturbed energy contains a third order term. In other words, our solution
might be eective in order to obtain the contribution from high order approximations. The possible physical reason
is that a redened form of the solution is obtained.
In special, when we allow the H

1
to have the diagonal elements, the improved second order approximation becomes
E
T,
E

+ h

1
+ G
(2)

+ G
(3)

. (126)
19
Likewise, if we redene

E
i
= E
i
+ h
i
1
+ G
(2)
i
+ G
(3)
i
+ G
(4)
i
. (127)
Thus, only considering the rst order approximation, we can obtain
E
T,
E

+ h

1
+ G
(2)

+ G
(3)

+ G
(4)

. (128)
In fact, the reason is our conjecture in the previous section. The correct form of redened

E
i
should be
E
T,
E

+ h

1
+ G
(2)

+ G
(3)

+ G
(4)

+ G
(5)

+ . (129)
This implies that our improved scheme absorbs the partial even whole signicant contributions from the high order
approximations. In addition, based on the fact that the improved second approximation is actually zero, it is possible
that this implies our solution will fade down more rapidly than the solution in the usual perturbative theory.
Actually, the main advantage of our solution is in dynamical development. The contributions from the high order
approximations play more important roles in the relevant physical problems such as the entanglement dynamics and
decoherence process. For the improved perturbed energy, its high order part has obvious physical meaning. But,
for the improved form of perturbed state, we nd them to be the same as the existed perturbation theory up to the
second approximation.
VI. EXAMPLE AND APPLICATION
In order to concretely illustrate that our exact solution and the improved scheme of perturbation theory are indeed
more eective and more accurate, let us study a simple example: two-state system, which appears in the most of
quantum mechanics textbooks. Its Hamiltonian can be written as
H =
_
E
1
V
12
V
21
E
2
_
, (130)
where we have used the the basis formed by the unperturbed energy eigenvectors, that is

1
_
=
_
1
0
_
,

2
_
=
_
0
1
_
. (131)
In other words:
H
0
|

= E

, ( = 1, 2) (132)
where
H
0
=
_
E
1
0
0 E
2
_
. (133)
Thus, this means the perturbing Hamiltonian is taken as
H
1
=
_
0 V
12
V
21
0
_
. (134)
This two state system has the following eigen equation
H|

= E
T

. (135)
It is easy to obtain its solution: corresponding eigenvectors and eigenvalues

1
_
=
1
_
4|V |
2
+ (
21
+
T
21
)
2
_

21
+
T
21
2V
21
_
, (136)

2
_
=
1
_
4|V |
2
+ (
21

T
21
)
2
_

21

T
21
2V
21
_
; (137)
20
E
T
1
=
1
2
_
E
1
+ E
2

T
21
_
, E
T
2
=
1
2
_
E
1
+ E
2
+
T
21
_
; (138)
where |V | = |V
12
| = |V
21
|,
21
= E
2
E
1
,
T
21
= E
T
2
E
T
1
=
_
4|V |
2
+
2
21
, and we have set E
2
> E
1
without loss of
generality.
Obviously the transition probability from state 1 to state 2 is
P
T
(1 2) =

e
iHt

1
_

2
=

1,2=1

2
|
1

1
|e
iHt
|
2

2
|
1

2
= |V |
2
sin
2
_

T
21
t/2
_
(
T
21
/2)
2
. (139)
In the usual perturbation theory, up to the second order approximation, the well-known perturbed energies are
E
P
1
= E
1

|V |
2

21
, E
P
2
= E
1
+
|V |
2

21
. (140)
While, under the rst order approximation, the transition probability from state 1 to state 2 is
P(1 2) = |V |
2
sin
2
(
21
t/2)
(
21
/2)
2
. (141)
Using our improved scheme, only to the rst order approximation, we get the corresponding perturbed energies

E
1
= E
1

|V |
2

21
+
|V |
4

3
21
,

E
2
= E
1
+
|V |
2

21

|V |
4

3
21
, (142)
where we have used the facts that
G
(2)
1
=
|V |
2

21
= G
(2)
2
, G
(3)
1
= G
(3)
2
= 0, G
(4)
1
=
|V |
4

3
21
= G
(4)
2
. (143)
Obviously, under the rst order approximation, our scheme yields the transition probability from state 1 to state 2 as
P
I
(1 2) = |V |
2
sin
2
(
21
t/2)
(
21
/2)
2
. (144)
where
21
=

E
2

E
1
. Therefore we can say our scheme is more eective. Moreover, we notice that
E
T
1
= E
1

|V |
2

21
+
|V |
4

3
21
+O(|V |
6
) =

E
1
+O(|V |
6
) = E
P
1
+
|V |
4

3
21
+O(|V |
6
), (145)

E
T
2
= E
1
+
|V |
2

21

|V |
4

3
21
+O(|V |
6
) =

E
2
+O(|V |
6
) = E
P
2

|V |
4

3
21
+O(|V |
6
). (146)
and
P
T
(1 2) = |V |
2
sin
2
(
21
t/2)
(
21
/2)
2
+|V |
2
_
sin (
21
t)
2(
21
/2)
2

sin
2
(
21
t/2)
(
21
/2)
3
_
(
21

21
) +O[(
21

21
)
2
] (147)
= P
I
(1 2) |V |
2
sin
2
(
21
t/2)
(
21
/2)
3
(
21

21
) +O[(
21

21
)
2
] (148)
= P(1 2) +|V |
2
_
sin (
21
t)
2(
21
/2)
2

sin
2
(
21
t/2)
(
21
/2)
3
_
(
21

21
) +O[(
21

21
)
2
]. (149)
Therefore, we can say that our scheme is more accurate.
VII. DISCUSSION AND CONCLUSION
In this paper, our improved scheme of perturbation theory is proposed based on our exact solution in general
quantum systems [3]. Because our exact solution has a general term that is a c-number function and proportional
to power of the perturbing Hamiltonian, this provides the probability considering the partial contributions from the
high order even all order approximations. While our dynamical rearrangement and summation method helps us to
21
realize this probability. Just as the contributions from the high order even all order approximations are absorbed to
the lower order approximations, our scheme becomes an improved one.
It must be emphasized that our improved scheme of perturbation theory is proposed largely dependent on the facts
that we nd and develop a series of skills and methods. From that the Hamiltonian redivision skill overcomes the
aw in the usual perturbation theory, improves the calculation precision, extends the applicable range and removes
the possible degeneracies to that the perturbing Hamiltonian matrix product decomposition method separates the
contraction terms and anti-contraction terms, eliminates the apparent divergences in the power series of the perturbing
Hamiltonian and provides the groundwork of dynamical rearrangement and summation, we have seen these ideas,
skills and methods to be very useful.
Actually, the start point that delays to introduce the perturbing parameter as possible plays an important even key
role in our improved scheme of perturbation theory. It enlightens us to seek for the above skills and methods.
From our exact solution transferring to our improved scheme of perturbation theory we does not directly use the
cut-o approximation, but rst deals with the power series of perturbation so that the contributions from some high
order even all order approximations can be absorbed into the lower orders than the cut-o order as possible. Hence, our
improved scheme of perturbation theory is physically reasonable, mathematically clear and methodologically skillful.
This provides the guarantee achieving high eciency and high precision. Through nding the improved forms of
perturbed solutions of dynamics, we generally demonstrate this conclusion. Furthermore, we prove the correctness
of this conclusion via calculating the improved form of transition probability, perturbed energy and perturbed state.
Specially, we obtain the revised Fermis golden rule. Moreover, we illustrate the advantages of our improved scheme
in an easy understanding example of two-state system. All of this implies the physical reasons and evidences why our
improved scheme of perturbation theory is actually calculable, operationally ecient, conclusively more accurate.
From the features of our improved scheme, we believe that it will have interesting applications in the calculation of
entanglement dynamics and decoherence process as well as the other physical quantities dependent on the expanding
coecients.
In fact, a given lower order approximation of improved form of the perturbed solution absorbing the partial con-
tributions from the higher order even all order approximations is obtained by our dynamical rearrangement and
summation method, just like Fynmann gures summation that has been done in the quantum eld theory. It
is emphasized that these contributions have to be signicant in physics. Considering time evolution form is our
physical ideas and absorbing the high order approximations with the factors t
a
e
iE
i
t
, (a = 1, 2, ) to the improved
lower order approximations denitely can advance the precision. Therefore, using our dynamical rearrangement and
summation method is appropriate and reasonable in our view.
For a concrete example, except for some technological and calculational works, it needs the extensive physical
background knowledge to account for the signicance of related results. That is, since the dierences of the related
conclusions between our improved scheme and the usual perturbation theory are in high order approximation parts,
we have to study the revisions (dierences) to nd out whether they are important or unimportant to the studied
problems. In addition, our conjecture about the perturbed energy is based on physical symmetry and mathematical
consideration, it is still open at the strict sense. As to the degenerate cases including specially, vanishing the o-
diagonal element of the perturbing Hamiltonian matrix between any two degenerate levels, we have discussed how
to deal with them, except for the very complicated cases that the degeneracy can not be completely removed by
the diagonalization of the degenerate subspaces trick and the Hamiltonian redivision skill as well as the o-diagonal
element of the perturbing Hamiltonian matrix between any two degenerate levels are not vanishing when the remained
degeneracies are allowed.
It must be emphasized that the study on the time evolution operator plays a central role in quantum dynamics
and perturbation theory. Because of the universal signicance of our general and explicit expression of the time
evolution operator, we wish that it will have more applications in quantum theory. Besides the above studies through
the perturbative method, it is more interesting to apply our exact solution to the formalization study of quantum
dynamics in order to further and more powerfully show the advantages of our exact solution.
In summary, our results can be thought of as theoretical developments of perturbation theory, and they are helpful
for understanding the theory of quantum mechanics and providing some powerful tools for the calculation of physical
quantities in general quantum systems. Together with our exact solution [3] and open system dynamics [6], they
can nally form the foundation of theoretical formulism of quantum mechanics in general quantum systems. Further
study on quantum mechanics of general quantum systems is on progressing.
Acknowledgments
We are grateful all the collaborators of our quantum theory group in the Institute for Theoretical Physics of our
university. This work was funded by the National Fundamental Research Program of China under No. 2001CB309310,
22
and partially supported by the National Natural Science Foundation of China under Grant No. 60573008.
APPENDIX A: THE CALCULATIONS OF THE HIGH ORDER TERMS
Since we have taken the H
1
only with the o-diagonal part, it is enough to calculate the contributions from them. In
Sec. VI the contributions from the rst, second and third order approximations have been given. In this appendix, we
would like to nd the contributions from the fourth to the sixth order approximations. The calculational technologies
used by us are mainly to the limit process, dummy index changing and summation, as well as the replacement
g
ij
1

ij
= g
ij
1
since g
ij
1
has been o-diagonal. These calculations are not dicult, but are a little lengthy.
1. l = 4 case
For the fourth order approximation, its contributions from the rst decompositions consists of eight terms:
A

4
= A

4
(ccc) + A

4
(ccn) + A

4
(cnc) + A

4
(ncc)
+A

4
(cnn) + A

4
(ncn) + A

4
(nnc) + A

4
(nnn). (A1)
Its the former four terms have no the nontrivial second contractions, and its the fth and seven terms have one
nontrivial second contraction as follows
A

4
(cnn) = A

4
(cnn, kc) + A

4
(cnn; kn), (A2)
A

4
(ncn) = A

4
(ncn, c) + A

4
(ncn, n), (A3)
A

4
(nnc) = A

4
(nnc, ck) + A

4
(nnc, nk). (A4)
In addition, the last term in eq.(A1) has two nontrivial second contractions, and its fourth term has also the third
contraction. Hence
A

4
(nnn) = A

4
(nnn, cc) + A

4
(nnn, cn)A

4
(nnn, nc) + A

4
(nnn, nn), (A5)
A

4
(nnn, nn) = A

4
(nnn, nn, c) + A

4
(nnn, nn, n). (A6)
All together, we have the fteen terms that are the contributions from whole contractions and anti-contractions of
the fourth order approximation.
First, let us calculate the former four terms only with the rst contractions and anti-contractions, that is, with
more than two functions (or less than two functions)
A

