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An extended gaussian overlap approximation in the generator coordinate method
Volume 152B, number 5,6
PHYSICS LETTERS
14 March 1985
AN EXTENDED GAUSSIAN OVERLAP APPROXIMATION IN THE GENERATOR COORDINATE METHOD A. GOZD2 1 Institute of Theoretical Physics 2, University of Regensburg, D-8400 Regensburg, West Germany Received 17 October 1984 Revised manuscript received 20 December 1984
A derivation of a SchrSdinger-type collective hamiltonian for a wide class of the overlap functions is shown. The presented approach is an extension of the gaussian overlap approximation in the generator coordinate method.
In the generator coordinate method (GCM) of Griffin, Hill and Wheeler [1 ], one considers a family of many-particle wave functions la) which depend parametrically on one or several collective variables a i (i = 1 , 2 ..... n). With the help of these so-called generating functions, an approximate wave function of the many-particle system is constructed by taking superpositions of the form Iff) = f d a f ( a ) l a ) .
(1)
The weight functionf(a) of the ansatz (1) is determined from the variational principle for the expectation value of the total many-body hamiltonian H: 6 ~------T (~1 '
(2)
which leads to the Hill-Wheeler integral equation for f ( a ) [ 1,2]. Its solution is in general rather awkward. Instead, one can easily obtain a collective hamiltonian using the so-called gaussian overlap approximation (GOA) [1,3,4]. In this approximation the overlap function (ala') must be a gaussian as a function of differences of the collective variables, s k = a k - a 'k. It is justified for rather heavy systems and only one collective variable 1 On leave of absence from the Physics Department of The Maria Sldodowska-Curie University, 20-031 Lublin, Poland. 2 Work supported in part by GSI, Darmstadt, West Germany. 0370-2693/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
while using the BCS-type generating functions [ 2 - 4 ] . For either lighter nuclei or a few collective variables the gaussian overlap approximation usually breaks down and one cannot use this very useful, fully quantal approach. It is the purpose of this paper to present a derivation of a collective hamiltonian, with a more general form of the overlap function and all advantages of the GOA formalism. For this goal we propose to replace the overlap (ala') by the model overlap function
(ala') ~ (ala')G = exp(-l ~ [p(a,a')] 2 ) .
(3)
K=I
The real functions F K should satisfy the following three conditions: (i) The set of the functions FK(a, a0) stands for the one-to-one correspondence between (a K) and new variables {~K = pK(a ,a0)} ' (ii) PK(a, a0) + FK(a0,a ') = FK(a,a'),
(iii) r~(a, a') = -r~(a', a), for a fixed (but, in principle, arbitrary) basic point a 0 and arbitrary points a and a'. A possible choice for the functions F K in the ansatz (3) is the following: n
P~(a, a0) =
=
fw(q)dq u,
(4)
C
where C denotes an arbitrary but f i x e d path of integration between the endpoints a 0 and a. Elementary 281
Volume 152B, number 5,6
PHYSICS LETTERS
properties of the line integrals insure that the conditions (ii) and (iii) are always fulfilled but (i) must be checked in each case separately. The representation (4) of the functions F ~ is quite general. It can be used in practice with an appropriate choice of the matrix (w~) to reproduce the overlaps not only of gaussian shapes. To introduce a metric structure in a collective space, it is useful to define a metric tensor 7uv(a). However, in the GCM, there is no physical prescription for this fundamental tensor i.e. one can suppose that a choice ofTu v is arbitrary, but the exponential argument must be a scalar in the collective space. To obtain the simplest picture, we assume that in the a K = PK(a, a0) coordinates the metric tensor ~uv(a) is represented by the Kronecker-delta function
~(~)=~
.
(5)
It is easy to prove that it corresponds to the following metric tensor in the collective variables {a K}: n
?uv(a) = ~ 3PK(a'ao) 3P•(a,ao) ~=I 3au aav
(7)
a
= [7(~)] l/2(a - a'),
(8)
a'
where ~ is a point in the interval (a',a) obtained by the Cauchy theorem. Usually, one assumes the approxima_1 l tion ~ - i ( a + a) to end up with a standard form of the GOA [2,3,6,7]. A natural, but awkward way to calculate the metric tensor 7uv(a) from the evaluated functions P~ can be often avoided using the following approximation:
282
( 3 _ ~
3
s=0
.
