Download Perturbation Theory for Quasidegenerate System in Quantum

534
Progress of Theoretical Physics, VoL 72, No.3, September 1984
Perturbation Theory for Quasidegenerate System
in Quantum Mechanics
Kenji SUZUKI and Ryoji
OKAMOTO
Department of Physics, Kyushu Institute of Technology
Kitakyushu 804
(Received March 29, 1984)
Quantum mechanical perturbation theory is formulated for a quasidegenerate system. The equation
of des Cloizeaux and Brandow, which has been a basic equation for determining the usual non-Hermitian
effective interaction in the degenerate perturbation theory, is generalized to the quasidegenerate case. It
is proved that the generalized equation can be solved exactly and the usual non-Hermitian effective
interaction can be given in an explicit form even though unperturbed energies are not degenerate. The
structure of Hermitian forin is also discussed.
§ 1.
Introduction
Perturbation theory has been receiving increasing attention for many years particularly in nuclear and chemical physics_ The purpose of the present study of perturbation theory is to construct an effective Hamiltonian acting in a model space (or zero-order
eigenspace). In previous works in Refs. 1) and 2), a general and unified form of the
effective Hamiltonian was given which includes almost all of the effective Hamiltonians
studied so far, namely the standard non-Hermitian form, the Hermitian form and Kato's
definition, as special cases.
Formulations of perturbation theory are classified into two categories, degenerate and
. quasidegenerate theories, according to whether unperturbed energies of model-space
states are degenerate or not. For a degenerate case, general solutions were obtained for
both the standard non-Hermitian and Hermitian forms. 2 )
A central concern of this work is to extend the theory in Refs. 1) and 2) to the case
that unperturbed energies of model-space states are not degenerate but quasidegenerate.
In the formulation of quasidegenerate perturbation theory we make only one assumption
that an unperturbed model-space Hamiltonian represents a good approximation of the full
Hamiltonian, and we do not need any other restrictive condition for unperturbed energies
or the dimensionality of the model space.' Therefore, we can say that the quasidegenerate
theory deals with the most general situation in quantum-mechanical perturbation theory.
Compared with the degenerate theory, an important significance, of the quasidegenerate
theory is that it has a possibility of constructing an effective Hamiltonian in a wider model
space than that in the degenerate case.
Recent progress of the quasidegenerate perturbation theory was due to Lindgren3 ) and
Shavitt and Redmon. 4 ) They gave some recursion formulae for calculating the effective
Hamiltonian. However, anyone has not succeeded in finding an explicit and exact
expression of the effective Hamiltonian for the quasidegenerate case. Among various
definitions of the effective Hamiltonian, we shall discuss two types of them, the standard
non-HerI)1itian form and the Hermitian form, because the former has a very simple
structure (probably the simplest) and the latter has an advantage of being Hermitian.
Perturbation Theory for Quasidegenerate System
535
In §2, we shall give a general equation for obtaining the effective Hamiltonian. In §3,
the degenerate perturbation theory studied in Refs.l) and 2) is outlined. In §4, a general
theory is formulated for a quasidegenerate case. A general solution for the standard nonHermitian form is given. In §5 , a method is given for constructing the Hermitian
effective Hamiltonian. First several terms in the perturbation expansion of the Hermitian form are presented. A general method is given for converting the non- Hermitian
form to an equivalent Hermitian form. In §6, some conclusions obtained in this study are
given.
§ 2.
General equation for effective Hamiltonian
We shall outline the theory given in Refs. 1) and 2). The entire Hilbert space is
partitioned into the model space (P space) and its complement (Q space). We define
projection operators P and Q which project a state onto the model space and the
complement, respectively.
We introduce an operator w which satisfies
QHP+ QHQw-wPHP-wPHQw=O,
(2·1)
where H is the Hamiltonian of a quantum system. The operator w acts as a mapping of
a P -space state to a Q-space state, that is, w has the following properties:
w=QwP,
(2·2)
wQ=Pw=O.
(2·3)
From these properties of w we see that the operator w t w acts only in the P space.
With w we define two operators acting in the P space as
${(m, n)=(P+w t wtmH(l +w)(P+w t w)n,
(2·4)
x(m, n)=(P+w t w)n-m .