4
(ccc) =

1, ,5
_
5

i=1
(1)
i1
e
iE
i
t
d
i
(E[, 4])
_
_
_
4

j=1
g
jj+1
1
_
_
_
3

k=1

k+2
_

1
_
3e
iEt
(E

E
1
)
4

3e
iE
1
t
(E

E
1
)
4
(it)
2e
iEt
(E

E
1
)
3
(it)
e
iE
1
t
(E

E
1
)
3
+
(it)
2
2
e
iEt
(E

E
1
)
2
_
|g
1
1
|
4

. (A7)
A

4
(ccn) =

1, ,5
_
5

i=1
(1)
i1
e
iE
i
t
d
i
(E[, 4])
_
_
_
4

j=1
g
jj+1
1
_
_

13

24

35

1
_

e
iEt
(E

E
1
)
2
(E

)
2

2e
iEt
(E

E
1
)
3
(E

e
iE
1
t
(E

E
1
)
2
(E
1
E

)
2
+
2e
iE
1
t
(E

E
1
)
3
(E
1
E

)
+
e
iE

t
(E

)
2
(E
1
E

)
2
+ (it)
e
iEt
(E

E
1
)
2
(E

)
+(it)
e
iE
1
t
(E

E
1
)
2
(E
1
E

)
_
|g
1
1
|
2
g
1
1
g
1

. (A8)
23
A

4
(cnc) =

1, ,5
_
5

i=1
(1)
i1
e
iE
i
t
d
i
(E[, 4])
_
_
_
4

j=1
g
jj+1
1
_
_

13

24

35

1,2
_
e
iEt
(E

E
1
) (E

E
2
)
3
+
e
iEt
(E

E
1
)
2
(E

E
2
)
2
+
e
iEt
(E

E
1
)
3
(E

E
2
)

e
iE
1
t
(E

E
1
)
3
(E
1
E
2
)
+
e
iE
2
t
(E

E
2
)
3
(E
1
E
2
)
(it)
e
iEt
(E

E
1
) (E

E
2
)
2
(it)
e
iEt
(E

E
1
)
2
(E

E
2
)
+
(it)
2
2
e
iEt
(E

E
1
) (E

E
2
)
_
|g
1
1
|
2
|g
2
1
|
2

12

. (A9)
A

4
(ncc) =

1, ,5
_
5

i=1
(1)
i1
e
iE
i
t
d
i
(E[, 4])
_
_
_
4

j=1
g
jj+1
1
_
_

13

24

35

1
_
e
iEt
(E

E
1
)
2
(E

)
2
+
2e
iE
1
t
(E

E
1
) (E
1
E

)
3

e
iE
1
t
(E

E
1
)
2
(E
1
E

)
2

2e
iE

t
(E

) (E
1
E

)
3

e
iE

t
(E

)
2
(E
1
E

)
2
(it)
e
iE
1
t
(E

E
1
) (E
1
E

)
2
(it)
e
iE

t
(E

) (E
1
E

)
2
_

g
1

2
g
1
1
g
1

. (A10)
Then, we calculate the three terms with the single rst contraction, that is, with one function. Because one
function can not eliminate the whole apparent singularity, we also need to nd out the nontrivial second contraction-
and/or anti-contraction terms.
A

4
(cnn, kc) =

1, ,5
_
5

i=1
(1)
i1
e
iE
i
t
d
i
(E[, 4])
_
_
_
4

j=1
g
jj+1
1
_
_

13

24

35

2

1

1
_

2e
iEt
(E

E
1
) (E

)
3

e
iEt
(E

E
1
)
2
(E

)
2
+
e
iE
1
t
(E

E
1
)
2
(E
1
E

)
2

e
iE

t
(E

)
2
(E
1
E

)
2

2e
iE

t
(E

)
3
(E
1
E

)
+ (it)
e
iEt
(E

E
1
) (E

)
2
(it)
e
iE

t
(E

)
2
(E
1
E

)
_

2
g
1
1
g
1

1
. (A11)
A

4
(cnn, kn) =

1, ,5
_
5

i=1
(1)
i1
e
iE
i
t
d
i
(E[, 4])
_
_
_
4

j=1
g
jj+1
1
_
_

13

24

35

2

1

1,2
_

e
iEt
(E

E
1
) (E

E
2
) (E

)
2

e
iEt
(E

E
1
) (E

E
2
)
2
(E

e
iEt
(E

E
1
)
2
(E

E
2
) (E

)
+
e
iE
1
t
(E

E
1
)
2
(E
1
E
2
) (E
1
E

e
iE
2
t
(E

E
2
)
2
(E
1
E
2
) (E
2
E

)
+
e
iE

t
(E

)
2
(E
1
E

) (E
2
E

)
+(it)
e
iEt
(E

E
1
) (E

E
2
) (E

)
_
|g
1
1
|
2
g
2
1
g
2

1

12

. (A12)
24
A

4
(ncn, c) =

1, ,5
_
5

i=1
(1)
i1
e
iE
i
t
d
i
(E[, 4])
_
_
_
4

j=1
g
jj+1
1
_
_

13

24

35

1,2
_

e
iEt
(E

E
1
)
2
(E

E
2
)
2

2e
iEt
(E

E
1
)
3
(E

E
2
)

e
iE
1
t
(E

E
1
)
2
(E
1
E
2
)
2
+
2e
iE
1
t
(E

E
1
)
3
(E
1
E
2
)
+
e
iE
2
t
(E

E
2
)
2
(E
1
E
2
)
2
+ (it)
e
iEt
(E

E
1
)
2
(E

E
2
)
+(it)
e
iE
1
t
(E

E
1
)
2
(E
1
E
2
)
_
|g
1
1
|
2
|g
12
1
|
2

. (A13)
A

4
(ncn, n) =

1, ,5
_
5

i=1
(1)
i1
e
iE
i
t
d
i
(E[, 4])
_
_
_
4

j=1
g
jj+1
1
_
_

13

24

35

1,2
_
e
iEt
(E

E
1
)
2
(E

E
2
) (E

)
+
e
iE
1
t
(E

E
1
) (E
1
E
2
) (E
1
E

)
2
+
e
iE
1
t
(E

E
1
) (E
1
E
2
)
2
(E
1
E

e
iE
1
t
(E

E
1
)
2
(E
1
E
2
) (E
1
E

e
iE
2
t
(E

E
2
) (E
1
E
2
)
2
(E
2
E

)
+
e
iE

t
(E

) (E
1
E

)
2
(E
2
E

)
(it)
e
iE
1
t
(E

E
1
) (E
1
E
2
) (E
1
E

)
_
|g
12
1
|
2
g
1
1
g
1

1

2

. (A14)
A

4
(nnc, ck) =

1, ,5
_
5

i=1
(1)
i1
e
iE
i
t
d
i
(E[, 4])
_
_
_
4

j=1
g
jj+1
1
_
_

13

24

35

14

1
_

2e
iEt
(E

E
1
) (E

)
3

e
iEt
(E

E
1
)
2
(E

)
2
+
e
iE
1
t
(E

E
1
)
2
(E
1
E

)
2

e
iE

t
(E

)
2
(E
1
E

)
2

2e
iE

t
(E

)
3
(E
1
E

)
+ (it)
e
iEt
(E

E
1
) (E

)
2
(it)
e
iE

t
(E

)
2
(E
1
E

)
_

2
g
1
1
g
1

1
. (A15)
A

4
(nnc, nk) =

1, ,5
_
5

i=1
(1)
i1
e
iE
i
t
d
i
(E[, 4])
_
_
_
4

j=1
g
jj+1
1
_
_

13

24

35

14

1,2
_
e
iEt
(E

E
1
) (E

E
2
) (E

)
2

e
iE
1
t
(E

E
1
) (E
1
E
2
) (E
1
E

)
2
+
e
iE
2
t
(E

E
2
) (E
1
E
2
) (E
2
E

)
2

e
iE

t
(E

) (E
1
E

) (E
2
E

)
2

e
iE

t
(E

) (E
1
E

)
2
(E
2
E

e
iE

t
(E

)
2
(E
1
E

) (E
2
E

)
(it)
e
iE

t
(E

) (E
1
E

) (E
2
E

)
_

g
2

2
g
1
1
g
1

1

2

12

. (A16)
Finally, we calculate the A

4
(nnn) by considering the two second decompositions, that is, its former three terms
25
A

4
(nnn, cc) =

1, ,5
_
5

i=1
(1)
i1
e
iE
i
t
d
i
(E[, 4])
_
_
_
4

j=1
g
jj+1
1
_
_
_
3

k=1

k+2
_

14

25

1
_

2e
iEt
(E

E
1
) (E

)
3

e
iEt
(E

E
1
)
2
(E

)
2
+
e
iE
1
t
(E

E
1
)
2
(E
1
E

)
2

e
iE

t
(E

)
2
(E
1
E

)
2

2e
iE

t
(E

)
3
(E
1
E

)
+ (it)
e
iEt
(E

E
1
) (E

)
2
(it)
e
iE

t
(E

)
2
(E
1
E

)
_
g

1
g

1
1
g
1
1
g

1
. (A17)
A

4
(nnn, cn) =

1, ,5
_
5

i=1
(1)
i1
e
iE
i
t
d
i
(E[, 4])
_
_
_
4

j=1
g
jj+1
1
_
_
_
3

k=1

k+2
_

14

25

1,2
_

e
iEt
(E

E
1
) (E

E
2
) (E

)
2

e
iEt
(E

E
1
) (E

E
2
)
2
(E

e
iEt
(E

E
1
)
2
(E

E
2
) (E

)
+
e
iE
1
t
(E

E
1
)
2
(E
1
E
2
) (E
1
E

e
iE
2
t
(E

E
2
)
2
(E
1
E
2
) (E
2
E

)
+
e
iE

t
(E

)
2
(E
1
E

) (E
2
E

)
+(it)
e
iEt
(E

E
1
) (E

E
2
) (E

)
_
g
1
1
g
12
1
g
2
1
g

1

1

2
. (A18)
A

4
(nnn, nc) =

1, ,5
_
5

i=1
(1)
i1
e
iE
i
t
d
i
(E[, 4])
_
_
_
4

j=1
g
jj+1
1
_
_
_
3

k=1

k+2
_

14

25

1,2
_
e
iEt
(E

E
1
) (E

E
2
) (E

)
2

e
iE
1
t
(E

E
1
) (E
1
E
2
) (E
1
E

)
2
+
e
iE
2
t
(E

E
2
) (E
1
E
2
) (E
2
E

)
2

e
iE

t
(E

) (E
1
E

) (E
2
E

)
2

e
iE

t
(E

) (E
1
E

)
2
(E
2
E

e
iE

t
(E

)
2
(E
1
E

) (E
2
E

)
(it)
e
iE

t
(E

) (E
1
E

) (E
2
E

)
_
g

1
g

1
1
g
12
1
g
2

1

1

2
. (A19)
while the fourth term has the third decomposition, that is
A

4
(nnn, nn, c) =

1, ,5
_
5

i=1
(1)
i1
e
iE
i
t
d
i
(E[, 4])
_
_
_
4

j=1
g
jj+1
1
_
_
_
3

k=1

k+2
_

1,2,3
_

e
iEt
(E

E
1
) (E

E
2
) (E

E
3
)
2

e
iEt
(E

E
1
) (E

E
2
)
2
(E

E
3
)

e
iEt
(E

E
1
)
2
(E

E
2
) (E

E
3
)
+
e
iE
1
t
(E

E
1
)
2
(E
1
E
2
) (E
1
E
3
)

e
iE
2
t
(E

E
2
)
2
(E
1
E
2
) (E
2
E
3
)
+
e
iE
3
t
(E

E
3
)
2
(E
1
E
3
) (E
2
E
3
)
+(it)
e
iEt
(E

E
1
) (E

E
2
) (E

E
3
)
_
g
1
1
g
12
1
g
23
1
g
3
1

2

13

. (A20)
26
A

4
(nnn, nn, n) =

1, ,5
_
5

i=1
(1)
i1
e
iE
i
t
d
i
(E[, 4])
_
_
_
4

j=1
g
jj+1
1
_
_
_
3

k=1

k+2
_

1,2,3
_
e
iEt
(E

E
1
) (E

E
2
) (E

E
3
) (E

e
iE
1
t
(E

E
1
) (E
1
E
2
) (E
1
E
3
) (E
1
E

)
+
e
iE
2
t
(E

E
2
) (E
1
E
2
) (E
2
E
3
) (E
2
E

e
iE
3
t
(E

E
3
) (E
1
E
3
) (E
2
E
3
) (E
3
E

)
+
e
iE

t
(E

) (E
1
E

) (E
2
E

) (E
3
E

)
_
g
1
1
g
12
1
g
23
1
g
3

1

2

13

1

2
. (A21)
Now, all 15 contractions and/or anti-contractions in the fourth order approximation have been calculated out.
In order to absorb the contributions from the fourth order approximation to the improved forms of lower order
perturbed solutions, we rst decompose A

4
, which is a summation of all above terms, into the three parts according
to their factor forms in e
iE
i
t
, (it)e
iE
i
t
and (it)
2
e
iE
i
t
/2, that is
A

4
= A

4
(e) + A

4
(te) + A

4
(t
2
e). (A22)
Secondly, we decompose its every term into three parts according to the factor forms in e
iEt
, e
iE
1
t
(

1
e
iE
1
t
)
and e
iE

t
, that is
A

4
(e) = A

4
(e
iEt
) + A

4
(e
iE
1
t
) + A

4
(e
iE

t
), (A23)
A

4
(te) = A

4
(te
iEt
) + A

4
(te
iE
1
t
) + A

4
(te
iE

t
), (A24)
A

4
(t
2
e) = A

4
(t
2
e
iEt
) + A

4
(t
2
e
iE
1
t
) + A

4
(t
2
e
iE

t
). (A25)
Finally, we again decompose every term in the above equations into the diagonal and o-diagonal parts about and

, that is
A

4
(e
iE
i
t
) = A

4
(e
iE
i
t
; D) + A

4
(e
iE
i
t
; N), (A26)
A

4
(te
iE
i
t
) = A

4
(te
iE
i
t
; D) + A

4
(te
iE
i
t
; N), (A27)
A

4
(t
2
e
iE
i
t
) = A

4
(t
2
e
iE
i
t
; D) + A

4
(t
2
e
iE
i
t
; N). (A28)
where E
i
takes E

, E
1
and E

.
If we do not concern the improved forms of perturbed solutions equal to or higer than the fourth order one, we
only need to write down the second and third terms in eq.(A22) and calculate their diagonal and o-diagonal parts
respectively. Based on the calculated results above, it is easy to obtain
A