(9)
is=0
The expression (9) for 7u v can be easily obtained from eq. (3) with a = q + s/2 and a' = q - s/2. It is a straightforward extension of the approximation always used in the GOA [2,3]. To derive the collective hamiltonian one can start from the expression for the expectation value of the many-body hamiltonian H (qJIHI~) =
f f da da' f(a)*h(a,a')(ala') G f(a'),
(10)
where the GCM wave function is given by the equation
f da1d a2
...
dan f( al , a2 ..... an)lala2...an), (11)
and the reduced energy overlap is written as
This special case corresponds to the local gaussian approximation considered by Onishi and Une [5] where the assumption that the line integral in (4) is independent of a path relates to the integrability condition in the solution of their eigenequation for the overlap function. Obviously, in the one-dimensional case eq. (5) can always be used and one can write
f D(q)] 1/2 dq
~-~'-(q + s/2lq - s/2>G)
a u as v
(6)
K
r(a,a') =
7uv(q) = - ( ~
IV> =
Within the ansatz (4) but with path independent line integral the metric tensor 7uv(a) can be directly expressed in terms of the matrix w:
vu (a) ='23
14 March 1985
h(a, a') = (a Inla'>/
(12)
After the transformation of eq. (10) to the new coordinate system {a K} in which
(a,a')G=(OdO/)G=exp(--l~(aK--o/~)2),
(13)
the standard derivation, as in the GOA, of a collective hamiltonian can be applied [2,4,8]. Because (13) is of gaussian shape with a constant width the reduced energy overlap h'(a, a') can be replaced by an infinite order differential operator. However, for a collective motion, the expansion (see eq. (10.130) of ref. [2] ) up to second order for the differential operator seems to be sufficient [6,9]. After the inverse transformation to the collective variables {a K} we obtain the following collective hamiltonian:
1 0
r-[
1 ]ll' b
~coll =-
(14) where 3, - det(7i/), the inverse mass tensor is given by the formula
Volume 152B, number 5,6
(
1
)
2-~(q) p.v
Aal~
=-4
PHYSICS LETTERS
lIzxa,Ah(a'a') __
Aa v
7
A_,V i.Ja=a,=q a
(15)
and the potential energy is the difference o f the expectation value (q IHlq) and the so-called zero point energy [10,11]
V(q) = (qlnlq) - e 0
(16)
with
eo = - !7~v2
14 March 1985
is a slowly varying function, as it is expected in collective phenomena, one can obtain a Schr6dinger-type collective hamiltonian. In this way we end up with an approach which has a wide range of applicability, with all advantages o f a simple gaussian overlap approximation. The author would like to thank Professor K. Pomorski, E. Werner and M. Brack for stimulating discussion and for many helpful comments on this subject.
References A ( A h ( a , q) t Aq~ Aa v I a=q
+53'1~ V ( ~ a t~A A h ( a ,vq ) ]
(17)
a=q Here A/Aau denotes the covariant derivatives and 7~ v stands for the inverse (contravariant components) o f the metric tensor 7~v" To Summarize the results o f the paper we draw some conclusions. The model overlap function (3) is flexible enough to reproduce a wide class of the overlap functions. The approximation (9) often allows to skip the difficult problem o f calculations of the functions F ~. In the case when the reduced energy overlap
[1] J.J. Griffin and J.A. Wheeler, Phys. Rev. 108 (1957) 311. [2] P. Ring and P. Schuck, The nuclear many-body problem (Springer, Berlin, 1980), and references therein. [3] D.M. Brink and A. Weiguny, Nucl. Phys. A120 (1968) 59. [4] B. Banerjee and D.M. Brink, Z. Phys. 258 (1973) 46. [5] N. Onishi and T. Une, Prog. Theor. Phys. 53 (1975) 504. [6] P.G. Reinhaxd, Nucl. Phys. A261 (1976) 291. [7] L.S. Ferreira and M.H. Caldeira, Nucl. Phys. A189 (1972) 250. [8] B. Giraud and B. Grammaticos, Nucl. Phys. A233 (1974) 373; A~255 (1975) 141. [9] P.G. Reinhard et al., Z. Phys. A317 (1984) 339. [10] M. Girod and B. Grammaticos,.Nucl. Phys. A330 (1979) 40. [11] P.G. Reinhard, Nucl. Phys. A252 (1975) 120;A306 (1978) 19.
