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Matrix elements of integrable nuclear tensor operators
Nuclear Physics 70 (1965) 219--224; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher
MATRIX ELEMENTS OF INTEGRABLE NUCLEAR TENSOR OPERATORS P. J. ROBERTS Department of Natural Philosophy, Glasgow University, Scotland
Received 2 February 1965 Abstract: Explicit expressions are given for the matrix elements of two-particle nuclear tensor operators present in the asymptotic one-pion exchange potential. The functional forms of the potentials V(r) are not prescribed beyond the requirement that they should be integrable. A basis of orbitals with radial factors of the form r 2n+! exp (--ar 2) is used. The results are given in closed form and are very much simpler than previous formulae given by Nesbet and Talmi.
1. Introduction
The computation of matrix elements of operators that occur in a realistic Schr6dinger Hamiltonian is usually performed in a basis of orbitals chosen for ease o f evaluation of these matrix elements and for their qualitative resemblance to the Hartree-Fock orbitals of the system. As shown by Talmi x), when these orbitals are the harmonic oscillator eigenfunctions, it is possible to evaluate the matrix elements of two-particle operators (the well-known Slater parameters) by transforming to a co-ordinate system in which the interparticle co-ordinates occur as variables of integration. This transformation makes five of the six integrations trivial, and leaves only a one-dimensional quadrature over the interparticle distance, which can be performed numerically if necessary. The radial interparticle potential function only enters the calculation in the final quadrature, which is convenient since our present knowledge of this potential is incomplete. Computer programmes based on the Talmi transformation are committed to a definite f o r m of the interparticle potential only through a single subroutine which carries out the final quadrature. Actual calculations with harmonic oscillator, or Gaussian, orbitals by the Talmi procedure are hampered by the fact that it is necessary first to calculate the wellknown Talmi coefficients (also called the Talmi transformation brackets). The evaluation of these coefficients, even after ten years of effort by nuclear physicists, is still a matter of great difficulty, as is evidenced by the large number of papers dealing with their calculation which have been published since 1952. It is rather astonishing that this is the case, since we shall show in this paper that Talmi's analysis is completely unnecessary, a far shorter and less complicated analysis being ready to hand. 219
220
P.J. ROBERTS
The present paper describes a method of evaluating matrix elements of two-particle operators in a physically realistic nuclear Hamiltonian by using a basis set of orbitals recently proposed by Nesbet 2). These orbitals resemble the eigenfunctions of the harmonic oscillator in having Gaussian exponential factors in the radial functions, and the analysis given herein for the Nesbet basis can be applied identically to an orthonormal basis set of harmonic oscillator functions. Our analysis will be quite unlike both the Talmi 1) analysis for the scalar interaction, and the analysis of Nesbet 2) for the tensor interaction, and will involve no more formidable mathematics than the theory of complex Fourier transforms. Because our results are given in a simple closed form, any computer programme based on them will be very much less complicated than that outlined by Nesbet 2). The only forms of the nuclear two-body interactions treated in the present paper are those that occur in the one-pion exchange potential; integrable velocity-independent scalar and tensor interactions. Although this asymptotic internucleon interaction is now fairly well established from meson theory, the inner region is still indefinitely determined 3). The radial potential function V(r) is left indefinite except for the requirement of integrability. This excludes the hard-core interaction which is infinite in a finite region, but includes many functional forms that are large and positive at shoit range and possibly singular at an isolated point in configuration space. Such a potential, giving results reasonably in agreement with experiment, was proposed by Goldhammer 4) for calculating the binding energy of 0 16 and the doublet splitting in the lp shell caused by the tensor interaction. 2. Matrix Elements o f a Tensor Operator
The basis set of orbitals which we shall use are the Gaussian functions proposed by Nesbet 2) for building up nuclear Hartree-Fock functions. These have the form R,(r) = r 2n'+'' exp (-air2)y(li, m,; 0, q~),
(1)
where n and l are positive integers or zero, a is a variation parameter, and Y is the complex spherical harmonic normalized to unity, as defined by Edmonds 5). In the form given, these orbitals are unnormalized. Here l and m are the orbital angular momentum quantum numbers. Nesbet 2) has given the usual form of the nuclear tensor operator, which includes the co-ordinates of both spin and configuration space. Since the spin matrix elements are essentially trivial, we assume in what follows that they have been evaluated, so that we only consider the configuration space co-ordinates. In these co-ordinates the usual nuclear two-particle operator is just a special case of the general operator
V(r) = V(r)Y(L, M; ~, •),
(2)
where r = r 1 - r 2 is the vector connecting two nucleons denoted by the subscripts 1 and 2, and (r, g, 8) are the spherical polar co-ordinates of this vector.