4
_
te
iEt
; D
_
= (it) e
iEt
_

1
2 |g
1
1
|
4
(E

E
1
)
3

1,2
|g
1
1
|
2
|g
2
1
|
2

12
(E

E
1
) (E

E
2
)
2

1,2
|g
1
1
|
2
|g
2
1
|
2

12
(E

E
1
)
2
(E

E
2
)
+

1,2
|g
1
1
|
2
|g
12
1
|
2

2
(E

E
1
)
2
(E

E
2
)
+

1,2,3
g
1
1
g
12
1
g
23
1
g
3
1

2

13
(E

E
1
) (E

E
2
) (E

E
3
)
_

. (A29)
Substituting the relation
12
= 1
12
, using the technology of index exchanging and introducing the denitions
of so-called the ath revision energy G
(a)

:
G
(2)

1
|g
1
1
|
2
E

E
1
(A30)
27
G
(4)

1,2,3
g
1
1
g
12
1
g
23
1
g
3
1

2
(E

E
1
) (E

E
2
) (E

E
3
)

1,2
g
1
1
g
1
1
g
2
1
g
2
1
(E

E
1
)
2
(E

E
2
)
, (A31)
we can simplify Eq. (A29) to the following concise form:
A

4
_
te
iEt
; D
_
= (it) e
iEt
_

1
G
(2)

(E

E
1
)
2
|g
1
1
|
2
G
(4)

. (A32)
Similar calculation and simplication lead to
A

4
_
te
iEt
; N
_
= (it)e
iEt
_
G
(3)

1
(E

)
+

1
G
(2)

(E

E
1
) (E

)
_
, (A33)
where
G
(3)

1,2
g
1
1
g
12
1
g
2
1
(E

E
1
) (E

E
2
)
. (A34)
For saving the space, the corresponding detail is omitted. In fact, it is not dicult, but it is necessary to be careful
enough, specially in the cases of higher order approximations.
In the same way, we can obtain:
A

4
_
te
iE
1
t
; D
_
= (it)

1
G
(2)
1
e
iE
1
t
(E

E
1
)
2
|g
1
1
|
2

, (A35)
A

4
_
te
iE
1
t
; N
_
= (it)

1
G
(2)
1
e
iE
1
t
(E

E
1
) (E
1
E

)
g
1
1
g
1

. (A36)
A

4
_
te
iE

t
; D
_
= 0, (A37)
A

4
_
te
iE

t
; N
_
= (it)e
iE

t
_

1
G
(3)

(E

)
g

1
+

1
G
(2)

(E

) (E
1
E

)
g
1
1
g
1

_
. (A38)
For the terms with the factor t
2
e , only one is nonzero, that is
A

4
_
t
2
e
_
= A

4
_
t
2
e
iEt
; D
_
=
(iG
(2)

t)
2
2!
e
iEt
. (A39)
since
A

4
_
t
2
e
iE
1
t
; D
_
= A

4
_
t
2
e
iE

t
; D
_
= 0, (A40)
A

4
_
t
2
e
iEt
; N
_
= A

4
_
t
2
e
iE
1
t
; D
_
= A

4
_
t
2
e
iE

t
; D
_
= 0. (A41)
We can see that these terms can be absorbed into (or merged with) the lower order approximations to obtain the
improved forms of perturbed solutions.
2. l=5 case
Now let we consider the case of the fth order approximation (l = 5). From eq.(42) it follows that the rst
decompositions of g-product have 2
4
= 16 terms. They can be divided into 5 groups
A

5
=
4

i=0
A

5
(i; ), (A42)
28
where i indicates the number of functions. Obviously
A

5
(0; ) = A

5
(cccc), (A43)
A

5
(1; ) = A

5
(cccn) + A

5
(ccnc) + A

5
(cncc) + A

5
(nccc), (A44)
A

5
(2; ) = A

5
(ccnn) + A

5
(cncn) + A

5
(cnnc)
+A

5
(nccn) + A

5
(ncnc) + A

5
(nncc), (A45)
A

5
(3; ) = A

5
(cnnn) + A

5
(ncnn) + A

5
(nncn) + A

5
(nnnc), (A46)
A

5
(4; ) = A

5
(nnnn). (A47)
Here, we have used the notations stated in Sec. VI.
By calculation, we obtain the A

5
(0, ) and every term of A

5
(1, ) have only nontrivial rst contractions and/or
anti-contractions. But, we can nd that every term of A

5
(2, ) can have one nontrivial second or third or fourth
contraction or anti-contraction, that is
A

5
(ccnn) = A

5
(ccnn, kkc) + A

5
(ccnn, kkn), (A48)
A

5
(cncn) = A

5
(cncn, kc) + A

5
(cncn, kc), (A49)
A

5
(cnnc) = A

5
(cnnc, kck) + A

5
(cnnc, kck), (A50)
A

5
(nccn) = A

5
(nccn, c) + A

5
(nccn, n), (A51)
A

5
(ncnc) = A

5
(ncnc, ck) + A

5
(ncnc, nk), (A52)
A

5
(nncc) = A

5
(nncc, ckk) + A

5
(nncc, nkk). (A53)
Similarly, every term of A

5
(3, ) can have two higher order contractions and/or anti-contractions:
A

5
(cnnn) = A

5
(cnnn, kcc) + A

5
(cnnn, kcn)
+A

5
(cnnn, knc) + A

5
(cnnn, knn), (A54)
A

5
(ncnn) = A

5
(ncnn, kkc, ck) + A

5
(ncnn, kkn, ck)
+A

5
(ncnn, kkc, nk) + A

5
(ncnn, kkn, nk), (A55)
A

5
(nncn) = A

5
(nncn, ckk, kc) + A

5
(nncn, ckk, kn)
+A

5
(nncn, nkk, kc) + A

5
(nncn, nkk, kn), (A56)
A

5
(nnnc) = A

5
(nnnc, cck) + A

5
(nnnc, cnk)
+A

5
(nnnc, nck) + A

5
(nnnc, nnk). (A57)
Moreover, their last terms, with two higher order anti-contractions, can have one nontrivial more higher contraction
or anti-contraction:
A

5
(cnnn, knn) = A

5
(cnnn, knn, kc) + A

5
(cnnn, knn.kn), (A58)
A

5
(ncnn, kkn, nk) = A

5
(ncnn, kkn, nk, c) + A

5
(ncnn, kkn, nk, n), (A59)
A

5
(nncn, nkk, kn) = A

5
(nncn, nkk, kn, c) + A

5
(nncn, nkk, kn, n), (A60)
A

5
(nnnc, nnk) = A

5
(nnnc, nnk, ck) + A

5
(nnnc, nnk, nk). (A61)
In the case of A

5
(nnnn), there are three terms corresponding to the second decompositions that result in
A

5
(nnnn) = A

5
(nnnn, ccc) + A

5
(nnnn, ccn) + A

5
(nnnn, cnc) + A

5
(nnnn, ncc)
+A

5
(nnnn, cnn) + A

5
(nnnn, ncn) + A

5
(nnnn, nnc) + A

5
(nnnn, nnn). (A62)
29
In the above expression, from the fth term to the seventh term have the third- or fourth- contraction and anti-
contraction, the eighth term has two third contractions and anti-contractions:
A

5
(nnnn, cnn) = A

5
(nnnn, cnn, kc) + A

5
(nnnn, cnn, kn), (A63)
A

5
(nnnn, ncn) = A

5
(nnnn, ncn, c) + A

5
(nnnn, ncn, n), (A64)
A

5
(nnnn, nnc) = A

5
(nnnn, nnc, ck) + A

5
(nnnn, nnc, nk), (A65)
A

5
(nnnn, nnn) = A

5
(nnnn, nnn, cc) + A

5
(nnnn, nnn, cn)
+A

5
(nnnn, nnn, nc) + A

5
(nnnn, nnn, nn). (A66)
In addition, A

5
(nnnn, nnn, nn) consists of the fourth contraction and anti-contraction
A

5
(nnnn, nnn, nn) = A

5
(nnnn, nnn, nn, c) + A

5
(nnnn, nnn, nn, n). (A67)
According to the above analysis, we obtain that the contribution from the ve order approximation is made of 52
terms after nding out all of contractions and anti-contractions.
Just like we have done in the l = 4 case, we decompose
A

5
= A

5
(e) + A

5
(te) + A

5
(t
2
e), (A68)
where
A

5
(e) = A

5
(e
iEt
) + A

5
(e
iE
1
t
) + A

5
(e
iE
2
t
) + A

5
(e
iE

t
), (A69)
A

4
(te) = A

5
(te
iEt
) + A

5
(te
iE
1
t
) + A

5
(te
iE
2
t
) + A

5
(te
iE

t
), (A70)
A

5
(t
2
e) = A

5
(t
2
e
iEt
) + A

5
(t
2
e
iE
1
t
) + A

5
(t
2
e
iE
2
t
) + A

5
(t
2
e
iE

t
). (A71)
While, every term in the above equations has its diagonal and o-diagonal parts about and

, that is
A

5
(e
iE
i
t
) = A

5
(e
iE
i
t
; D) + A

5
(e
iE
i
t
; N), (A72)
A

5
(te
iE
i
t
) = A

5
(te
iE
i
t
; D) + A

5
(te
iE
i
t
; N), (A73)
A

5
(t
2
e
iE
i
t
) = A

5
(t
2
e
iE
i
t
; D) + A

5
(t
2
e
iE
i
t
; N). (A74)
where E
i
takes E

, E
1
, E
2
and E

.
If we do not concern the improved forms of perturbed solution higher than the fourth order one, we only need to write
down the second and third terms in eq.(A68). We can calculate them and the results are put in the supplementary
of Ref. [10].
Based on these contraction- and anti contraction- expressions, we can, in terms of the rearrangement and summation,
obtain
A
5
(te
iEt
, D) = (iG
(3)

t)

1
e
iEt
(E

E
1
)
2
g
1
1
g
1
1

(iG
(2)

t)

1,2
_
e
iEt
(E

E
1
)
2
(E

E
2
)
+
e
iEt
(E

E
1
) (E

E
2
)
2
_
g
1
1
g
12
1
g
2
1

+ (iG
(5)

t)

(A75)
where
G
(5)

1,2,3,4
g
1
1
g
12
1
g
23
1
g
34
1
g
4
1

2

3
(E

E
1
) (E

E
2
) (E

E
3
) (E

E
4
)

1,2,3
_
g
1
1
g
2
1
g
1
1
g
23
1
g
3
1
(E

E
1
)
2
(E

E
2
) (E

E
3
)
+
g
1
1
g
2
1
g
1
1
g
23
1
g
3
1
(E

E
1
) (E

E
2
)
2
(E

E
3
)
+
g
1
1
g
2
1
g
1
1
g
23
1
g
3
1
(E

E
1
) (E

E
2
) (E

E
3
)
2
_
. (A76)
30
A
5
(te
iE
1
t
, D) = (iG
(3)

t)

1
e
iE
1
t
(E

E
1
)
2
g
1
1
g
1
1

+(iG
(2)

t)

1,2
e
iE
1
t
(E

E
1
)
2
(E
1
E
2
)
g
1
1
g
12
1
g
2
1

(A77)
A
5
(te
iE
2
t
, D) = (iG
(2)

t)

1,2
e
iE
2
t
(E

E
2
)
2
(E
1
E
2
)
g
1
1
g
12
1
g
2
1

(A78)
A
5
(te
iE

t
, D) = 0 (A79)
A
5
(te
iEt
, N) = (iG
(4)

t)
e
iEt
g

1
(E

)
+ (iG
(3)

t)

1
e
iEt
g
1
1
g
1

(E

E
1
) (E

)
(iG
(2)

t)

1
_
e
iEt
g
1
1
g
1
1
g

1
(E

E
1
)
2
(E

)
+
e
iEt
g
1
1
g
1
1
g

1
(E

E
1
) (E

)
2
_
+(iG
(2)

t)

1,2
e
iEt
g
1
1
g
12
1
g
2

1

2

(E

E
1
) (E

E
2
) (E

)
(A80)
A
5
(te
iE
1
t
, N) = (it)

1
G
(3)
1
e
iE
1
t
g
1
1
g
1

(E

E
1
) (E
1
E

)
(it)

1,2
G
(2)
1
e
iE
1
t
g
1
1
g
12
1
g
2

1

1

(E

E
1
) (E
1
E
2
) (E
1
E

)
(A81)
A
5
(te
iE
2
t
, N) = (it)

1,2
G
(2)
2
e
iE
2
t
g
1
1
g
12
1
g
2

1

2

(E

E
2
) (E
1
E
2
) (E
2
E

)
(A82)
A
5
(te
iE

t
, N) = (iG
(4)

t)
e
iE

t
g

1
(E

)
+ (iG
(3)

t)

1
e
iE

t
g
1
1
g
1

(E

) (E
1
E

)
+(iG
(2)

t)

1
_
e
iE

t
g

1
1
g
1

1
g

1
(E

)
2
(E
1
E

)
+
e
iE

t
g

1
1
g
1

1
g

1
(E

) (E
1
E

)
2
_
(iG
(2)

t)

