Continuum discretization in a basis of transformed harmonic-oscillator states

Continuum discretization in a basis of transformed harmonic-oscillator states F. Pe ´rez-Bernal,1I. Martel,1J. M. Arias,2and J. Go ´mez-Camacho2 1Departamento de Fı ´sica Aplicada e Ingenierı ´a Ele ´ctrica, Universidad de Huelva, 21071 Huelva, Spain 2Departamento de Fı ´sica Ato ´mica, Molecular y Nuclear, Facultad de Fı ´sica, Universidad de Sevilla, Apartado 1065, 41080 Sevilla, Spain 共Received 10 November 2000; published 18 April 2001兲 The inclusion of the continuum in the study of weakly bound systems is discussed. A transformed harmonicoscillator basis is introduced to provide an appropriate discrete and finite basis for treating the continuum part of the spectrum. As examples of application of the method the one-dimensional Poeschl-Teller and Morse potentials are worked out. The strength functions corresponding to different operators that couple the ground state to the continuum are investigated. It is found that the energy moments of those distributions are accurately reproduced with a small basis set. DOI: 10.1103/PhysRevA.63.052111 PACS number共s兲: 03.65.Ca, 21.60.⫺n, 31.15.⫺p, 34.10.⫹x I. INTRODUCTION A general time-independent quantum-mechanical potential gives rise to a Hamiltonian with both bound and unbound eigenstates. Usually, the Hamiltonian of the system has a finite number of bound eigenstates while the unbound ones form a continuum. Therefore, a calculation of the system properties in terms of eigenfunctions of Hinvolves a summation over the discrete states as well as an integration over the continuum ones. The last one is an involved task and normally the properties of the bound system are analyzed by just using the bound eigenstates, while the continuum ones are of special relevance for dispersion processes. However, the study of the effect of the continuum part of the spectrum for treating properties of the bound system has a long tradition in physics 共conversely, the effect of the bound states on dispersion processes has been widely investigated too兲. Recent examples can be found in nuclear 关1–6兴, molecular 关7–9兴, and atomic physics 关10–13兴. In particular, in nuclear physics the advent of the radioactive beam facilities has provided a variety of new nuclear structure problems 关14兴that include halo nuclei and neutron and proton rich nuclei close to the drip lines. All these systems are weakly bound and their proper treatment requires the inclusion in some way of the continuum part of the spectrum. This has been done in several ways, each one having its own advantages and drawbacks. Among them we cite: 共i兲The R-matrix method 关15兴in which the basic idea is to solve the many-body problem in a box and then make the matching with the adequate boundary conditions. 共ii兲The use of a Sturmian basis 关16–18兴, where one uses bound states of scaled potentials, which are orthogonal when weighted with the potentials. 共iii兲The Siegert pseudostate formulation 关19兴, which provides a finite basis representation of the outgoing wave solutions to the radial Schro ¨dinger equation for cutoff potentials. 共iv兲The use of Gamow states 关20兴, which are nonnormalizable solutions of the Schro ¨dinger equation corresponding to outgoing boundary conditions characterized by complex energies. 共v兲The method of continuum discretization coupled channels 共CDCC兲关21兴in which the continuum is discretized by means of taking fixed intervals, or bins, of k-values in the continuum states. 