(2·5)
It has been proved in Ref. 2) that if w is a solution to Eq. (2·1), the operator ${(m, n) can
be an effective Hamiltonian for every set of (m, n)in the sense that an eigenvalue Ek
determined through the eigenvalue equation
(2·6)
agrees with one of the eigenvalues of the full Hamiltonian H. The operator x(m, n) is
usually called a metric or an overlap operator.
As we have discussed in Ref. 2), various effective Hamiltonians can be classified by a
set of numbers (m, n). The most familiar effective Hamiltonian, which is written with
the wave operator Q as PHQ, corresponds to ${(O, 0), i.e.,
${(O, O)=PHQ ,
(2·7)
Q=P+w.
(2·8)
where Q is related with w as
The Hermitian effective Hamiltonian, which is often referred to as the canonical form or
K. Suzuki and R. Okamoto
536
the Van Vleck form, agrees with $C( -1/ 2, -1/ 2). Furthermore, the operator $C(O, -1)
corresponds to Kato's effective Hamiltonian. 5)
Among various effective Hamiltonians, $C(O, 0) and $C(-1/2, -1/2) have special
importance, because $C (0, 0) has probably the simplest structure and $C( -1/ 2, - 1/2) has
an advantage of being Hermitian. It is noted that the associated overlap operators for
both of $C(O, 0) and $C(-1/2, -1/2) are the identity operator in the P space, that is,
x(O, 0) = X( -1/ 2, -1/2) = P. On the other hand, Kato's effective Hamiltonian $C (0, -1)
is accompanied with a non-identity overlap x(O, -1)=(P+W t W)-l. The use of an
effective Hamiltonian with the identity overlap will of course make the problem simpler.
Therefore, in the present paper the structure of two effective Hamiltonians $C(O, 0) and
$C( -1/ 2, -1/2) will mainly be discussed.
The problem of constructing the effective Hamiltonian can be reduced to solving the
operator w in Eq. (2·1). Once the operator w is given, every type of effective
Hamiltonian can be calculated according to Eq. (2·4). The formal relation between
Eq. (2·1) for wand the definition of $C(m, n) in Eq. (2·4) is true irrespective of the natun:;
of the Hamiltonian H. We may also say that the definition of the effective Hamiltonian
$C(m, n) is always valid whether the unperturbed energies of the P-space states are
degenerate or not.
§ 3.
Degenerate perturbation theory
The perturbation theory has been studied extensively for the case that the P-space
states have degenerate unperturbed energies. For the degenerate case the solution w in
Eq. (2·1) was solved exactly.2) Some methods used in the degenerate perturbation theory
will also be useful in formulating a theory for a system with quasidegenerate unperturbed
energies.
We here review briefly the outline of the degenerate perturbation theory given in
Refs. 1-) and 2). The full Hamiltonian H is split into two parts
H=Ho+ V,
(3·1)
where Ho is an unperturbed Hamiltonian and V a perturbation. We assume that the
unperturbed energies of the P-space states are all degenerate, i.e.,
PHoP=EoP.
(3·2)
R=PVP+ PVQw
(3·3)
We define an operator R
which is the effective interaction in the standard perturbation theory and is related with
$C(O, 0) in Eq. (2·7) as
$C(O, O)=EoP+ R .
(3·4)
In the degenerate case, Eq. (2·1) for w can be written in terms of R as
(3·5)
Iterating Eq. (3·5) for w, we have
Perturbation Theory for Quasidegenerate System
w= ~o( - )n( Eo- ~HQ
r+l
537
(3·6)
QVP Rn.
Substituting w in Eq. (3'6) into Eq. (3'3), we have an equation for R
R=
i: QnR n ,
(3'7)
n=o
where
__ {
Qn-
PVP+PVQ(Eo-~HQ )QVP
for n=O,
n
(
1
(-) PVQ Eo-QHQ
for n;;;;l.
)n+1
QVP
(3'8)
According to Kuo 6) and his collaborators we call Qo the Q-box. Noting that QHQ
= QHoQ+ QVQ, we can write Qo in an expansion form as
-
.
1
1
1
Qo=PVP+ PVQ-QVP+ PVQ-QVQ-QVP+"',
e
e
e
(3'9)
where
(3·10)
The operator Qn for n~l can be written in the energy-derivative expression as
- _ 1 dnQo
Qn-nr dEon .
(3'11)
Equation (3'7) for R was also obtained by des Cloizeaux7l and Brandow8) in different
ways.