1,2
e
iE

t
g
1
1
g
12
1
g
2

1

1

(E

) (E
1
E

) (E
2
E

)
(A83)
For the parts with t
2
e, we have
A
5
(t
2
e
iEt
, D) =
(it)
2
2!
2G
(2)

G
(3)

e
iEt
, (A84)
A
5
(t
2
e
iE
1
t
, D) = A
5
(t
2
e
iE
2
t
, D) = A
5
(t
2
e
iE

t
, D) = 0. (A85)
A
5
(t
2
e
iEt
, N) =
(it)
2
2!
_
G
(2)

_
2
e
iEt
E

1
, (A86)
A
5
(t
2
e
iE
2
t
, N) = A
5
(t
2
e
iE

t
, N) = 0, (A87)
A
5
(t
2
e
iE

t
, N) =
(it)
2
2!
_
G
(2)

_
2
e
iE

t
E

1
. (A88)
It is clear that the above diagonal and o-diagonal part about A

5
(te) and A

5
(te) indeed has the expected
forms and can be absorbed reasonably into the lower order approximations in order to obtain the improved forms of
perturbed solutions.
31
3. l = 6 case
Now let we consider the case of the sixth order approximation (l = 6). From eq.(42) it follows that the rst
decompositions of g-product have 2
5
= 32 terms. Like the l = 5 case, they can be divided into 6 groups
A

6
=
4

i=0
A

6
(i; ), (A89)
where i indicates the number of functions. Obviously
A

6
(0; ) = A

6
(ccccc), (A90)
A

6
(1; ) = A

6
(ccccn) + A

6
(cccnc) + A

6
(ccncc)
+A

6
(cnccc) + A

6
(ncccc), (A91)
A

6
(2; ) = A

6
(cccn) + A

6
(ccncn) + A

6
(cnccn) + A

6
(ncccn)
+A

6
(ccnnc) + A

6
(cncnc) + A

6
(nccnc) + A

6
(cnncc)
+A

6
(ncncc) + A

6
(nnccc), (A92)
A

6
(3; ) = A

6
(ccnnn) + A

6
(cncnn) + A

6
(cnncn) + A

6
(cnnnc)
+A

6
(nccnn) + A

6
(ncncn) + A

6
(ncnnc) + A

6
(nnccn)
+A

6
(nncnc) + A

6
(nnncc), (A93)
A

6
(4; ) = A

6
(cnnnn) + A

6
(ncnnn) + A

6
(nncnn)
+A

6
(nnncn) + A

6
(nnnnc) (A94)
A

6
(5; ) = A

6
(nnnnn). (A95)
Furthermore considering the high order contraction or anti-contraction, we have
A