MATRIX ELEMENTS
221
All matrix elements (hi] VUk) of V(r) in the basis of orbitals given in eq. (1) can be written as
(hi[ V] jk) = f dr 1dr 2 R~(r,)Rj(rl)V(r)R*(r2)Rk(r2).
(3)
A product of two spherical harmonics is just a sum of spherical harmonics 5), which we may write as
Y(l, m; O, c~)Y(L, M; O, ~)* = Z [(2k+ 1)/4rc]~ k
x Ck(l, m; L, M)Y(k, m - M ; O, (o), (4) where the C k are the Gaunt coefficients tabulated by Condon and Shortley 6). The latter are zero unless the following conditions are observed:
l + L + k = even integer,
[1-L[ ~ k < l+L.
(5)
By substituting eq. (4) into eq. (3), the latter equation reduces to a linear combination of reduced matrix elements:
(hil VI jk) = ~ [(2/+ 1)(22 + 1)]~r(4n)- 1Cl(lj, ms; Ih, mh) l, 3.
x Ca(Ik, mk; li, m,)fdrxdr2r 2n+t exp (--ar2)r(l, m; 1) x V(r)r 2N+x exp ( - br2)y(2, #; 2),
(6)
where
a = ah+aj,
b = a~+ak,
N = n,+nk+½(l~+lk--2), #
n = nh+nj+½(lh+li--l), m = ms--rob,
= mk--ml,
and we have used an obvious nomenclature for the spherical harmonics belonging to particles 1 and 2. The reduced matrix elements remaining in eq. (6) are of the general f o r m
I = f drx dr2f(rl)g(r2)V(r),
(7)
where f and g are the Nesbet Gaussians defined by eq. (1). This integral may also be written
I by making a simple change of variables.
(8)
222
V. J. ROBERTS
By using the Fourier convolution theorem in three dimensions on the second integral, it is a simple matter to rewrite eq. (8) in the form I = f dr V(r)F+(r),
(9)
where we define the complex Fourier transforms
f +(r) = 'drf(r) exp (+_ iX. r),
F(K) =
(10) (11)
Sincefand 9 are the Gaussian functions defined by eq. (1), their Fourier transforms are also Gaussian functions, and are easily evaluated by using the expansion of a plane wave in spherical harmonics s). Thus, if we define f ( r , ) = r 2n+l 1 exp ( - ar2)y(l, rn; 1),
(12)
g(r2) = r 2s+z exp ( - b r 2 ) e ( 2 , #; 2),
(13)
as in eq. (6), then
F(K) = ~(l+~),,(2 +~)N(-½i)t(½i)~a-"-t-~b -N-a-~ × ~
N ~ {( - n)s(- N)t/(l + ~)s(2 + ½)ts !t !(4a)~(4b) t}
s=O t=O
x K 2s+2'+t+a exp (-cK2)r(1, m; u, v)Y(2, kt; u, v),
(14)
where
c = 1/4a + 1/4b, and (K, u, v) are the spherical polar co-ordinates of the vector K. If we combine the spherical harmonics as in eq. (4), then F(K) may be seen to be a finite sum of the Nesbet Gaussians defined by eq. (1). Thus the Fourier transform F+(r) is also a finite sum of the Gaussian functions, and may be evaluated in exactly the same way as F(K). Since the Fourier transform of a spherical harmonic is just a spherical harmonic, the angular integrations over (a, r) are easily carried out in eq. (9), remembering that V(r) is given by eq. (2). This leaves us only with one integration involving the variable r, and we may finally write I = ~ 6 = 4 - v ( - 1)M+' +Va(rn + It, - M)(l + ½).(2~+ ½)N x (22+ 1)~ra-"-t-~b-S-a-}c-L-~Ca(L,
- M ; l, m)
N q X ~ Z Z {( -- n)s( -- N)t( -- q)u(L + ~)~/s !t !u !(l + ½)s(2 + ~)t(L + ~)u cq(4a)S(4b)t(4c) u} s=O t=O u=0
x
£
dr V(r)r 2"+r+ 2 exp ( - r2/4c),
(15)
MATRIX ELEMENTS
223
where p = ½(l+2+L),
q = s+t+½(l+2--L).
It is easily seen from this that the integral I will vanish unless l + 2 + L = even integer, and 11-21 < L < 1+2,
m + # + M = 0,
l _-< L + 2 .
Finally, if we substitute eq. (15) into eq. (6), we have
(hilVljl¢) = Z [(2/+1)(22+1)]÷(4~) -1 C'(lj, ?nj'~ lh, mh)Ca(lk, ink; l,, m,)I.