Equation (3'7) for R can be solved exactly and R is given byl),2)
R=
i: i: i: ... i: f(ml,
n=lml=om2=O
mn=O
m2, ''', mn)Qm'Qm.... Qmn'
(3'12)
where f( ml, ''', mn) is a coefficient defined by
for n=l,
for n=2,
(3'13)
In the above, r;(x) denotes the step function
r;(x
)={ ~
forx;;;;O,
forx<O.
(3'14)
It is noted that /(ml, "', mn) takes the value 1 or 0, because it is given by a product of
only the Kronecker a functions and the step functions. The solution R expressed with the
f coefficient is essentially equivalent to that obtained from the general expansion rules of
des Cloizeaux,1) Lee,9) and Krenciglowa and KUO.lO) Substituting R in Eq. (3'12) into
Eq. (3·6), we can write w as
K. Suzuki and R. Okamoto
538
With the solution OJ, the effective Hamiltonian ..9C(m, n) in Eq. (2·4) and the overlap
operator x(m, n) can be given, in principle, in terms ofthe Q-box for every set of (m, n).
In particular, an expansion formula for the Hermitian form ..9C( -1/ 2, -1/ 2) has been
given in Ref. 2).
§ 4.
4.1.
Properties
01 I
Quasidegenerate perturbation theory
coefficient
Before discussing the quasidegenerate perturbation tht:0ry, we shall refer to the
properties of the I coefficient defined in Eq. (3'13). IUs very convenient to introduce a
concept of allowed sequence. Let (ml, m2, "', mn) be a sequence of n zero-or-positive
integers. A set (ml, m2, "', mn) is called an allowed sequence if the coefficient I( ml, m2,
"', mn) takes the value 1. If /(ml, m2, "', mn)=I, QmJJmz"Qmn appears as an allowed
term in the expansion of R in Eq. (3·12).
We now introduce a simplified notation for writing a sequence:
(4'1)
(In Ref. 2) the notation [m]n = (ml, m2, "', mn) was used, but in the present paper we shall
use mainly another notation m[i, j].) The / coefficient satisfies the following formula:
/(m[I, rl])/(m[rl + 1, r2])"'/(m[rn-1 + 1, rn])= /(n, m[I, rn]).
(4'2)
The above relation means that if n sequences {m[Yi-I+I, r;]}, I~i~n and ro=O, are all
allowed, then a combined sequence
(n, m[I, rl], m[rl+I, r2], "', m[rn-I+I, rn])
can also be allowed. It should be noted that when we want to construct a new allowed
sequence from n allowed sequences, we must add the number n as the first term. This
property was first proved by des Cloizeaux.7) The other properties of the / coefficient
have been discussed in Ref. 2).
The converse relation of Eq. (4·2) is also valid, that is, for a given allowed sequence
(ml, m2, "', mn)=m[I, n] there exists a set of mi positive integers (rl, r2, "', rm.)
satisfying
(4·4)
such that m[2, n] is decomposed into mi allowed sequences, or equivalently l(m[I, n]) is
written in a product form as
(4'5)
Since m[2, rl] is allowed, we can again decompose m[3, rl] into m2 allowed sequences.
Continuing this decomposition procedure, we finally reach a theorem: Let (ml, m2, "', mn)
Perturbation Theory- for Quasidegenerate System
539
be an allowed sequence. If mi ~ 1, 1 ~ i ~ n, there exists a number p, i + 1 ~ P ~ n, such that
the sequence m[i + 1, p] is decomposed into mi allowed subsequences as
m[i+1, P]=(m[i+1, rl], m[rl+1, r2], "', in[rmi-1+1, rm.]).
(p=rmJ
(4·6)
The decomposition of m[i + 1, p] in Eq. (4·6) is specified by a set of numbers (rl, r2, "',
rmJ. The set (rl, r2, "., r mi ) can be considered to be a function of a sequence m[i, n]
which is a subsequence of a given sequence m[l, n]. We thus write
(4·7)
If mi=O, we also define
r(m[i, n])=O
for mi=O,
(4·8)
where 0 denotes the empty set. For illustration the sets (rl, r2, ... ) are given for simple
cases in Table I.
4.2.