6
(cccnn) = A

6
(cccnn, kkkc) + A

6
(cccnn, kkkn), (A96)
A

6
(ccncn) = A

6
(ccncn, kkc) + A

6
(ccncn, kkn), (A97)
A

6
(ccnnc) = A

6
(ccnnc, kkck) + A

6
(ccnnc, kknk), (A98)
A

6
(cnccn) = A

6
(cnccn, kc) + A

6
(cnccn, kn), (A99)
A

6
(cncnc) = A

6
(cncnc, kck) + A

6
(cncnc, knk), (A100)
A

6
(cnncc) = A

6
(cnncc, kckk) + A

6
(cnncc, knkk), (A101)
A

6
(ncccn) = A

6
(ncccn, c) + A

6
(ncccn, n), (A102)
A

6
(nccnc) = A

6
(nccnc, ck) + A

6
(nccnc, nk), (A103)
A

6
(ncncc) = A

6
(ncncc, ckk) + A

6
(ncncc, nkk), (A104)
A

6
(nnccc) = A

6
(nnccc, ckkk) + A

6
(nnccc, nkkk) (A105)
A

6
(ccnnn) = A

6
(ccnnn, kkcc) + A

6
(ccnnn, kkcn) + A

6
(ccnnn, kknc)
+A

6
(ccnnn, kknn, kkc) + A

6
(ccnnn, kknn, kkn), (A106)
32
A

6
(cncnn) = A

6
(cncnn, kkkc, kck) + A

6
(cncnn, kkkc, knk) + A

6
(cncnn, kkkn, kck)
+A

6
(cncnn, kkkn, knk, kc) + A

6
(cncnn, kkkn, knk, kn), (A107)
A

6
(cnncn) = A

6
(cnncn, kckk, kkc) + A

6
(cnncn, kckk, kkn) + A

6
(cnncn, knkk, kkc)
+A

6
(cnncn, knkk, kkn, kc) + A

6
(cnncn, knkk, kkn, kn), (A108)
A

6
(cnnnc) = A

6
(cnnnc, kcck) + A

6
(cnnnc, kcnk) + A

6
(cnnnc, knck)
+A

6
(cnnnc, knnk, kck) + A

6
(cnnnc, knnk, knk), (A109)
A

6
(nccnn) = A

6
(nccnn, kkkc, ck) + A

6
(nccnn, kkkc, nk) + A

6
(nccnn, kkkn, ck)
+A

6
(nccnn, kkkn, nk, c) + A

6
(nccnn, kkkn, nk, n), (A110)
A

6
(ncncn) = A

6
(ncncn, ckc) + A

6
(ncncn, ckn) + A

6
(ncncn, nkc)
+A

6
(ncncn, nkn, c) + A

6
(ncncn, nkn, n), (A111)
A

6
(ncnnc) = A

6
(ncnnc, kkck, ckk) + A

6
(ncnnc, kkck, nkk) + A

6
(ncnnc, kknk, ckk)
+A

6
(ncnnc, kknk, nkk, ck) + A

6
(ncnnc, kknk, nkk, nk), (A112)
A

6
(nnccn) = A

6
(nnccn, cc) + A

6
(nnccn, cn) + A

6
(nnccn, nc)
+A

6
(nnccn, nn, c) + A

6
(nnccn, nn, n), (A113)
A

6
(nncnc) = A

6
(nncnc, ckkk, kck) + A

6
(nncnc, ckkk, knk) + A

6
(nncnc, nkkk, kck)
+A

6
(nncnc, nkkk, knk, ck) + A

6
(nncnc, nkkk, knk, nk), (A114)
A

6
(nnncc) = A

6
(nnncc, cckk) + A

6
(nnncc, cnkk) + A

6
(nnncc, nckk)
+A

6
(nnncc, nnkk, c) + A

6
(nnncc, nnkk, n) (A115)
A

6
(cnnnn) = A

6
(cnnnn, kccc) + A

6
(cnnnn, kccn) + A

6
(cnnnn, kcnc)
+A

6
(cnnnn, kncc) + A

6
(cnnnn, kcnn) + A

6
(cnnnn, kncn)
+A

6
(cnnnn, knnc) + A

6
(cnnnn, knnn), (A116)
A

6
(cnnnn, kcnn) = A

6
(cnnnn, kcnn, kkc) + A

6
(cnnnn, kcnn, kkn), (A117)
A

6
(cnnnn, kncn) = A

6
(cnnnn, kncn, kc) + A

6
(cnnnn, kncn, kn), (A118)
A

6
(cnnnn, knnc) = A

6
(cnnnn, knnc, kck) + A

6
(cnnnn, knnc, knk), (A119)
A

6
(cnnnn, knnn) = A

6
(cnnnn, knnn, kcc) + A

6
(cnnnn, knnn, kcn)
+A

6
(cnnnn, knnn, knc) + A

6
(cnnnn, knnn, knn, kc)
+A

6
(cnnnn, knnn, knn, kn). (A120)
A

6
(ncnnn) = A

6
(ncnnn, kkcc, ckk) + A

6
(ncnnn, kkcc, nkk)
+A

6
(ncnnn, kkcn, ckk) + A

6
(ncnnn, kknc, ckk)
+A

6
(ncnnn, kkcn, nkk) + A

6
(ncnnn, kknc, nkk)
+A

6
(ncnnn, kknn, ckk) + A

6
(ncnnn, kknn, nkk), (A121)
A

6
(ncnnn, kkcn, nkk) = A

6
(ncnnn, kkcn, nkk, c) + A

6
(ncnnn, kkcn, nkk, n), (A122)
A

6
(ncnnn, kknc, nkk) = A

6
(ncnnn, kknc, nkk, ck) + A

6
(ncnnn, kknc, nkk, nk), (A123)
A

6
(ncnnn, kknn, ckk) = A

6
(ncnnn, kknn, ckc) + A

6
(ncnnn, kknn, ckn), (A124)
33
A

6
(ncnnn, kknn, nkk) = A

6
(ncnnn, kknn, nkc, ck) + A

6
(ncnnn, kknn, nkc, nk)
+A

6
(ncnnn, kknn, nkn, ck) + A

6
(ncnnn, kknn, nkn, nk, c)
+A

6
(ncnnn, kknn, nkn, nk, n) (A125)
A

6
(nncnn) = A

6
(nncnn, ckkc, kck) + A

6
(nncnn, ckkc, knk)
+A

6
(nncnn, ckkn, kck) + A

6
(nncnn, nkkc, kck)
+A

6
(nncnn, ckkn, knk) + A

6
(nncnn, nkkc, knk)
+A

6
(nncnn, nkkn, kck) + A

6
(nncnn, nkkn, knk), (A126)
A

6
(nncnn, ckkn, knk) = A

6
(nncnn, ckkn, knk, kc) + A

6
(nncnn, ckkn, knk, kn), (A127)
A

6
(nncnn, nkkc, knk) = A

6
(nncnn, nkkc, knk, ck) + A

6
(nncnn, nkkc, knk, nk), (A128)
A

6
(nncnn, nkkn, kck) = A

6
(nncnn, nkkn, kck, c) + A

6
(nncnn, nkkn, kck, n), (A129)
A

6
(nncnn, nkkn, knk) = A

6
(nncnn, nkkn, knk, cc) + A

6
(nncnn, nkkn, knk, cn)
+A

6
(nncnn, nkkn, knk, nc) + A

6
(nncnn, nkkn, knk, nn, c)
+A

6
(nncnn, nkkn, knk, nn, n) (A130)
A

6
(nnncn) = A

6
(nnncn, cckk, kkc) + A

6
(nnncn, cckk, kkn)
+A

6
(nnncn, cnkk, kkc) + A

6
(nnncn, nckk, kkc)
+A

6
(nnncn, cnkk, kkn) + A

6
(nnncn, nckk, kkn)
+A

6
(nnncn, nnkk, kkc) + A

6
(nnncn, nnkk, kkn), (A131)
A

6
(nnncn, cnkk, kkn) = A

6
(nnncn, cnkk, kkn, kc) + A

6
(nnncn, cnkk, kkn, kn), (A132)
A

6
(nnncn, nckk, kkn) = A

6
(nnncn, nckk, kkn, c) + A

6
(nnncn, nckk, kkn, n), (A133)
A

6
(nnncn, nnkk, kkc) = A

6
(nnncn, nnkk, ckc) + A

6
(nnncn, nnkk, nkc), (A134)
A

6
(nnncn, nnkk, kkn) = A

6
(nnncn, nnkk, ckn, kc) + A

6
(nnncn, nnkk, ckn, kn)
+A

6
(nnncn, nnkk, nkn, kc) + A

6
(nnncn, nnkk, nkn, kn, c)
+A

6
(nnncn, nnkk, nkn, kn, n). (A135)
A

6
(nnnnc) = A

6
(nnnnc, ccck) + A

6
(nnnnc, ccnk) + A

6
(nnnnc, cnck)
+A

6
(nnnnc, ncck) + A

6
(nnnnc, cnnk) + A

6
(nnnnc, ncnk)
+A

6
(nnnnc, nnck) + A

6
(nnnnc, nnnk), (A136)
A

6
(nnnnc, cnnk) = A

6
(nnnnc, cnnk, kck) + A

6
(nnnnc, cnnk, knk), (A137)
A

6
(nnnnc, ncnk) = A

6
(nnnnc, ncnk, ck) + A

6
(nnnnc, ncnk, nk), (A138)
A

6
(nnnnc, nnck) = A

6
(nnnnc, nnck, c) + A

6
(nnnnc, nnck, n), (A139)
A

6
(nnnnc, nnnk) = A

6
(nnnnc, nnnk, cck) + A

6
(nnnnc, nnnk, cnk)
+A

6
(nnnnc, nnnk, nck) + A

6
(nnnnc, nnnk, nnk, ck)
+A

6
(nnnnc, nnnk, nnk, nk). (A140)
34
A

6
(nnnnn) = A

6
(nnnnn, cccc) + A

6
(nnnnn, cccn) + A

6
(nnnnn, ccnc)
+A

6
(nnnnn, cncc) + A

6
(nnnnn, nccc) + A

6
(nnnnn, ccnn)
+A

6
(nnnnn, cncn) + A

6
(nnnnn, cnnc) + A

6
(nnnnn, nccn)
+A

6
(nnnnn, ncnc) + A

6
(nnnnn, nncc) + A

6
(nnnnn, cnnn)
+A

6
(nnnnn, ncnn) + A

6
(nnnnn, nncn) + A

6
(nnnnn, nnnc)
+A

6
(nnnnn, nnnn), (A141)
A

6
(nnnnn, ccnn) = A

6
(nnnnn, ccnn, kkc) + A

6
(nnnnn, ccnn, kkn), (A142)
A

6
(nnnnn, cncn) = A

6
(nnnnn, cncn, kc) + A

6
(nnnnn, cncn, kn), (A143)
A

6
(nnnnn, cnnc) = A

6
(nnnnn, cnnc, kck) + A

6
(nnnnn, cnnc, knk), (A144)
A

6
(nnnnn, nccn) = A

6
(nnnnn, nccn, c) + A

6
(nnnnn, nccn, n), (A145)
A

6
(nnnnn, ncnc) = A

6
(nnnnn, ncnc, ck) + A

6
(nnnnn, ncnc, nk), (A146)
A

6
(nnnnn, nncc) = A

6
(nnnnn, nncc, ckk) + A

6
(nnnnn, nncc, nkk), (A147)
A

6
(nnnnn, cnnn) = A

6
(nnnnn, cnnn, kcc) + A

6
(nnnnn, cnnn, kcn)
+A

6
(nnnnn, cnnn, knc) + A

6
(nnnnn, cnnn, knn, kc)
+A

6
(nnnnn, cnnn, knn, kn), (A148)
A

6
(nnnnn, ncnn) = A

6
(nnnnn, ncnn, kkc, ck) + A

6
(nnnnn, ncnn, kkc, nk)
+A

6
(nnnnn, ncnn, kkn, ck) + A

6
(nnnnn, ncnn, kkn, nk, c)
+A

6
(nnnnn, ncnn, kkn, nk, n), (A149)
A

6
(nnnnn, nncn) = A

6
(nnnnn, nncn, ckk, kc) + A

6
(nnnnn, nncn, ckk, kn)
+A

6
(nnnnn, nncn, nkk, kc) + A

6
(nnnnn, nncn, nkk, kn, c)
+A

6
(nnnnn, nncn, nkk, kn, n), (A150)
A

6
(nnnnn, nnnc) = A

6
(nnnnn, nnnc, cck) + A

6
(nnnnn, nnnc, cnk)
+A

6
(nnnnn, nnnc, nck) + A

6
(nnnnn, nnnc, nnk, ck)
+A

6
(nnnnn, nnnc, nnk, nk), (A151)
A

6
(nnnnn, nnnn) = A

6
(nnnnn, nnnn, ccc) + A

6
(nnnnn, nnnn, ccn)
+A

6
(nnnnn, nnnn, cnc) + A

6
(nnnnn, nnnn, ncc)
+A

6
(nnnnn, nnnn, cnn) + A

6
(nnnnn, nnnn, ncn)
+A

6
(nnnnn, nnnn, nnc) + A

6
(nnnnn, nnnn, nnn), (A152)
A

6
(nnnnn, nnnn, cnn)=A

6
(nnnnn, nnnn, cnn, kc) + A

6
(nnnnn, nnnn, cnn, kn), (A153)
A

6
(nnnnn, nnnn, ncn)=A

6
(nnnnn, nnnn, ncn, c) + A

6
(nnnnn, nnnn, ncn, n), (A154)
A

6
(nnnnn, nnnn, nnc)=A

6
(nnnnn, nnnn, nnc, ck) + A

6
(nnnnn, nnnn, nnc, nk), (A155)
A

6
(nnnnn, nnnn, nnn) = A

6
(nnnnn, nnnn, nnn, cc) + A

6
(nnnnn, nnnn, nnn, cn)
+A

6
(nnnnn, nnnn, nnn, nc) + A

6
(nnnnn, nnnn, nnn, nn, c)
+A

6
(nnnnn, nnnn, nnn, nn, n). (A156)
Thus, we obtain that the contribution from the six order approximation is made of 203 terms after nding out all
of contractions and anti-contractions.
35
Just like we have done in the l = 4 or 5 cases, we decompose
A

6
= A

6
(e) + A

6
(te) + A

6
(t
2
e) + A

6
(t
3
e) (A157)
= A

6
(e, te) + A

6
(t
2
e, t
3
e), (A158)
To our purpose, we only calculate the second term A

6
(t
2
e, t
3
e) in eq.(A158). Without loss of generality, we decompose
it into
A

6
(t
2
e, t
3
e) = A

6
(t
2
e
iEt
, t
3
e
iEt
) + A

6
(t
2
e
iE
1
t
, t
3
e
iE
1
t
)
+A

6
(t
2
e
iE

t
, t
3
e
iE

t
). (A159)
While, every term in the above equations has its diagonal and o-diagonal parts about and

, that is
A

6
(t
2
e
iE
i
t
) = A

6
(t
2
e
iE
i
t
; D) + A

6
(t
2
e
iE
i
t
; N), (A160)
A

6
(t
3
e
iE
i
t
) = A

6
(t
3
e
iE
i
t
; D) + A

6
(t
3
e
iE
i
t
; N). (A161)
where E
i
takes E

, E
1
and E

.
Based on our calculations, we nd that there are nonvanishing 91 terms and vanishing 112 terms with t
2
e, t
3
e factor
parts in all of 203 contraction- and anti contraction- expressions (see in the supplementary Ref. [10]). Therefore we
can, in terms of rearrangement and summation, obtain the following concise forms:
A
6
(t
2
e
iEt
, D) =
(it)
2
2!
_
G
(3)

_
2
e
iEt

+
(it)
2
2!
2G
(2)

G
(4)

e
iEt

(it)
2
2!

1
_
G
(2)

_
2
e
iEt
(E

E
1
)
2
g
1
1
g
1
1

. (A162)
A
6
(t
2
e
iE
1
t
, D) =
(it)
2
2!

1
_
G
(2)

_
2
e
iE
1
t
(E

E
1
)
2
g
1
1
g
1
1

. (A163)
A
6
(t
2
e
iE

t
, D) = 0. (A164)
A
6
(t
2
e
iEt
, N) =
(it)
2
2!
2G
(2)

G
(3)

e
iEt
(E

)
g

1
+
(it)
2
2!

1
_
G
(2)

_
2
e
iEt
(E

E
1
) (E

)
g
1
1
g
1

. (A165)
A
6
(t
2
e
iE
1
t
, N) =
(it)
2
2!

1
_
G
(2)
1
_
2
e
iE
1
t
(E

E
1
) (E
1
E

)
g
1
1
g
1

. (A166)
A
6
(t
2
e
iE

t
, N) =
(it)
2
2!
2G
(2)

G
(3)

e
iE

t
(E

)
g

1
+
(it)
2
2!

1
_
G
(2)

_
2
e
iE

t
(E

) (E
1
E

)
g
1
1
g
1

. (A167)
Their forms are indeed the same as expected and can be absorbed reasonably into the lower order approximations in
order to obtain the improved forms of perturbed solutions.
[1] E. Schrodinger, Ann. Phys. 79, 489-527(1926)
36
[2] P. A. M. Dirac, The principles of Quantum Mechanics, 4th edn(revised), Oxford: Clarendon Press (1974)
[3] An Min Wang, Quantum mechanics in general quantum systems (I): exact solution and perturbation theory, quant-
ph/0611216. Its earlier version is quant-ph/0602055
[4] R. P. Feynman and A. R. Hibbs, Path Intergral and Quantum Mechanics, NcGraw-Hill (1965)
[5] C. J. Joachain, Quantum collision theory, North-Holland Publishing (1975)
[6] An Min Wang, Quantum mechanics in general quantum systems (III): open system dynamics, quant-ph/0601051
[7] J. von Neumann, The Mathematical Foundations of Quantum Mechanics, Princeton: University Press (1955)
[8] E. Fermi, Nuclear Physics, Chicago: University of Chicago Press (1950)
[9] Jian Tang and An Min Wang, quant-ph/0611195
[10] An Min Wang, Supplementary in Quantum mechanics in general quantum systems (II): perturbation theory, quant-
ph/0611217. Its earlier version is quant-ph/0602055
Supplement
1 l = 5 case
In the following, we respectively calculate the 52 component expressions of A