(16)
1, A
As was promised in sect. 1, eqs. (15) and (16) together demonstrate that the matrix elements of the general tensor operator in configuration space with a basis of the Gaussian orbitals proposed by Nesbet 2) are easily reduced in closed form to a finite number of quadratures involving the radial potential function V(r). Nesbet 2) has proposed the alternative methods of Gauss-Laguerre and Gauss-Hermite quadrature for evaluating these integrals numerically. The method chosen will depend upon the form of the potential function V(r). Calculation of the matrix elements defined by eq. (3) can be carried out in exactly the same fashion as before when the Nesbet Gaussians from eq. (1) are replaced by the eigenfunctions of a three-dimensional harmonic oscillator in spherical polar co-ordinates, as used by Talmi 1). These orbitals have the advantage over the Nesbet Gaussians of forming a complete orthonormal set. The Fourier transform of a harmonic oscillator function is just an oscillator function, exactly as with the Nesbet orbitals, so that the analysis of the reduced matrix elements given by eq. (7) is exactly the same as before. When the nuclear two-particle interaction is merely a scalar operator, exactly the same analysis holds, provided we set L = M = 0 in eqs. (2) and (15), which results in substantial simplification of the latter equation.
3. Conclusion
Our final formulae for the matrix elements of the nuclear two-particle tensor or scalar operator, the former being a generalization of the usual tensor operator 7), have been derived with the minimum of mathematics in a unified manner. Any computer programme based on eqs. (15) and (16) will thus be as efficient as possible. I would like to express my thanks to Professor J. C. Gunn for his support and encouragement during the completion of this research.
224
v . J . ROBERTS
References 1) 2) 3) 4) 5)
I. Talmi, Helv. Phys. Acta 25 (1952) 185 T. K. Nesbet, I. Math. Phys. 4 (1963) 1262 N. K. Glendenning and G. Kramer, Phys. Rev. 126 (1962) 2159 P. Goldhammer, Phys. Rev. 116 (1959) 676; 125 (1962) 660 A. R. Edmonds, Angular momentum in quantum mechanics (Princeton University Press, Princeton, 1957) 6) E. U. Condon and G. H. Shortley, The theory of atomic spectra (Cambridge University Press, New York, 1953) 7) J. P. Elliott, Proc. Roy. Soc. A218 (1953) 345
MATRIX ELEMENTS OF INTEGRABLE NUCLEAR TENSOR OPERATORS P. J. ROBERTS Department of Natural Philosophy, Glasgow University, Scotland
Received 2 February 1965 Abstract: Explicit expressions are given for the matrix elements of two-particle nuclear tensor operators present in the asymptotic one-pion exchange potential. The functional forms of the potentials V(r) are not prescribed beyond the requirement that they should be integrable. A basis of orbitals with radial factors of the form r 2n+! exp (--ar 2) is used. The results are given in closed form and are very much simpler than previous formulae given by Nesbet and Talmi.
1. Introduction
The computation of matrix elements of operators that occur in a realistic Schr6dinger Hamiltonian is usually performed in a basis of orbitals chosen for ease o f evaluation of these matrix elements and for their qualitative resemblance to the Hartree-Fock orbitals of the system. As shown by Talmi x), when these orbitals are the harmonic oscillator eigenfunctions, it is possible to evaluate the matrix elements of two-particle operators (the well-known Slater parameters) by transforming to a co-ordinate system in which the interparticle co-ordinates occur as variables of integration. This transformation makes five of the six integrations trivial, and leaves only a one-dimensional quadrature over the interparticle distance, which can be performed numerically if necessary. The radial interparticle potential function only enters the calculation in the final quadrature, which is convenient since our present knowledge of this potential is incomplete. Computer programmes based on the Talmi transformation are committed to a definite f o r m of the interparticle potential only through a single subroutine which carries out the final quadrature. Actual calculations with harmonic oscillator, or Gaussian, orbitals by the Talmi procedure are hampered by the fact that it is necessary first to calculate the wellknown Talmi coefficients (also called the Talmi transformation brackets). The evaluation of these coefficients, even after ten years of effort by nuclear physicists, is still a matter of great difficulty, as is evidenced by the large number of papers dealing with their calculation which have been published since 1952. It is rather astonishing that this is the case, since we shall show in this paper that Talmi's analysis is completely unnecessary, a far shorter and less complicated analysis being ready to hand. 219