Expansion formula for non-Hermitian effective interaction R
We assume that the P-space unperturbed Hamiltonian PHoP has quasidegenerate
eigenvalues. We write the projection operator P as
Table I. The set of numbers r(m[i, n]).- The symbol 0 denotes the empty set.
n
1
2
(0)
0,0)
3
(2, 0, 0)
0, 1, 0)
4
m[i, n]
m[l, n]
(3, 0, 0, 0)
(2, 1, 0, 0)
(2, 0, 1, 0)
0, 2, 0, 0)
(1, 1, 1, 0)
1
1
2
1
2
3
1
2
3
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
(0)
(1,0)
(0)
(2, 0, 0)
(0, 0)
(0)
(1, 1, 0)
(1, 0)
(0)
(3, 0, 0, 0)
(0, 0, 0)
(0, 0)
(0)
(2, 1, 0, 0)
(1, 0, 0)
(0,0)
(0)
(2, 0, 1, 0)
(0, 1, 0)
(1, 0)
(0)
(1,2,0,0)
(2, 0, 0)
(0,0)
(0)
(1, 1, 1, 0)
0, 1, 0)
(1, 0)
(0)
r(m[i, n])
0
(2)
0
(2,3)
0
0
(3)
(3)
0
(2, 3, 4)
0
0
0
(3,4)
(3 )
0
0
(2, 4)
0
(4)
0
(4)
(3, 4)
0
0
(4)
(4)
(4)
0
K. Suzuki and R .. Okamoto
540
d
(4·9)
P=L:Pk
k=l
with
(4·10)
where Ik> is an unperturbed eigenstate of PHoP and d is the dimension of the P space.
Let EOk be the unperturbed energy of the state Ik>. We then write
(4·11)
Using Eqs. (2·1), (3·1) and (4·11), we have an equation for w
d . 1
d
1
w= k=l
L:-(k)
QVPkL:-(k)
wRPk ,
e
k=l e
(4·12)
e(k)=Eok - QHQ.
(4·13)
where
For the degenerate case that e(k) does not depend on k, Eq. (4·12) is reduced to Eq. (3·5).
Iteration of Eq. (4 ·12) leads us, to
(4·14)
where we have used a symbol defined generally as
b
b
b
b
L: =L:L:"'L:
k[l,n]=a
k,=akz=a
kn=a
(4·15)
e(k[l, n])=e(kl)e(k2)···e(kn).
(4·16)
and e(k[l, n]) denotes
Substituting w in Eq. (4·14) into Eq. (3·3), we have an equation for R
(4·17)
where
for n=l,
(4·18)
for
n~2.
It is noted that S(kl, k2, "', k n ) is Hermitian and it does not depend on the order of
(kl, k2, "', kn) because the operators {e(k;)} are mutually commutable. For example, we
write
(4·19)
The inverse of e(k) is expanded into
Perturbation Theory jor Quasidegenerate System
111
11
1
1
--=-+-QVQ-+-QVQ-QVQ-+···
e(k)
eOk
eOk
eOk
eOk
eOk
eOk
541
(4·20 )
with
(4·21)
From Eq. (4·20) we see that the operator S(kI, k2, ... , k n ) can easily be expanded into the
series in which each term has only energy denominators given with unperturbed energies.
For the degenerate case, S(kI, k2, ... , kn) agrees with the Q-box Qn-I and Eq. (4·17) is
reduced to Eq. (3·7) of des Cloizeaux 7 ) and Brandow. B) Therefore, Eq. (4·17) can be
considered as a natural extension of Eq. (3·7) to the quasidegenerate case.
Equation (4·17) for R can be solved exactly as
'"
R= L:
'"
d
d
d
L:
L:
L:
L:
n=Im[l,n] =Ok[l,n] = ll,[l,m,] = 112[1,m2]= 1
d
L:
In[l,mn]=l
n
_
8(kI, 11[1, mI], k2, Ml, m2]"··, kn, In[1, mn]){II S(k i , Ii [1, m;])Pk,},
i=l
(4·22)
where we have used the symbol defined in Eq. (4 ·15). The notation Ii [1, m;] means a set
of mi integers written with double subscripts as
(4·23)
If mi=O, 1;[1, mi] means the empty set.
{S(ki)
S(k i , 1;[1, m;])= S-(k. t.
z,
11,
The operator S(ki, li[l, m;]) denotes
t·12, ... , zt.m t )
for mi=O,
for mi~l.
(4·24)
The 8 is a coefficient given by
n
=j(m[l, n])IIo(l;[l, m;], k{r(m[i, n])}),
i=l
(4·25)
where k{r(m[i, n])} means a subset of (kI, k2, ... , k n ) determined with the set r(m[i, n])
=(rI, r2, ... , rmJ in Eq. (4·7) as
k{r(m[i, n])}=(kr" kr., ... , krq).