5
, and put the second and third
terms in eq.(A68) together as A

5
(te, t
2
e).
A

5
(cccc; te, t
2
e) =
_
(it)
3e
iEt
+ 3e
iE

t
(E

)
4
+
(it)
2
2
e
iEt
(E

)
3

(it)
2
2
e
iE

t
(E

)
3
_

4
g

1
. (168)
A

5
(cccn; te, t
2
e) =

1
_
(it)
2e
iEt
(E

E
1
)
3
(E

)
(it)
e
iEt
(E

E
1
)
2
(E

)
2
(it)
e
iE
1
t
(E

E
1
)
3
(E
1
E

)
+
(it)
2
2!
e
iEt
(E

E
1
)
2
(E

)
_
|g
1
1
|
4
g

1

1
. (169)
A

5
(ccnc; te, t
2
e) =

1
_
(it)
e
iEt
(E

E
1
) (E

)
3
(it)
e
iE

t
(E

)
2
(E
1
E

)
2
(it)
2e
iE

t
(E

)
3
(E
1
E

(it)
2
2!
e
iE

t
(E

)
2
(E
1
E

)
_

g
1

2
g

1

1
. (170)
A

5
(cncc; te, t
2
e) =

1
_
(it)
2e
iEt
(E

E
1
) (E

)
3
(it)
e
iEt
(E

E
1
)
2
(E

)
2
+(it)
e
iE

t
(E

)
3
(E
1
E

)
+
(it)
2
2!
e
iEt
(E

E
1
) (E

)
2
_
|g
1
1
|
2

2
g

1

1
. (171)
37
A

5
(nccc; te, t
2
e) =

1
_
(it)
e
iE
1
t
(E

E
1
) (E
1
E

)
3
(it)
2e
iE

t
(E

) (E
1
E

)
3
(it)
e
iE

t
(E

)
2
(E
1
E

)
2

(it)
2
2!
e
iE

t
(E

) (E
1
E

)
2
_

g
1

4
g

1

1
. (172)
A

5
(ccnn, kkc; te, t
2
e)
=

1,2
_
(it)
e
iEt
(E

E
1
)
2
(E

E
2
)
2
(it)
2e
iEt
(E

E
1
)
3
(E

E
2
)
(it)
e
iE
1
t
(E

E
1
)
3
(E
1
E
2
)
+
(it)
2
2!
e
iEt
(E

E
1
)
2
(E

E
2
)
_
|g
1
1
|
2
g
1
1
g
12
1
g
2
1

. (173)
A

5
(ccnn, kkn; te, t
2
e) =

1,2
_
(it)
e
iEt
(E

E
1
)
2
(E

E
2
) (E

)
+(it)
e
iE
1
t
(E

E
1
)
2
(E
1
E
2
) (E
1
E

)
_
|g
1
1
|
2
g
1
1
g
12
1
g
2

1

1

2

. (174)
A

5
(cncn, kc; te, t
2
e)
=

1
_
(it)
2e
iEt
(E

E
1
) (E

)
3
(it)
e
iEt
(E

E
1
)
2
(E

)
2
+(it)
e
iE

t
(E

)
3
(E
1
E

)
+
(it)
2
2!
e
iEt
(E

E
1
) (E

)
2
_
|g
1
1
|
2

2
g

1

1
. (175)
A

5
(cncn, kn; te, t
2
e) =

1,2
_
(it)
e
iEt
(E

E
1
) (E

E
2
) (E

)
2
(it)
e
iEt
(E

E
1
) (E

E
2
)
2
(E

)
(it)
e
iEt
(E

E
1
)
2
(E

E
2
) (E

)
+
(it)
2
2!
e
iEt
(E

E
1
) (E

E
2
) (E

)
_
|g
1
1
|
2
|g
2
1
|
2
g

1

12

1

2
. (176)
A

5
(cnnc, kck; te, t
2
e)
=

1
_
(it)
e
iEt
(E

E
1
)
2
(E

)
2
+ (it)
e
iE
1
t
(E

E
1
)
2
(E
1
E

)
2
+(it)
e
iE

t
(E

)
2
(E
1
E

)
2
_
|g
1
1
|
2

g
1

2
g

1
. (177)
38
A

5
(cnnc, knk; te, t
2
e) =

1,2
_
(it)
e
iEt
(E

E
1
) (E

E
2
) (E

)
2
+(it)
e
iE

t
(E

)
2
(E
1
E

) (E
2
E

)
_
|g
1
1
|
2

g
2

2
g

1

1

12

2
. (178)
A

5
(nccn, c; te, t
2
e)
=

1,2
_
(it)
e
iEt
(E

E
1
)
2
(E

E
2
)
2
+ (it)
e
iE
1
t
(E

E
1
)
2
(E
1
E
2
)
2
+(it)
e
iE
2
t
(E

E
2
)
2
(E
1
E
2
)
2
_
|g
12
1
|
2
g
1
1
g
12
1
g
2
1

. (179)
A

5
(nccn, n; te, t
2
e) =

1,2
_
(it)
e
iE
1
t
(E

E
1
) (E
1
E
2
)
2
(E
1
E

)
(it)
e
iE
2
t
(E

E
2
) (E
1
E
2
)
2
(E
2
E

)
_
|g
12
1
|
2
g
1
1
g
12
1
g
2

1

1

2

. (180)
A

5
(ncnc, ck; te, t
2
e)
=

1
_
(it)
e
iEt
(E

E
1
) (E

)
3
(it)
e
iE

t
(E

)
2
(E
1
E

)
2
(it)
2e
iE

t
(E

)
3
(E
1
E

(it)
2
2!
e
iE

t
(E

)
2
(E
1
E

)
_

g
1

2
g

1

1
. (181)
A

5
(ncnc, nk; te, t
2
e) =

1,2
_
(it)
e
iE

t
(E

) (E
1
E

) (E
2
E

)
2
(it)
e
iE

t
(E

) (E
1
E

)
2
(E
2
E

)
(it)
e
iE

t
(E

)
2
(E
1
E

) (E
2
E

(it)
2
2!
e
iE

t
(E

) (E
1
E

) (E
2
E

)
_

g
1

g
2

2
g

1

1

12
. (182)
A

5
(nncc, ckk; te, t
2
e)
=

1,2
_
(it)
2e
iEt
(E

E
1
) (E

E
2
)
3
(it)
e
iEt
(E

E
1
)
2
(E

E
2
)
2
+(it)
e
iE
2
t
(E

E
2
)
3
(E
1
E
2
)
+
(it)
2
2!
e
iEt
(E

E
1
) (E

E
2
)
2
_
|g
2
1
|
2
g
1
1
g
12
1
g
2
1

. (183)
39
A

5
(nncc, nkk; te, t
2
e) =

1,2
_
(it)
e
iE
2
t
(E

E
2
) (E
1
E
2
) (E
2
E

)
2
+(it)
e
iE

t
(E

) (E
1
E

) (E
2
E

)
2
_

g
2

2
g
1
1
g
12
1
g
2

1

1

2

. (184)
A

5
(cnnn, kcc; te, t
2
e)
=

1,2
_
(it)
2e
iEt
(E

E
1
)
3
(E

E
2
)
(it)
e
iEt
(E

E
1
)
2
(E

E
2
)
2
(it)
e
iE
1
t
(E

E
1
)
3
(E
1
E
2
)
+
(it)
2
2!
e
iEt
(E

E
1
)
2
(E

E
2
)
_
|g
1
1
|
2
g
2
1
g
21
1
g
1
1

. (185)
A

5
(cnnn, kcn; te, t
2
e) =

1,2
_
(it)
e
iEt
(E

E
1
)
2
(E

E
2
) (E

)
+(it)
e
iE
1
t
(E

E
1
)
2
(E
1
E
2
) (E
1
E

)
_
|g
1
1
|
2
g
2
1
g
21
1
g
1

1

2

. (186)
A

5
(cnnn, knc; te, t
2
e) =

1,2,3
_
(it)
e
iEt
(E

E
1
)
2
(E

E
2
) (E

E
3
)
(it)
e
iEt
(E

E
1
) (E

E
2
)
2
(E

E
3
)
(it)
e
iEt
(E

E
1
) (E

E
2
) (E

E
3
)
2
+
(it)
2
2!
e
iEt
(E

E
1
) (E

E
2
) (E

E
3
)
_
|g
1
1
|
2
g
2
1
g
23
1
g
3
1

12

13

. (187)
A

5
(cnnn, knn, kc; te, t
2
e) =

1,2
_
(it)
e
iEt
(E

E
1
) (E

E
2
) (E

)
2
+(it)
e
iE

t
(E

)
2
(E
1
E

) (E
2
E

)
_

2
g
1
1
g
12
1
g
2

1

2

1
. (188)
A

5
(cnnn, knn, kn; te, t
2
e)
=

1,2,3
(it)
e
iEt
|g
1
1
|
2
g
2
1
g
23
1
g
3

1

3

12

13

1

2

(E

E
1
) (E

E
2
) (E

E
3
) (E

)
. (189)
40
A

5
(ncnn, kkc, ck; te, t
2
e)
=

1
_
(it)
e
iEt
(E

E
1
)
2
(E

)
2
+ (it)
e
iE
1
t
(E

E
1
)
2
(E
1
E

)
2
+(it)
e
iE

t
(E

)
2
(E
1
E

)
2
_
|g
1
1
|
2

g
1

2
g

1
. (190)
A

5
(ncnn, kkc, nk; te, t
2
e) =

1,2
_
(it)
e
iE
1
t
(E

E
1
) (E
1
E
2
) (E
1
E

)
2
+(it)
e
iE

t
(E

) (E
1
E

)
2
(E
2
E

)
_

g
1

2
g
1
1
g
12
1
g
2

1

2

. (191)
A

5
(ncnn, kkn, ck; te, t
2
e) =

1,2
_
(it)
e
iEt
(E

E
1
)
2
(E

E
2
) (E

)
+(it)
e
iE
1
t
(E

E
1
)
2
(E
1
E
2
) (E
1
E

)
_
|g
1
1
|
2
|g
12
1
|
2
g

1

2

1

2
. (192)
A

5
(ncnn, kkn, nk, c; te, t
2
e) =

1,2,3
_
(it)
e
iEt
(E

E
1
)
2
(E

E
2
) (E

E
3
)
+(it)
e
iE
1
t
(E

E
1
)
2
(E
1
E
2
) (E
1
E
3
)
_
|g
12
1
|
2
g
1
1
g
13
1
g
3
1

2

23

. (193)
A

5
(ncnn, kkn, nk, n; te, t
2
e)
=

1,2,3
(it)
e
iE
1
t
|g
12
1
|
2
g
1
1
g
13
1
g
3

1

2

1

23

(E

E
1
) (E
1
E
2
) (E
1
E
3
) (E
1
E

)
. (194)
A

5
(nncn, ckk; kc, te, t
2
e)
=

1
_
(it)
e
iEt
(E

E
1
)
2
(E

)
2
+ (it)
e
iE
1
t
(E

E
1
)
2
(E
1
E

)
2
+(it)
e
iE

t
(E

)
2
(E
1
E

)
2
_
|g
1
1
|
2

g
1

2
g

1
. (195)
A

5
(nncn, ckk, kn; te, t
2
e) =

1,2
_
(it)
e
iEt
(E

E
1
) (E

E
2
)
2
(E

)
(it)
e
iE
2
t
(E

E
2
)
2
(E
1
E
2
) (E
2
E

)
_
|g
2
1
|
2
g
1
1
g
12
1
g
2

1
. (196)
41
A

5
(nncn, nkk, kc; te, t
2
e) =

1,2
_
(it)
e
iE
1
t
(E

E
1
) (E
1
E
2
) (E
1
E

)
2
+(it)
e
iE

t
(E

) (E
1
E

)
2
(E
2
E

)
_
|g
12
1
|
2

g
1

2
g

1

1

2
. (197)
A

5
(nncn, nkk, kn, c; te, t
2
e) =

1,23
_
(it)
e
iEt
(E

E
1
) (E

E
2
)
2
(E

E
3
)
(it)
e
iE
2
t
(E

E
2
)
2
(E
1
E
2
) (E
2
E
3
)
_
|g
23
1
|
2
g
1
1
g
12
1
g
2
1

3

13

. (198)
A

5
(nncn, nkk, kn, n; te, t
2
e)
=

1,2,3
(it)
e
iE
2
t
|g
23
1
|
2
g
1
1
g
12
1
g
2

1

2

13

1

3

(E

E
2
) (E
1
E
2
) (E
2
E
3
) (E
2
E

)
. (199)
A

5
(nnnc, cck; te, t
2
e)
=

1,2
_
(it)
e
iEt
(E

E
1
)
2
(E

E
2
)
2
(it)
2e
iEt
(E

E
1
)
3
(E

E
2
)
(it)
e
iE
1
t
(E

E
1
)
3
(E
1
E
2
)
+
(it)
2
2!
e
iEt
(E

E
1
)
2
(E

E
2
)
_
|g
1
1
|
2
g
1
1
g
12
1
g
2
1

. (200)
A

5
(nnnc, cnk; te, t
2
e) =

1,2,3
_
(it)
e
iEt
(E

E
1
) (E

E
2
) (E

E
3
)
2
(it)
e
iEt
(E

E
1
) (E

E
2
)
2
(E

E
3
)
(it)
e
iEt
(E

E
1
)
2
(E

E
2
) (E

E
3
)
+
(it)
2
2!
e
iEt
(E

E
1
) (E

E
2
) (E

E
3
)
_
|g
3
1
|
2
g
1
1
g
12
1
g
2
1

13

23

. (201)
A

5
(nnnc, nck; te, t
2
e) =

1,2
_
(it)
e
iE
1
t
(E

E
1
) (E
1
E
2
) (E
1
E

)
2
+(it)
e
iE

t
(E

) (E
1
E

)
2
(E
2
E

)
_

g
1

2
g
1
1
g
12
1
g
2

1

2

. (202)
42
A

5
(nnnc, nnk, ck; te, t
2
e) =

1,2
_
(it)
e
iEt
(E

E
1
) (E

E
2
) (E

)
2
+(it)
e
iE

t
(E

)
2
(E
1
E

) (E
2
E

)
_

2
g
1
1
g
12
1
g
2

1

2

1
. (203)
A

5
(nnnc, nnk, nk; te, t
2
e)
=

1,2,3
(it)
e
iE

g
3

2
g
1
1
g
12
1
g
2

1

2

13

1

23
(E

) (E
1
E

) (E
2
E

) (E
3
E

)
. (204)
A

5
(nnnn, ccc; te, t
2
e)
=

1
_
(it)
e
iEt
(E

E
1
)
2
(E

)
2
+ (it)
e
iE
1
t
(E

E
1
)
2
(E
1
E

)
2
+(it)
e
iE

t
(E

)
2
(E
1
E

)
2
_
|g
1
1
|
2
_
g
1
1
g
1

1
_
2
g

1
. (205)
A

5
(nnnn, ccn; te, t
2
e) =

1,2
_
(it)
e
iEt
(E

E
1
)
2
(E

E
2
) (E

)
+(it)
e
iE
1
t
(E

E
1
)
2
(E
1
E
2
) (E
1
E

)
_
(g
1
1
g
12
1
g
2
1
) g
1
1
g
1

2
. (206)
A

5
(nnnn, cnc; te, t
2
e) =

1,2
_
(it)
e
iEt
(E

E
1
) (E

E
2
) (E

)
2
+(it)
e
iE

t
(E

)
2
(E
1
E

) (E
2
E

)
_

_
g
1
1
g
1

1
__
g
2
1
g
2

1
_
g

1

12
. (207)
A

5
(nnnn, ncc; te, t
2
e) =

1,2
_
(it)
e
iE
1
t
(E

E
1
) (E
1
E
2
) (E
1
E

)
2
+(it)
e
iE

t
(E

) (E
1
E

)
2
(E
2
E

)
_

_
g

2
1
g
21
1
g
1

1
_
g
1
1
g
1

1

2

. (208)
A

5
(nnnn, cnn, kc; te, t
2
e) =

1,2
_
(it)
e
iEt
(E

E
1
) (E

E
2
) (E

)
2
+(it)
e
iE

t
(E

)
2
(E
1
E

) (E
2
E

)
_

_
g

1
1
g
1
1
__
g
2
1
g
2

1
_
g

1

12
. (209)
43
A

5
(nnnn, cnn, kn; te, t
2
e)
=

1,2,3
(it)
e
iEt
g
1
1
g
12
1
g
2
1
g
3
1
g
3

13

23

1

2

(E

E
1
) (E

E
2
) (E

E
3
) (E

)
. (210)
A

5
(nnnn, ncn, c; te, t
2
e) =

1,23
_
(it)
e
iEt
(E

E
1
)
2
(E

E
2
) (E

E
3
)
+(it)
e
iE
1
t
(E

E
1
)
2
(E
1
E
2
) (E
1
E
3
)
_
|g
1
1
|
2
g
12
1
g
23
1
g
31
1

2

. (211)
A

5
(nnnn, ncn, n; te, t
2
e)
=

1,2,3
(it)
e
iE
1
t
g
1
1
g
12
1
g
23
1
g
31
1
g
1

2

3

(E

E
1
) (E
1
E
2
) (E
1
E
3
) (E
1
E

)
. (212)
A

5
(nnnn, nnc, ck; te, t
2
e) =

1,2
_
(it)
e
iEt
(E

E
1
) (E

E
2
) (E

)
2
+(it)
e
iE

t
(E

)
2
(E
1
E

) (E
2
E

)
_

_
g
1
1
g
1

1
__
g

2
1
g
2
1
_
g

1

12
. (213)
A

5
(nnnn, nnc, nk; te, t
2
e)
=

1,2,3
(it)
e
iE

t
g

2
1
g
23
1
g
3

1
g
1
1
g
1

12

13
(E

) (E
1
E

) (E
2
E

) (E
3
E

)
. (214)
A

5
(nnnn, nnn, cc; te, t
2
e) =

1,2
_
(it)
e
iEt
(E

E
1
) (E

E
2
) (E

)
2
+(it)
e
iE

t
(E

)
2
(E
1
E

) (E
2
E

)
_
g

1
g

1
1
g
12
1
g
2
1
g

1

1

2
. (215)
A

5
(nnnn, nnn, cn; te, t
2
e)
=

1,2,3
(it)
e
iEt
g
1
1
g
12
1
g
23
1
g
3
1
g

1

2

13

1

2

3

(E

E
1
) (E

E
2
) (E

E
3
) (E

)
. (216)
A

5
(nnnn, nnn, nc; te, t
2
e)
=

1,2,3
(it)
e
iE

t
g

1
1
g
12
1
g
23
1
g
3

1
g

1

1

13

(E

) (E
1
E

) (E
2
E

) (E
3
E

)
. (217)
A

5
(nnnn, nnn, nn, c; te, t
2
e)
=

1,2,3,4
(it)
e
iEt
g
1
1
g
12
1
g
23
1
g
34
1
g
4
1

2

13

14

24

(E

E
1
) (E

E
2
) (E

E
3
) (E

E
4
)
. (218)
A

5
(nnnn, nnn, nn, n; te, t
2
e) = 0 (219)
44
D. l = 6 case
Based on our calculations, we nd that there are nonvanishing 91 terms and vanishing 112 terms with t
2
e, t
3
e factor
parts in all of 203 contraction- and anti contraction- expressions. In the following, we respectively calculate them
term by term, and we only write down the non-zero expressions for saving space.
A
6
(ccccc; t
2
e, t
3
e) =