共vi兲The expansion of the single-particle wave functions in a harmonic-oscillator basis 关22兴. This last method has become very popular since it provides a simple complete discrete basis. However, for weakly bound systems the Gaussian asymptotic behavior of the harmonic-oscillator wave functions is a poor representation of the continuum. Thus, methods based in a general localscaling point transformation to the harmonic-oscillator functions 关23–27兴have drawn considerable attention recently 关1–3兴. The so-called transformed harmonic-oscillator basis 共THO兲retains the simplicity of the harmonic-oscillator expansion and includes the correct asymptotic behavior. In this paper we discuss a way of defining a THO basis designed to take into account the continuum by an appropriate discretization. We present the method and apply it to two analytic one-dimensional potentials of interest in molecular physics: the Morse potential and the Poeschl-Teller potential. The method can be equally applied to three-dimensional potentials. The paper is structured as follows. First, in Sec. II, the formalism is presented and the transformation to introduce a THO basis is proposed. In Sec. III, the application of the formalism to the Poeschl-Teller and Morse potentials is worked out. Finally, in Sec. IV the outlook and conclusions of this paper are presented. II. FORMALISM OF THE TRANSFORMED HARMONICOSCILLATOR STATES „THO…IN ONE-DIMENSIONAL HAMILTONIANS In this section we will apply the formalism of transformed harmonic-oscillator states to weakly bound systems. Such systems as the deuteron, halo nuclei, or Van der Waals molecules are of current interest. We consider the onedimensional Hamiltonian given by h⫽⫺ ប2 2 ␮ d2 dr2⫹v共r兲,共1兲 where ris the relative coordinate of two particles, ␮ is the reduced mass, and v(r) is the interaction between both parPHYSICAL REVIEW A, VOLUME 63, 052111 1050-2947/2001/63共5兲/052111共9兲/$20.00 ©2001 The American Physical Society63 052111-1 ticles. If distances are measured in units of ␣ ⫺1such that x ⫽ ␣ ris dimensionless and energies are given in units of ប2 ␣ 2/ ␮ , the preceding Hamiltonian can be written as h⫽⫺ 1 2 d2 dx2⫹v共x兲,共2兲 which will be the Hamiltonian used hereafter. Note that h and v(x) are then dimensionless quantities. In order to maximize the continuum contribution we assume that the system has just one bound state, ␺ B(x), though the present formalism can be easily extended to systems with several bound states as well as to three-dimensional systems: h ␺ B共x兲⫽eB ␺ B共x兲.共3兲 The Hamiltonian hhas also an infinite number of eigenstates in the continuum that, however, are not normalizable. Our objective is to develop a procedure that allows a convenient description of the states in the continuum by means of a finite number of normalizable states. The general formalism presented herewith is applied in the next section to two cases of interest in molecular physics, the Poeschl-Teller 关28兴, and Morse 关29兴potentials. A. Coordinate transformation Let us consider the one dimensional harmonic-oscillator basis ␾ n HO共s兲⫽NnHn共s兲exp共⫺s2/2兲,共4兲 where Hn(s) are the usual Hermite polynomials and Nn ⫽( 冑 ␲ 2nn!)⫺1/2 the corresponding normalization constants. This basis is orthogonal and forms a complete set for all functions that are square integrable in s. Besides, if we make an arbitrary change of coordinates, given by the monotonously increasing function x⫽x(s), and its inverse s⫽s(x), the functions ␸ n THO共x兲⫽ 冑 ds dx ␾ n HO关s共x兲兴共5兲 are orthogonal and form a complete set of all the functions that are square integrable in x. These functions are called THO states. The transformation x(s) is arbitrary in principle. However, it can be chosen in order to describe properly the properties of bound states in finite potentials. So, for small values of s, the harmonic-oscillator may be a reasonable approximation for the potential v(x), and thus xshould depend linearly on s. However, for large values of s, the harmonic-oscillator wave functions behave as exp(⫺s2/2), while the bound wave function in v(x) behaves as exp(⫺qx), where q2/2⫽eB. So, for large s,qx has to be proportional to s2/2. If the bound state wave function ␺ B(x) is known, the transformation x(s) can be completely determined by requiring that ␸ 0 THO共x兲⫽ ␺ B共x兲.