(q=mi)
(4·26)
The 0 function on the r.h.s. of Eq. (4·25) means a product of the Kronecker 0 functions
defined with the notations in Eqs. (4·23) and (4·26) as
(4· 27)
Since the 8 coefficient is wyitten by a product of only the j coefficient and the 0 functions,
it takes only 0 or 1. The proof of Eq. (4·22) for R is given in the Appendix.
A general expression of R in Eq. (4·22) is rather complicated. Equivalent rules for
generating the terms contained in R are given as follows:
(1)
Consider an allowed sequence (mI, m2, ... , mn)(=m[l, n]).
(2)
If mi~l, find a number p and decompose (mi+I, mi+2, ... , mp) into mi allowed
K. Suzuki and R. Okamoto
542
sequences. This decomposition procedure determines uniquely a set of numbers
r{m[i, n]} in Eq. (4'7) which we denote as
(4'28)
r{m[i, n]}=(ril, riZ, "', rimJ.
If mi=O, set r{m[i, n]}=O as defined in Eq. (4'8).
Follow procedure (2) for all m/s and determine the sets r{m[i, n]},
Consider finally
(3)
1~ i~
n.
(4·29)
where a= ml, b= m2, "', z= mn. The term given in Eq. (4'29) appears as an allowed term
in the expansion orR in Eq. (4'22).
An essential point of the rules for generating the terms contained in R is to determine
the set r{m[i, n]} in Eq. (4'28). Some examples of r{m[i, n]} are given in Table 1. If
r{m[i, n]} is known, the corresponding term in R can be calculated according to
Eq. (4·29). The effective interaction R can be written explicitly through fourth order in
S as
R=
"2.SIPI
+ "2.
SI2PlS2P2+ 1,2,3
"2. {S123PIS2P2S3P3+ S13PIS23P2S3P3}
I
1,2
(4'30)
+SI4PIS234P2S3P3S4P4+S14PIS24P2S34P3S4P4}+'" ,
where we have used the notations
Sij"'l=S(ki, k j ,
(4'31)
kl ),
"',
(4'32)
and
"2.
l,2,···,n
d
=
d
d
"2. "2.' .. kn=
"2. 1
(4'33)
kI = 1 k2= 1
4.3. Solution for the operator (J)
Since the operator R is solved exactly, the operator (J) can also be given exactly.
Substituting R in Eq. (4·22) into Eq. (4·14), we have
d
. .
00
00
d
d
d
(J)= "2. (l/e(k))QVPk+ "2. "2.
"2.
"2. '''''2.
k=1
n=2m[I,n]=Ok[I,n]=ll,[I,m,]=1 I nll,mn]=1
(4'34)
Perturbation Theory for Quasidegenerate System
543
where the coefficient 8 and e( Ml, ml]) have been defined in Eqs. (4·25) and (4 ·16),
respectively. If we substitute 8 in Eq. (4·25) into Eq. (4·34), w can be written in a more
compact form as
1
n
-
.
x e(kl)e(l[I, ml]) QVPk.{}lS(kj , k{r(m[j, n])})PkJ}
(4·35)
where 8 has been defined in Eq. (4·27).
For the construction of the Hermitian effective interaction, we need an expression of
the operator w t w. Since w is known, w t w can also be given exactly. It is quite interesting that w t w can be given in terms of only the operators 5 and P k • The expression of
w t w is rather complicated, but it is given explicitly as
d
00
wtw=-~~
~
~
P=IQ=l m [l,P+q]=Ok[l,P+q]=l
P-2
_
f(m[l, P])f(m[P+ 1, p+q]){II Pkp_JS(k p - j , k{r(m[p- j, p])})}
j=O
(4·36)
where, if P = 1 (or q = 1), the product {II1;;;:g···}( or {IIJ4"}) means merely the identity
operator. First few terms in the expansion of w t ware given by
w t w= - ~PlSI2P2- ~ {P1 SIP2S!23P3 +h.c.}
1,2
1,2,3
(4·37)
where we have used the notations in Eqs. (4·31)~(4·33). With the solutions for wand
w t w, a general effective Hamiltonian &(m, n) can be calculated according to Eq. (2·4).
§ 5.