1
_
(it)
3
3!
e
iEt
(E

E
1
)
3

(it)
2
2
3e
iEt
(E

E
1
)
4
+
(it)
2
2
e
iE
1
t
(E

E
1
)
4
_
|g
1
1
|
6

. (220)
A
6
(ccccn; t
2
e, t
3
e) =

1
_
(it)
2
2
e
iEt
(E

E
1
)
3
(E

(it)
2
2
e
iE
1
t
(E

E
1
)
3
(E
1
E

)
_
|g
1
1
|
4
g
1
1
g
1

. (221)
A
6
(cccnc; t
2
e, t
3
e) =

1,2
_
(it)
3
3!
e
iEt
(E

E
1
)
2
(E

(it)
2
2
e
iEt
(E

E
1
)
2
(E

E
2
)
2

(it)
2
2
e
iEt
(E

E
1
)
3
(E

E
2
)
_
|g
1
1
|
4
|g
2
1
|
2

12

. (222)
A
6
(ccncc; t
2
e, t
3
e) =

1
(it)
2
2
e
iE
1
t
(E

E
1
)
2
(E
1
E

)
2
|g
1
1
|
2

g
1

2
g
1
1
g
1

. (223)
A
6
(cnccc; t
2
e, t
3
e) =

1,2
_
(it)
3
3!
e
iEt
(E

E
1
) (E

E
2
)
2

(it)
2
2
e
iEt
(E

E
1
) (E

E
2
)
3

(it)
2
2
e
iEt
(E

E
1
)
2
(E

E
2
)
2
_
|g
1
1
|
2
|g
2
1
|
4

12

. (224)
A
6
(ncccc; t
2
e, t
3
e) =

1
_

(it)
2
2
e
iE
1
t
(E

E
1
) (E
1
E

)
3
+
(it)
2
2
e
iE

t
(E

) (E
1
E

)
3
_

g
1

4
g
1
1
g
1

. (225)
A
6
(cccnn, kkkc; t
2
e, t
3
e) =

1
_
(it)
2
2
e
iEt
(E

E
1
) (E

)
3
+
(it)
2
2
e
iE

t
(E

)
3
(E
1
E

)
_

4
g
1
1
g
1

1
. (226)
A
6
(cccnn, kkkn; t
2
e, t
3
e) =

1,2
(it)
2
2
e
iEt
|g
1
1
|
4
g
2
1
g
2

12

(E

E
1
)
2
(E

E
2
) (E

)
. (227)
45
A
6
(ccncn, kkc; t
2
e, t
3
e) =

1,2
_
(it)
2
2
e
iEt
(E

E
1
)
3
(E

(it)
2
2
e
iE
1
t
(E

E
1
)
3
(E
1
E
2
)
_
|g
1
1
|
4
|g
12
1
|
2

. (228)
A
6
(ccncn, kkn; t
2
e, t
3
e) =

1,2
(it)
2
2
e
iE
1
t
|g
1
1
|
2
|g
12
1
|
2
g
1
1
g
1

1

2

(E

E
1
)
2
(E
1
E
2
) (E
1
E

)
. (229)
A
6
(ccnnc, kkck; t
2
e, t
3
e) =

1
(it)
2
2
e
iEt
|g
1
1
|
2

2
g
1
1
g
1

1
(E

E
1
)
2
(E

)
2
. (230)
A
6
(cnccn, kc; t
2
e, t
3
e) =

1
(it)
2
2
e
iEt
|g
1
1
|
2

2
g
1
1
g
1

1
(E

E
1
)
2
(E

)
2
. (231)
A
6
(cnccn, kn; t
2
e, t
3
e) =

1,2
(it)
2
2
e
iEt
|g
1
1
|
2
|g
2
1
|
2
g
2
1
g
2

12

(E

E
1
) (E

E
2
)
2
(E

)
. (232)
A
6
(cncnc, kck; t
2
e, t
3
e) =

1,2
_
(it)
3
3!
e
iEt
(E

E
1
)
2
(E

E
2
)

(it)
2
2
e
iEt
(E

E
1
)
2
(E

E
2
)
2

(it)
2
2
e
iEt
(E

E
1
)
3
(E

E
2
)
_
|g
1
1
|
4
|g
2
1
|
2

12

. (233)
A
6
(cncnc, knk; t
2
e, t
3
e) =

1,2,3
_
(it)
3
3!
e
iEt
(E

E
1
) (E

E
2
) (E

E
3
)

(it)
2
2
e
iEt
(E

E
1
) (E

E
2
) (E

E
3
)
2

(it)
2
2
e
iEt
(E

E
1
) (E

E
2
)
2
(E

E
3
)

(it)
2
2
e
iEt
(E

E
1
)
2
(E

E
2
) (E

E
3
)
_
|g
1
1
|
2
|g
2
1
|
2
|g
3
1
|
2

12

13

23

. (234)
A
6
(cnncc, kckk; t
2
e, t
3
e) =

1
(it)
2
2
e
iE

g
1

2
g
1
1
g
1

1
(E

)
2
(E
1
E

)
2
. (235)
A
6
(ncccn, c; t
2
e, t
3
e) =

1,2
(it)
2
2
e
iE
1
t
|g
1
1
|
2
|g
12
1
|
4

(E

E
1
)
2
(E
1
E
2
)
2
. (236)
46
A
6
(ncccn, n; t
2
e, t
3
e) =

1,2
(it)
2
2
e
iE
1
t
|g
12
1
|
4
g
1
1
g
1

(E

E
1
) (E
1
E
2
)
2
(E
1
E

)
. (237)
A
6
(nccnc, ck; t
2
e, t
3
e) =

1
(it)
2
2
e
iE

g
1

2
g
1
1
g
1

1
(E

)
2
(E
1
E

)
2
. (238)
A
6
(nccnc, nk; t
2
e, t
3
e) =

1,2
(it)
2
2
e
iE

g
1

g
2

2
g
1
1
g
1

12
(E

) (E
1
E

)
2
(E
2
E

)
. (239)
A
6
(ncncc, ckk; t
2
e, t
3
e) =

1,2
_
(it)
2
2
e
iEt
(E

E
1
)
3
(E

E
2
)