283
PHYSICS LETTERS
14 March 1985
AN EXTENDED GAUSSIAN OVERLAP APPROXIMATION IN THE GENERATOR COORDINATE METHOD A. GOZD2 1 Institute of Theoretical Physics 2, University of Regensburg, D-8400 Regensburg, West Germany Received 17 October 1984 Revised manuscript received 20 December 1984
A derivation of a SchrSdinger-type collective hamiltonian for a wide class of the overlap functions is shown. The presented approach is an extension of the gaussian overlap approximation in the generator coordinate method.
In the generator coordinate method (GCM) of Griffin, Hill and Wheeler [1 ], one considers a family of many-particle wave functions la) which depend parametrically on one or several collective variables a i (i = 1 , 2 ..... n). With the help of these so-called generating functions, an approximate wave function of the many-particle system is constructed by taking superpositions of the form Iff) = f d a f ( a ) l a ) .
(1)
The weight functionf(a) of the ansatz (1) is determined from the variational principle for the expectation value of the total many-body hamiltonian H: 6
(2)
which leads to the Hill-Wheeler integral equation for f ( a ) [ 1,2]. Its solution is in general rather awkward. Instead, one can easily obtain a collective hamiltonian using the so-called gaussian overlap approximation (GOA) [1,3,4]. In this approximation the overlap function (ala') must be a gaussian as a function of differences of the collective variables, s k = a k - a 'k. It is justified for rather heavy systems and only one collective variable 1 On leave of absence from the Physics Department of The Maria Sldodowska-Curie University, 20-031 Lublin, Poland. 2 Work supported in part by GSI, Darmstadt, West Germany. 0370-2693/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
while using the BCS-type generating functions [ 2 - 4 ] . For either lighter nuclei or a few collective variables the gaussian overlap approximation usually breaks down and one cannot use this very useful, fully quantal approach. It is the purpose of this paper to present a derivation of a collective hamiltonian, with a more general form of the overlap function and all advantages of the GOA formalism. For this goal we propose to replace the overlap (ala') by the model overlap function
(ala') ~ (ala')G = exp(-l ~ [p(a,a')] 2 ) .
(3)
K=I
The real functions F K should satisfy the following three conditions: (i) The set of the functions FK(a, a0) stands for the one-to-one correspondence between (a K) and new variables {~K = pK(a ,a0)} ' (ii) PK(a, a0) + FK(a0,a ') = FK(a,a'),
(iii) r~(a, a') = -r~(a', a), for a fixed (but, in principle, arbitrary) basic point a 0 and arbitrary points a and a'. A possible choice for the functions F K in the ansatz (3) is the following: n
P~(a, a0) =
=
fw(q)dq u,
(4)
C
where C denotes an arbitrary but f i x e d path of integration between the endpoints a 0 and a. Elementary 281
Volume 152B, number 5,6
PHYSICS LETTERS
properties of the line integrals insure that the conditions (ii) and (iii) are always fulfilled but (i) must be checked in each case separately. The representation (4) of the functions F ~ is quite general. It can be used in practice with an appropriate choice of the matrix (w~) to reproduce the overlaps not only of gaussian shapes. To introduce a metric structure in a collective space, it is useful to define a metric tensor 7uv(a). However, in the GCM, there is no physical prescription for this fundamental tensor i.e. one can suppose that a choice ofTu v is arbitrary, but the exponential argument must be a scalar in the collective space. To obtain the simplest picture, we assume that in the a K = PK(a, a0) coordinates the metric tensor ~uv(a) is represented by the Kronecker-delta function
~(~)=~
.
(5)
It is easy to prove that it corresponds to the following metric tensor in the collective variables {a K}: n
?uv(a) = ~ 3PK(a'ao) 3P•(a,ao) ~=I 3au aav
(7)
a
= [7(~)] l/2(a - a'),
(8)
a'
where ~ is a point in the interval (a',a) obtained by the Cauchy theorem. Usually, one assumes the approxima_1 l tion ~ - i ( a + a) to end up with a standard form of the GOA [2,3,6,7]. A natural, but awkward way to calculate the metric tensor 7uv(a) from the evaluated functions P~ can be often avoided using the following approximation:
282
( 3 _ ~
3
s=0
.