Construction of Hermitian effective interaction
The Hermitian effective interaction denoted by W is introduced through
&(-1/2, -1/2)=PHoP+W.
(5·1)
For the quasidegenerate PHoP, W is written in terms of wand R in Eq. (3·3) as
W=(P+w t W)I/2R(P+w t wtl/2- [PHoP, (p+w t W)I/2](P+wt wt 1 / 2 •
(5·2)
This expression of W is of Brandow. 8 )
Another expression of W is given through a unitary transformation of the
Hamiltonian as
544
K. Suzuki and R. Okamoto
(5·3)
where G is related with w as
G = arctanh(w- w t )=
~
(- )n {w(w t w)n - h.c.}.
n=o2n+ 1
(5·4)
The above relation was obtained by Shavitt and Redmon,4) Westhaus l l ) and one of the
authors (K. S.)12).
It has been proved that the term PHoP in Eq. (5·2) or (5·3) can be eliminated and W
is written in terms of only wand R as 4),12)
00
00
W = L: L: F(P, q )(w t w)PR(w t w)q + h.c. ,
(5·5)
P=Oq=o
where F(p, q) is a coefficient defined as 4)
1/2
F(p, q)= { (- )p+q(q-p)(2p-1)!!(2q-1)!!
2(p + q )(p + q + 1 )(2p )!!(2q )!!
for p=q=O,
otherwise.
(5·6)
The expression of W in Eq. (5·5) is always valid whether the unperturbed energies of
P HoP are degenerate or not.
Among various effective interactions, the Hermitian form will of course be desirable.
In the degenerate case, a general expansion formula for the Hermitian form W in
Eq. (5·5) has been given,2) although it was somehow complicated.
For obtaining the expansion formula for W in the quasidegenerate case, the expression of W in Eq. (5·5) will also be useful. Since Rand w t ware given as in Eqs. (4·22)
and (4·36), respectively, the Hermitian effective interaction W can be written in terms of
the operators Sand P k • An exact expression of W is possible, but it seems impractical
because of its complexity. It will be a rather easy task to obtain first several terms in the
expansion of W. Using Eqs. (4·30) for R, (4·37) for w t wand (5·5) for W, we have
Perturbation Theory for Quasidegenerate System
545
(5·7)
where we have used the same notations as in Eq. (4·30) for R.
Another way of constructing the Hermitian form is as follows: We first calculate
{j)t (j) in Eq. (4·36) or (4·37).
The calculation of {j)t (j) will be much simpler than the direct
calculation of W. Consider an eigenvalue equation
(5·8)
Since (j)t (j) is a P-space operator, the eigenstate lai> belongs to the P space. All the
eigenvalues of (j) t (j) must be zero-or-positive.1 2) For this reason, we have written the
eigenvalue of (j) t (j) as /1/. It has been proved that, if we use a basis {Iai>}, the series for
W in Eq. (5·5) can be summed up and it becomes 12)
The above expression of W implies that we need only the operators Rand (j) t (j) for
constructing W. Since the exact expressions of Rand (j) t (j) have been known, we may
say that the problem of construCting the Hermitian form W for a quasidegenerate system
has been solved in a formal point of view.
§ 6.
Conclusions
We have studied a general perturbation theory for a quasidegenerate system. We
have found that Eq. (3·7) of des Cloizeaux 7 ) and Brandow,S) which has been a basic
equation in constructing their perturbation theories for a degenerate system, can naturally
be generalized to Eq. (4 ·17) for a quasidegenerate case. We have shown that this
generalized equation can be solved exactly and the standard non-Hermitian effective
interaction can be written explicitly in an operator form. The relation between the nonHermitian and Hermitian effeCtive interactions has been discussed. A method has been
given for converting the standard non-Hermitian form to an equivalent Hermitian form.
The whole results in the present formalism have been given by means of the operator
form, not by diagrams. Since we have not specified the full Hamiltonian, the present
theory will be applicable to every system subject to the non-relativistic quantum
mechanics. However, when we want to apply the theory to a many-body system, it will
of course be desirable to investigate how the present formulation relates with the usual
diagram method. This problem will be very interesting, but we here may conclude that
the problem of constructing the effeCtive interaction has been solved formally as far as we
are concerned with the time-independent and algebraic approach to the quasidegenerate
perturba,tion theory.