(it)
2
2
e
iE
1
t
(E

E
1
)
3
(E
1
E
2
)
_
|g
1
1
|
4
|g
12
1
|
2

. (240)
A
6
(ncncc, nkk; t
2
e, t
3
e) =

1,2
(it)
2
2
e
iE
1
t
|g
12
1
|
2

g
1

2
g
1
1
g
1

(E

E
1
) (E
1
E
2
) (E
1
E

)
2
. (241)
A
6
(nnccc, ckkk; t
2
e, t
3
e) =

1
_
(it)
2
2
e
iEt
(E

E
1
) (E

)
3
+
(it)
2
2
e
iE

t
(E

)
3
(E
1
E

)
_

4
g
1
1
g
1

1
. (242)
A
6
(nnccc, nkkk; t
2
e, t
3
e) =

1,2
(it)
2
2
e
iE

g
2

4
g
1
1
g
1

12
(E

) (E
1
E

) (E
2
E

)
2
. (243)
A
6
(ccnnn, kkcc; t
2
e, t
3
e) =

1
_
(it)
2
2
e
iEt
(E

E
1
) (E

)
3
+
(it)
2
2
e
iE

t
(E

)
3
(E
1
E

)
_

2
g

1
g

1
1
g
1
1
g

1
. (244)
A
6
(ccnnn, kkcn; t
2
e, t
3
e) =

1,2
(it)
2
2
e
iEt
|g
1
1
|
2
g
1
1
g
12
1
g
2
1
g

1

1

2

(E

E
1
)
2
(E

E
2
) (E

)
. (245)
A
6
(ccnnn, kknc; t
2
e, t
3
e) =

1,2
(it)
2
2
e
iE

2
g

1
g

1
1
g
12
1
g
2

1

1

2
(E

)
2
(E
1
E

) (E
2
E

)
. (246)
A
6
(ccnnn, kknn, kkc; t
2
e, t
3
e) =

1,2,3
(it)
2
2
e
iEt
|g
1
1
|
2
g
1
1
g
12
1
g
23
1
g
3
1

2

13

(E

E
1
)
2
(E

E
2
) (E

E
3
)
. (247)
47
A
6
(cncnn, kkkc, kck; t
2
e, t
3
e) =

1
(it)
2
2
e
iEt
|g
1
1
|
2

2
g
1
1
g
1

1
(E

E
1
)
2
(E

)
2
. (248)
A
6
(cncnn, kkkc, knk; t
2
e, t
3
e) =

1,2
(it)
2
2
e
iEt
|g
1
1
|
2

2
g
2
1
g
2

1

12

(E

E
1
) (E

E
2
) (E

)
2
. (249)
A
6
(cncnn, kkkn, kck; t
2
e, t
3
e) =

1,2
(it)
2
2
e
iEt
|g
1
1
|
2
|g
2
1
|
2
g
1
1
g
1

12

(E

E
1
)
2
(E

E
2
) (E

)
. (250)
A
6
(cncnn, kkkn, knk, kc; t
2
e, t
3
e) =

1,2
(it)
2
2
e
iEt
|g
1
1
|
2

2
g
2
1
g
2

1

12

(E

E
1
) (E

E
2
) (E

)
2
. (251)
A
6
(cncnn, kkkn, knk, kn; t
2
e, t
3
e)
=

1,2,3
(it)
2
2
e
iEt
|g
1
1
|
2
|g
2
1
|
2
g
3
1
g
3

12

13

1

23

(E

E
1
) (E

E
2
) (E

E
3
) (E

)
. (252)
A
6
(cnncn, kckk, kkc; t
2
e, t
3
e) =

1,2
(it)
2
2
e
iEt
|g
1
1
|
2
|g
12
1
|
2
|g
2
1
|
2

(E

E
1
)
2
(E

E
2
)
2
(253)
A
6
(cnncn, knkk, kkc; t
2
e, t
3
e) =

1,2,3
(it)
2
2
e
iEt
|g
1
1
|
2
|g
2
1
|
2
|g
23
1
|
2

12

13

(E

E
1
) (E

E
2
)
2
(E

E
3
)
. (254)
A
6
(cnnnc, kcck; t
2
e, t
3
e) =

1
_
(it)
2
2
e
iEt
(E

E
1
) (E

)
3
+
(it)
2
2
e
iE

t
(E

)
3
(E
1
E

)
_

4
g
1
1
g
1

1
. (255)
A
6
(cnnnc, kcnk; t
2
e, t
3
e) =

1,2
(it)
2
2
e
iE

g
2

2
g
1
1
g
1

1

2

12
(E

)
2
(E
1
E

) (E
2
E

)
. (256)
A
6
(cnnnc, knck; t
2
e, t
3
e) =

1,2
(it)
2
2
e
iEt
|g
1
1
|
2

2
g
2
1
g
2

1

12

(E

E
1
) (E

E
2
) (E
2
E

)
2
. (257)
A
6
(nccnn, kkkc, ck; t
2
e, t
3
e) =

1
(it)
2
2
e
iE

g
1

2
g

1
g

1
1
g
1
1
g

1
(E

)
2
(E
1
E

)
2
. (258)
A
6
(nccnn, kkkc, nk; t
2
e, t
3
e) =

1,2
(it)
2
2
e
iE

g
1

2
g

1
g

1
1
g
12
1
g
2

1

1

2
(E

) (E
1
E

)
2
(E
2
E

)
. (259)
48
A
6
(ncncn, ckc; t
2
e, t
3
e) =

1
(it)
2
2
e
iE
1
t
|g
1
1
|
2

g
1

2
g
1
1
g
1

(E

E
1
)
2
(E
1
E

)
2
. (260)
A
6
(ncncn, ckn; t
2
e, t
3
e) =

1,2
(it)
2
2
e
iE
1
t
|g
1
1
|
2
|g
12
1
|
2
g
1
1
g
1

1

2

(E

E
1
)
2
(E
1
E
2
) (E
1
E

)
. (261)
A
6
(ncncn, nkc; t
2
e, t
3
e) =

1,2
(it)
2
2
e
iE
1
t

g
1

2
|g
12
1
|
2
g
1
1
g
1

(E

E
1
) (E
1
E
2
) (E
1
E

)
2
. (262)
A
6
(ncncn, nkn, c; t
2
e, t
3
e) =

1,2,3
(it)
2
2
e
iE
1
t
|g
1
1
|
2
|g
12
1
|
2
|g
13
1
|
2

23

(E

E
1
)
2
(E
1
E
2
) (E
1
E
3
)
. (263)
A
6
(ncncn, nkn, n; t
2
e, t
3
e)
=

1,2,3
(it)
2
2
e
iE
1
t
|g
12
1
|
2
|g
13
1
|
2
g
1
1
g
1

1

2

23

2

3

(E

E
1
) (E
1
E
2
) (E
1
E
3
) (E
1
E

)
. (264)
A
6
(ncnnc, kkck, ckk; t
2
e, t
3
e) =

1,2
(it)
2
2
e
iEt
|g
1
1
|
2
|g
2
1
|
2
|g
12
1
|
2

(E

E
1
)
2
(E

E
2
)
2
. (265)
A
6
(ncnnc, kknk, ckk; t
2
e, t
3
e) =

1,2,3
(it)
2
2
e
iEt
|g
1
1
|
2
|g
3
1
|
2
|g
12
1
|
2

13

23

(E

E
1
)
2
(E

E
2
) (E

E
3
)
. (266)
A
6
(nnccn, cc; t
2
e, t
3
e) =

1
(it)
2
2
e
iEt
|g
1
1
|
2
g

1
g

1
1
g
1
1
g

1
(E

E
1
)
2
(E

)
2
. (267)
A
6
(nnccn, cn; t
2
e, t
3
e) =

1,2
(it)
2
2
e
iEt
|g
2
1
|
2
g
1
1
g
12
1
g
2
1
g

1

1

2

(E

E
1
) (E

E
2
)
2
(E

)
. (268)
A
6
(nncnc, ckkk, kck; t
2
e, t
3
e) =

1
(it)
2
2
e
iE

g
1

2
g
1
1
g
1

1
(E

)
2
(E
1
E

)
2
. (269)
A
6
(nncnc, ckkk, knk; t
2
e, t
3
e) =

1,2
(it)
2
2
e
iE

g
2

2
g
1
1
g
1

1

2

12
(E

)
2
(E
1
E

) (E
2
E

)
. (270)
A
6
(nncnc, nkkk, kck; t
2
e, t
3
e) =

1,2
(it)
2
2
e
iE

g
1

g
2

2
g
1
1
g
1

1

2

12

(E

) (E
1
E

)
2
(E
2
E

)
. (271)
A
6
(nncnc, nkkk, knk, ck; t
2
e, t
3
e) =

1,2
(it)
2
2
e
iE

g
2

2
g
1
1
g
1

1

2

12
(E

)
2
(E
1
E

) (E
2
E

)
. (272)
49
A
6
(nncnc, nkkk, knk, nk; t
2
e, t
3
e)
=

1,2,3
(it)
2
2
e
iE

g
2

g
3

2
g
1
1
g
1

1

2

12

13

23
(E

) (E
1
E

) (E
2
E

) (E
3
E

)
. (273)
A
6
(nnncc, cckk; t
2
e, t
3
e) =

1
_
(it)
2
2
e
iEt
(E

E
1
) (E

)
3
+
(it)
2
2
e
iE

t
(E

)
3
(E
1
E

)
_

2
g

1
g

1
1
g
1
1
g

1
. (274)
A
6
(nnncc, cnkk; t
2
e, t
3
e) =

1,2
(it)
2
2
e
iEt

2
g
1
1
g
12
1
g
2
1
g

1

1

2

(E

E
1
) (E

E
2
) (E

)
2
. (275)
A
6
(nnncc, nckk; t
2
e, t
3
e) =

1,2
(it)
2
2
e
iE

g
2

2
g

1
g

1
1
g
12
1
g
2

1

1

2
(E

) (E
1
E

) (E
2
E

)
2
. (276)
A
6
(nnncc, nnkk, c; t
2
e, t
3
e) =

1,2,3
(it)
2
2
e
iEt
|g
3
1
|
2
g
1
1
g
12
1
g
23
1
g
3
1

2

13

(E

E
1
) (E

E
2
) (E

E
3
)
2
. (277)
A
6
(cnnnn, kccc; t
2
e, t
3
e) =

1
(it)
2
2
e
iEt
|g
1
1
|
2
g

1
g

1
1
g
1
1
g

1
(E

E
1
)
2
(E

)
2
. (278)
A
6
(cnnnn, kccn; t
2
e, t
3
e) =

1,2
(it)
2
2
e
iEt
|g
1
1
|
2
g
2
1
g
21
1
g
1
1
g

1

1

2

(E

E
1
)
2
(E

E
2
) (E

)
. (279)
A
6
(cnnnn, kncc; t
2
e, t
3
e) =

1,2
(it)
2
2
e
iEt
|g
1
1
|
2
g

1
g

2
1
g
2
1
g

1

1

12
(E

E
1
) (E

E
2
) (E

)
2
. (280)
A
6
(cnnnn, kcnn, kkc; t
2
e, t
3
e) =

1,2,3
(it)
2
2
e
iEt
|g
1
1
|
2
g
2
1
g
21
1
g
13
1
g
3
1

23

(E

E
1
)
2
(E

E
2
) (E

E
3
)
. (281)
A
6
(cnnnn, kncn, kc; t
2
e, t
3
e) =

1,2
(it)
2
2
e
iEt

2
g
1
1
g
12
1
g
2
1
g

1

1

2

(E

E
1
) (E

E
2
) (E

)
2
. (282)
A
6
(cnnnn, kncn, kn; t
2
e, t
3
e)
=

1,2,3
(it)
2
2
e
iEt
|g
1
1
|
2
g
2
1
g
23
1
g
3
1
g

1

12

13

1

2

3

(E

E
1
) (E

E
2
) (E

E
3
) (E

)
. (283)
A
6
(cnnnn, knnn, kcc; t
2
e, t
3
e) =

1,2,3
(it)
2
2
e
iEt
|g
1
1
|
2
g
2
1
g
23
1
g
31
1
g
1
1

3

12

(E

E
1
)
2
(E

E
2
) (E

E
3
)
. (284)
50
A
6
(cnnnn, knnn, knc; t
2
e, t
3
e)
=

1,2,3,4
(it)
2
2
e
iEt
|g
1
1
|
2
g
2
1
g
23
1
g
34
1
g
4
1

3

12

13

14

24

(E

E
1
) (E

E
2
) (E

E
3
) (E

E
4
)
. (285)
A
6
(ncnnn, kkcc, ckk; t
2
e, t
3
e) =

1
(it)
2
2
e
iE

g
1

2
g
1
1
g
1

1
(E

)
2
(E
1
E

)
2
. (286)
A
6
(ncnnn, kkcc, nkk; t
2
e, t
3
e) =

1,2
(it)
2
2
e
iE

g
1

2
g

1
g

2
1
g
21
1
g
1

1

1

2
(E

) (E
1
E

)
2
(E
2
E

)
. (287)
A
6
(ncnnn, kknc, ckk; t
2
e, t
3
e) =

1,2
(it)
2
2
e
iE

g
1

2
g
2
1
g
2

1

1

12
(E

)
2
(E
1
E

) (E
2
E

)
. (288)
A
6
(ncnnn, kknc, nkk, ck; t
2
e, t
3
e) =

1,2
(it)
2
2
e
iE

g
1

2
g

1
g

2
1
g
2
1
g

1

1

12
(E

)
2
(E
1
E

) (E
2
E

)
. (289)
A
6
(ncnnn, kknc, nkk, nk; t
2
e, t
3
e)
=

1,2,3
(it)
2
2
e
iE

g
1

2
g

1
g

2
1
g
23
1
g
3

1

1

12

13
(E

) (E
1
E

) (E
2
E

) (E
3
E

)
. (290)
A
6
(nncnn, ckkc, kck; t
2
e, t
3
e) =

1,2
(it)
2
2
e
iEt
|g
1
1
|
2
|g
2
1
|
2
|g
12
1
|
2

(E

E
1
)
2
(E

E
2
)
2
. (291)
A
6
(nncnn, ckkc, knk; t
2
e, t
3
e) =

1,2,3
(it)
2
2
e
iEt
|g
2
1
|
2
g
1
1
g
12
1
g
23
1
g
3
1

13

(E

E
1
) (E

E
2
)
2
(E

E
3
)
. (292)
A
6
(nnncn, cckk, kkc; t
2
e, t
3
e) =

1
(it)
2
2
e
iEt
|g
1
1
|
2

2
g
1
1
g
1

1
(E

E
1
)
2
(E

)
2
. (293)
A
6
(nnncn, cckk, kkn; t
2
e, t
3
e) =

1,2
(it)
2
2
e
iEt
|g
1
1
|
2
g
1
1
g
12
1
g
2
1
g

1

1

2

(E

E
1
)
2
(E

E
2
) (E

)
. (294)
A
6
(nnncn, cnkk, kkc; t
2
e, t
3
e) =

1,2
(it)
2
2
e
iEt
|g
2
1
|
2

2
g
1
1
g
1

1

12

(E

E
1
) (E

E
2
) (E

)
2
. (295)
A
6
(nnncn, cnkk, kkn, kc; t
2
e, t
3
e) =

1,2
(it)
2
2
e
iEt
|g
2
1
|
2
g

1
g

1
1
g
1
1
g

1

12

(E

E
1
) (E

E
2
) (E

)
2
. (296)
51
A
6
(nnncn, cnkk, kkn, kn; t
2
e, t
3
e)
=

1,2,3
(it)
2
2
e
iEt
|g
3
1
|
2
g
1
1
g
12
1
g
2
1
g

1

13

1

23

2

3

(E

E
1
) (E

E
2
) (E

E
3
) (E

)
. (297)
A
6
(nnnnc, ccck; t
2
e, t
3
e) =

1
(it)
2
2
e
iE

g
1

2
g

1
g

1
1
g
1
1
g

1
(E

)
2
(E
1
E

)
2
. (298)
A
6
(nnnnc, ccnk; t
2
e, t
3
e) =

1,2
(it)
2
2
e
iE

g
2

2
g

1
g

1
1
g
1
1
g

1

2

12
(E

)
2
(E
1
E

) (E
2
E

)
. (299)
A
6
(nnnnc, ncck; t
2
e, t
3
e) =

1,2
(it)
2
2
e
iE

g
1

2
g

1
g

1
1
g
12
1
g
2

1

1

2
(E

) (E
1
E

)
2
(E
2
E

)
. (300)
A
6
(nnnnc, ncnk, ck; t
2
e, t
3
e) =

1,2
(it)
2
2
e
iE

2
g

1
g

1
1
g
12
1
g
2

1

1

2
(E

)
2
(E
1
E

) (E
2
E

)
. (301)
A
6
(nnnnc, ncnk, nk; t
2
e, t
3
e)
=

1,2,3
(it)
2
2
e
iE

g
3

2
g

1
g

1
1
g
12
1
g
2

1

1

13

23
(E

) (E
1
E

) (E
2
E

) (E
3
E

)
. (302)
A
6
(nnnnc, nnck, c; t
2
e, t
3
e) =

1,2,3
(it)
2
2
e
iEt
|g
2
1
|
2
g
1
1
g
12
1
g
23
1
g
3
1

13

(E

E
1
) (E

E
2
)
2
(E

E
3
)
. (303)
A
6
(nnnnc, nnnk, cck; t
2
e, t
3
e) =

1,2,3
(it)
2
2
e
iEt
|g
1
1
|
2
g
1
1
g
12
1
g
23
1
g
3
1

2

13

(E

E
1
)
2
(E

E
2
) (E

E
3
)
. (304)
A
6
(nnnnc, nnnk, cnk; t
2
e, t
3
e)
=

1,2,3,4
(it)
2
2
e
iEt
|g
4
1
|
2
g
1
1
g
12
1
g
23
1
g
3
1

2

13

14

24

34

(E

E
1
) (E

E
2
) (E

E
3
) (E

E
4
)
. (305)
A
6
(nnnnn, cccc; t
2
e, t
3
e) =

1,2
(it)
2
2
e
iEt
(g
1
1
g
12
1
g
2
1
)
2

(E

E
1
)
2
(E

E
2
)
2
. (306)
A
6
(nnnnn, ccnc; t
2
e, t
3
e) =

1,2,3
(it)
2
2
e
iEt
g
1
1
g
12
1
g
2
1
g
1
1
g
13
1
g
3
1

23

(E

E
1
)
2
(E

E
2
) (E

E
3
)
. (307)
A
6
(nnnnn, cncc; t
2
e, t
3
e) =

1,2,3
(it)
2
2
e
iEt
g
1
1
g
12
1
g
2
1
g
3
1
g
32
1
g
2
1

13

(E

E
1
) (E

E
2
)
2
(E

E
3
)
. (308)
A
6
(nnnnn, cnnc, kck; t
2
e, t
3
e) =

1,2,3
(it)
2
2
e
iEt
g
1
1
g
12
1
g
2
1
g
3
1
g
31
1
g
1
1

23

(E

E
1
)
2
(E

E
2
) (E

E
3
)
. (309)
A
6
(nnnnn, cnnc, knk; t
2
e, t
3
e)
=

1,2,3,4
(it)
2
2
e
iEt
g
1
1
g
12
1
g
2
1
g
3
1
g
34
1
g
4
1

13

14

23

24

(E

E
1
) (E

E
2
) (E

E
3
) (E

E
4
)
. (310)