220
P.J. ROBERTS
The present paper describes a method of evaluating matrix elements of two-particle operators in a physically realistic nuclear Hamiltonian by using a basis set of orbitals recently proposed by Nesbet 2). These orbitals resemble the eigenfunctions of the harmonic oscillator in having Gaussian exponential factors in the radial functions, and the analysis given herein for the Nesbet basis can be applied identically to an orthonormal basis set of harmonic oscillator functions. Our analysis will be quite unlike both the Talmi 1) analysis for the scalar interaction, and the analysis of Nesbet 2) for the tensor interaction, and will involve no more formidable mathematics than the theory of complex Fourier transforms. Because our results are given in a simple closed form, any computer programme based on them will be very much less complicated than that outlined by Nesbet 2). The only forms of the nuclear two-body interactions treated in the present paper are those that occur in the one-pion exchange potential; integrable velocity-independent scalar and tensor interactions. Although this asymptotic internucleon interaction is now fairly well established from meson theory, the inner region is still indefinitely determined 3). The radial potential function V(r) is left indefinite except for the requirement of integrability. This excludes the hard-core interaction which is infinite in a finite region, but includes many functional forms that are large and positive at shoit range and possibly singular at an isolated point in configuration space. Such a potential, giving results reasonably in agreement with experiment, was proposed by Goldhammer 4) for calculating the binding energy of 0 16 and the doublet splitting in the lp shell caused by the tensor interaction. 2. Matrix Elements o f a Tensor Operator
The basis set of orbitals which we shall use are the Gaussian functions proposed by Nesbet 2) for building up nuclear Hartree-Fock functions. These have the form R,(r) = r 2n'+'' exp (-air2)y(li, m,; 0, q~),
(1)
where n and l are positive integers or zero, a is a variation parameter, and Y is the complex spherical harmonic normalized to unity, as defined by Edmonds 5). In the form given, these orbitals are unnormalized. Here l and m are the orbital angular momentum quantum numbers. Nesbet 2) has given the usual form of the nuclear tensor operator, which includes the co-ordinates of both spin and configuration space. Since the spin matrix elements are essentially trivial, we assume in what follows that they have been evaluated, so that we only consider the configuration space co-ordinates. In these co-ordinates the usual nuclear two-particle operator is just a special case of the general operator
V(r) = V(r)Y(L, M; ~, •),
(2)
where r = r 1 - r 2 is the vector connecting two nucleons denoted by the subscripts 1 and 2, and (r, g, 8) are the spherical polar co-ordinates of this vector.
MATRIX ELEMENTS
221
All matrix elements (hi] VUk) of V(r) in the basis of orbitals given in eq. (1) can be written as
(hi[ V] jk) = f dr 1dr 2 R~(r,)Rj(rl)V(r)R*(r2)Rk(r2).
(3)
A product of two spherical harmonics is just a sum of spherical harmonics 5), which we may write as
Y(l, m; O, c~)Y(L, M; O, ~)* = Z [(2k+ 1)/4rc]~ k
x Ck(l, m; L, M)Y(k, m - M ; O, (o), (4) where the C k are the Gaunt coefficients tabulated by Condon and Shortley 6). The latter are zero unless the following conditions are observed:
l + L + k = even integer,
[1-L[ ~ k < l+L.
(5)
By substituting eq. (4) into eq. (3), the latter equation reduces to a linear combination of reduced matrix elements:
(hil VI jk) = ~ [(2/+ 1)(22 + 1)]~r(4n)- 1Cl(lj, ms; Ih, mh) l, 3.
x Ca(Ik, mk; li, m,)fdrxdr2r 2n+t exp (--ar2)r(l, m; 1) x V(r)r 2N+x exp ( - br2)y(2, #; 2),
(6)
where
a = ah+aj,
b = a~+ak,
N = n,+nk+½(l~+lk--2), #
n = nh+nj+½(lh+li--l), m = ms--rob,
= mk--ml,
and we have used an obvious nomenclature for the spherical harmonics belonging to particles 1 and 2. The reduced matrix elements remaining in eq. (6) are of the general f o r m
I = f drx dr2f(rl)g(r2)V(r),
(7)
where f and g are the Nesbet Gaussians defined by eq. (1). This integral may also be written
I by making a simple change of variables.