(9)
is=0
The expression (9) for 7u v can be easily obtained from eq. (3) with a = q + s/2 and a' = q - s/2. It is a straightforward extension of the approximation always used in the GOA [2,3]. To derive the collective hamiltonian one can start from the expression for the expectation value of the many-body hamiltonian H (qJIHI~) =
f f da da' f(a)*h(a,a')(ala') G f(a'),
(10)
where the GCM wave function is given by the equation
f da1d a2
...
dan f( al , a2 ..... an)lala2...an), (11)
and the reduced energy overlap is written as
This special case corresponds to the local gaussian approximation considered by Onishi and Une [5] where the assumption that the line integral in (4) is independent of a path relates to the integrability condition in the solution of their eigenequation for the overlap function. Obviously, in the one-dimensional case eq. (5) can always be used and one can write
f D(q)] 1/2 dq
~-~'-(q + s/2lq - s/2>G)
a u as v
(6)
K
r(a,a') =
7uv(q) = - ( ~
IV> =
Within the ansatz (4) but with path independent line integral the metric tensor 7uv(a) can be directly expressed in terms of the matrix w:
vu (a) ='23
14 March 1985
h(a, a') = (a Inla'>/
(12)
After the transformation of eq. (10) to the new coordinate system {a K} in which
(a,a')G=(OdO/)G=exp(--l~(aK--o/~)2),
(13)
the standard derivation, as in the GOA, of a collective hamiltonian can be applied [2,4,8]. Because (13) is of gaussian shape with a constant width the reduced energy overlap h'(a, a') can be replaced by an infinite order differential operator. However, for a collective motion, the expansion (see eq. (10.130) of ref. [2] ) up to second order for the differential operator seems to be sufficient [6,9]. After the inverse transformation to the collective variables {a K} we obtain the following collective hamiltonian:
1 0
r-[
1 ]ll' b
~coll =-
(14) where 3, - det(7i/), the inverse mass tensor is given by the formula
Volume 152B, number 5,6
(
1
)
2-~(q) p.v
Aal~
=-4
PHYSICS LETTERS
lIzxa,Ah(a'a') __
Aa v
7
A_,V i.Ja=a,=q a
(15)
and the potential energy is the difference o f the expectation value (q IHlq) and the so-called zero point energy [10,11]
V(q) = (qlnlq) - e 0
(16)
with
eo = - !7~v2
14 March 1985
is a slowly varying function, as it is expected in collective phenomena, one can obtain a Schr6dinger-type collective hamiltonian. In this way we end up with an approach which has a wide range of applicability, with all advantages o f a simple gaussian overlap approximation. The author would like to thank Professor K. Pomorski, E. Werner and M. Brack for stimulating discussion and for many helpful comments on this subject.
References A ( A h ( a , q) t Aq~ Aa v I a=q
+53'1~ V ( ~ a t~A A h ( a ,vq ) ]
(17)
a=q Here A/Aau denotes the covariant derivatives and 7~ v stands for the inverse (contravariant components) o f the metric tensor 7~v" To Summarize the results o f the paper we draw some conclusions. The model overlap function (3) is flexible enough to reproduce a wide class of the overlap functions. The approximation (9) often allows to skip the difficult problem o f calculations of the functions F ~. In the case when the reduced energy overlap
[1] J.J. Griffin and J.A. Wheeler, Phys. Rev. 108 (1957) 311. [2] P. Ring and P. Schuck, The nuclear many-body problem (Springer, Berlin, 1980), and references therein. [3] D.M. Brink and A. Weiguny, Nucl. Phys. A120 (1968) 59. [4] B. Banerjee and D.M. Brink, Z. Phys. 258 (1973) 46. [5] N. Onishi and T. Une, Prog. Theor. Phys. 53 (1975) 504. [6] P.G. Reinhaxd, Nucl. Phys. A261 (1976) 291. [7] L.S. Ferreira and M.H. Caldeira, Nucl. Phys. A189 (1972) 250. [8] B. Giraud and B. Grammaticos, Nucl. Phys. A233 (1974) 373; A~255 (1975) 141. [9] P.G. Reinhard et al., Z. Phys. A317 (1984) 339. [10] M. Girod and B. Grammaticos,.Nucl. Phys. A330 (1979) 40. [11] P.G. Reinhard, Nucl. Phys. A252 (1975) 120;A306 (1978) 19.
283