Appendix
We shall prove that the Qperator R in Eq. (4·22) can be a solution to Eq. (4·17), that
is, the r.h.s. of Eq. (4·17) which we denote as Y coincides with R. Let us write R in a
more compact form as
546
K. Suzuki and R. Okamoto
n
R=
~
_
/(m[l,nJ){II S(k j , k{r(m[j, n])} )Pk ,},
{n,m[l,nj,k[l,n]}
j=l
(A'1)
where
00
~
{n,m[1,n],k[1,n]}
=~
d
00
~
~.
n=lm[l,nj=Ok[l,n]=l
(A·2)
For the derivation of Eq. (A'1) we have used Eqs. (4·25) and (4'27).
We note that the second term in Yon the r.h.s. ofEq. (4·17) contains a product of R's.
For each R we denote the numbers n, m[l, n] and k[l, n] in the summation in Eq. (A'1)
as ni, m(i)[l, ni] and k(i)[1, ni], respectively. Substitution of R into Y gives
(A·3)
In the above, it is sufficient to consider only allowed sequences {m(i)[i, ni]} because
otherwise /(m(i)[l, niJ)=O and no contribution is added to Y.
Let us now prove that Y in Eq. (A, 3) becomes equal to R in Eq. (A·1). The
following three facts should be noticed:
(1) A product of the / coefficients in Eq. (A· 3) is reduced to a single / coefficient as
/(m(1)[l, n1J)/(m(2)[1, n2])···/(m(q)[1, nqJ)
=/(q , m 1
(1) m (1)
,
2,
•••
,
m n(1)
l'
•••
,
m 1(q) , m 2
(q), ... , m(q»)
nq,
(A·4)
where we have used the formula in Eq.(4·2). We alter numbering of the arguments
appeared in the / coefficient as m1=q, m2=m1(1), m3=m2(1), .. ', mN=m<,.q~. We then may
write
=m[l, N],
(A'5)
where
(A'6)
From Eqs. (A·4), (A'5) and (A'6) we have
/(m(1)[l, n1J)/(m(2)[1, n2J)"'/(m(q)[1, nq])= /(m[l, NJ).
(A'7)
(2) Let m(1)[l, n1], m(2)[1, n:i], "', m(q)[l, nq] in Eq. (A· 7) be all allowed. Then, they
correspond to decomposed sequences of m[2, N], which may be clear from Eqs. (A·4)
~ (A· 7). A set of numbers (n1, n2, "', nq) satisfy the condition for determining the set
r(m[l, NJ) in Eq. (4·7). Hence we have
(A·S)
In the second term on the r.h.s. of Eq. (A·3), a product of two projection operators appears
and it becomes
(s=ni, 1~ i~q)
(A·9)
547
Perturbation Theory for Quasidegenerate System
Similarly to Eq. (A -5) we renumber
(A -10)
Considering Eq. (A-g) for all i's and Eq. (A-10), we have a relation
(II, 12, --', lq)= l[l, q]
(A -11)
Recalling the rules for determining k{r(m[l,N])} in Eq. (4-26), we may write
(knl,knz,'--, knq)=k{r(m[l, N])}.
(A -12)
From Eqs. (A -11) and (A -12) it follows that
S(I, HI, q])=S(kl, k{r(m[l, N])}).
(A -13)
(3) According to the renumbering rules in Eqs. (A -5) and (A -10), the arguments of S in
Eq. (A-3) are altered to
(A-14)
where
for i=l,
for i;;;;;2.
(A -15)
We here note the following: A set of integers r(m[Ni+j, NiH]) in Eq. (A-14) are
determined only by a certain partial sequence m[Ni + j, tJ, t;;;;.Ni+l and remain unchanged
even though the allowed sequences m[Ni+l+1, N i+2], "', m[Nq +1, N] are added after
m[Ni+ j, NiH]. Hence, as long as NiH;;;;'N, we may write
(A -16)
Using Eqs. (A-n, (A-13) and (A-16), we finally obtain an expression of Y as
N
_
x {II
S(k i ,
i=2
co
=
L:
N=l
co
m
d
k{r(m[i, N])} )PkJ
N_
L:
L: f(m[l, N]){IIS(k
i , k{r(m[i, N])})PkJ
[I,N]=Ok[I,N]=1
i=1
(A -17)
which is just identical to R. This final result means that R can be a solution to Eq. (4 -17)
for determining the effective interaction in the quasidegenerate case.
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K. Suzuki and R. Okamoto
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