(8)
222
V. J. ROBERTS
By using the Fourier convolution theorem in three dimensions on the second integral, it is a simple matter to rewrite eq. (8) in the form I = f dr V(r)F+(r),
(9)
where we define the complex Fourier transforms
f +(r) = 'drf(r) exp (+_ iX. r),
F(K) =
(10) (11)
Sincefand 9 are the Gaussian functions defined by eq. (1), their Fourier transforms are also Gaussian functions, and are easily evaluated by using the expansion of a plane wave in spherical harmonics s). Thus, if we define f ( r , ) = r 2n+l 1 exp ( - ar2)y(l, rn; 1),
(12)
g(r2) = r 2s+z exp ( - b r 2 ) e ( 2 , #; 2),
(13)
as in eq. (6), then
F(K) = ~(l+~),,(2 +~)N(-½i)t(½i)~a-"-t-~b -N-a-~ × ~
N ~ {( - n)s(- N)t/(l + ~)s(2 + ½)ts !t !(4a)~(4b) t}
s=O t=O
x K 2s+2'+t+a exp (-cK2)r(1, m; u, v)Y(2, kt; u, v),
(14)
where
c = 1/4a + 1/4b, and (K, u, v) are the spherical polar co-ordinates of the vector K. If we combine the spherical harmonics as in eq. (4), then F(K) may be seen to be a finite sum of the Nesbet Gaussians defined by eq. (1). Thus the Fourier transform F+(r) is also a finite sum of the Gaussian functions, and may be evaluated in exactly the same way as F(K). Since the Fourier transform of a spherical harmonic is just a spherical harmonic, the angular integrations over (a, r) are easily carried out in eq. (9), remembering that V(r) is given by eq. (2). This leaves us only with one integration involving the variable r, and we may finally write I = ~ 6 = 4 - v ( - 1)M+' +Va(rn + It, - M)(l + ½).(2~+ ½)N x (22+ 1)~ra-"-t-~b-S-a-}c-L-~Ca(L,
- M ; l, m)
N q X ~ Z Z {( -- n)s( -- N)t( -- q)u(L + ~)~/s !t !u !(l + ½)s(2 + ~)t(L + ~)u cq(4a)S(4b)t(4c) u} s=O t=O u=0
x
£
dr V(r)r 2"+r+ 2 exp ( - r2/4c),
(15)
MATRIX ELEMENTS
223
where p = ½(l+2+L),
q = s+t+½(l+2--L).
It is easily seen from this that the integral I will vanish unless l + 2 + L = even integer, and 11-21 < L < 1+2,
m + # + M = 0,
l _-< L + 2 .
Finally, if we substitute eq. (15) into eq. (6), we have
(hilVljl¢) = Z [(2/+1)(22+1)]÷(4~) -1 C'(lj, ?nj'~ lh, mh)Ca(lk, ink; l,, m,)I.
(16)
1, A
As was promised in sect. 1, eqs. (15) and (16) together demonstrate that the matrix elements of the general tensor operator in configuration space with a basis of the Gaussian orbitals proposed by Nesbet 2) are easily reduced in closed form to a finite number of quadratures involving the radial potential function V(r). Nesbet 2) has proposed the alternative methods of Gauss-Laguerre and Gauss-Hermite quadrature for evaluating these integrals numerically. The method chosen will depend upon the form of the potential function V(r). Calculation of the matrix elements defined by eq. (3) can be carried out in exactly the same fashion as before when the Nesbet Gaussians from eq. (1) are replaced by the eigenfunctions of a three-dimensional harmonic oscillator in spherical polar co-ordinates, as used by Talmi 1). These orbitals have the advantage over the Nesbet Gaussians of forming a complete orthonormal set. The Fourier transform of a harmonic oscillator function is just an oscillator function, exactly as with the Nesbet orbitals, so that the analysis of the reduced matrix elements given by eq. (7) is exactly the same as before. When the nuclear two-particle interaction is merely a scalar operator, exactly the same analysis holds, provided we set L = M = 0 in eqs. (2) and (15), which results in substantial simplification of the latter equation.
3. Conclusion
Our final formulae for the matrix elements of the nuclear two-particle tensor or scalar operator, the former being a generalization of the usual tensor operator 7), have been derived with the minimum of mathematics in a unified manner. Any computer programme based on eqs. (15) and (16) will thus be as efficient as possible. I would like to express my thanks to Professor J. C. Gunn for his support and encouragement during the completion of this research.
224
v . J . ROBERTS
References 1) 2) 3) 4) 5)
I. Talmi, Helv. Phys. Acta 25 (1952) 185 T. K. Nesbet, I. Math. Phys. 4 (1963) 1262 N. K. Glendenning and G. Kramer, Phys. Rev. 126 (1962) 2159 P. Goldhammer, Phys. Rev. 116 (1959) 676; 125 (1962) 660 A. R. Edmonds, Angular momentum in quantum mechanics (Princeton University Press, Princeton, 1957) 6) E. U. Condon and G. H. Shortley, The theory of atomic spectra (Cambridge University Press, New York, 1953) 7) J. P. Elliott, Proc. Roy. Soc. A218 (1953) 345