共6兲 This condition together with Eq. 共5兲provide a basis set with the asymptotic behavior described above. Equation 共6兲is equivalent to the nonlinear equation 冕 ⫺⬁ x 兩 ␺ B共x⬘兲 兩 2dx⬘⫽ 冕 ⫺⬁ s 兩 ␾ 0 HO共s⬘兲 兩 2ds⬘⫽1⫹erf共s兲 2, 共7兲 which defines implicitly the function x⫽x(s) as well as its inverse. It should be noticed that the derivative can be written as dx共s兲 ds ⫽ 冉 ␾ 0 HO共s兲 ␺ B关x共s兲兴 冊 2 .共8兲 Once the wave function of the ground state provides the function x⫽x(s), we can employ the THO basis, Eq. 共5兲,to describe the continuum of our system. Note that, as the THO are orthogonal, and the n⫽0 state is the only bound state of the system, the states with n⭓1 describe the continuum. We expect that, as the dimension of the THO basis increases, the wave functions explore distances beyond the range of the potential and, at the same time, they have oscillations inside the potential. Thus, the THO basis allows for an appropriate description of long range phenomena, and at the same time it permits to describe accurately short-range effects. B. Diagonalization of the Hamiltonian. Width of the states. We evaluate the matrix elements of the Hamiltonian hin the THO basis. It should be noticed that the state ␸ 0 THO(x) ⫽ ␺ B(x) is an eigenstate of h, but this is not the case for the states with n⭓1. Let us consider the matrix element 具 THO,n 兩 共h⫺eB兲 兩 THO,m 典 ⫽ 冕 dx ␸ n THO共x兲共h⫺eB兲 ␸ m THO共x兲.共9兲 We can take into account that ␸ m THO(x) ⫽ ␲ 1/4NmHm关s(x)兴 ␸ 0 THO(x), and that (h⫺eB) ␸ 0 THO(x)⫽0, to write 具 THO,n 兩 共h⫺eB兲 兩 THO,m 典 ⫽ 冑 ␲ NnNm 2 冕 dx ␸ 0 THO共x兲关Hn关s共x兲兴, †共h⫺eB兲,Hm关s共x兲兴‡兴 ␸ 0 THO共x兲.共10兲 The double commutator is independent of the potential, and gives 关Hn关s共x兲兴,†共h⫺eB兲,Hm关s共x兲兴‡兴⫽dHn关s共x兲兴 dx dHm关s共x兲兴 dx ; 共11兲 writing the integral in terms of s, one gets F. PE ´REZ-BERNAL et al. PHYSICAL REVIEW A 63 052111 052111-2 具 THO,n 兩 共h⫺eB兲 兩 THO,m 典 ⫽2nNnmNm 冕 dsexp共⫺s2兲Hn⫺1共s兲Hm⫺1共s兲 冉 ds dx 冊 2 . 共12兲 This expression can be easily evaluated using Gaussian quadratures. Note that the only information required is the derivative of the function x(s), evaluated at the points sn, which define the quadrature. The matrix elements with n⫽0orm⫽0 vanish. This is due to the fact that the state of n⫽0 is an eigenstate of the Hamiltonian. Let us consider that we diagonalize the Hamiltonian in a Ndimensional basis of THO states, from i⫽0to i⫽N⫺1. The eigenstates of the Hamiltonian, in this restricted basis, are given by 兩 N,0 典 ⫽ 兩 THO,0 典 共13兲 兩 N,i 典 ⫽兺 j⫽1 N⫺1 兩 THO,j 典具 THO,j 兩 N,i 典 ,共14兲 where the states 兩 N,i 典 (i⫽1,...,N⫺1) represent the continuum states in the truncated Ndimensional THO basis. They can be expressed in the xrepresentation as 具 x 兩 N,i 典 ⫽ ␺ i N共x兲⫽ ␲ 1/4Pi N⫺1关s共x兲兴 ␸ 0 THO共x兲,共15兲 where Pi N⫺1(s) is a polynomial given by Pi N⫺1共s兲⫽兺 j⫽1 N⫺1 NjHj共s兲 具 THO,j 兩 N,i 典 .共16兲 The eigenvalues of the Hamiltonian, in the restricted basis, are related to the wave function through 共ei N⫺eB兲⫽1 2 ␦ ij 冕 dsexp共⫺s2兲 ⫻dPi N⫺1共s兲 ds dPj N⫺1共s兲 ds 冉 ds dx 冊 2 .共17兲 The use of the THO basis also allows to calculate the width of the states. In order to do so, we evaluate the matrix elements of the operator (h⫺eB)2in the basis 兩 N,i 典 . Note that if the states 兩 N,i 典 were the true eigenstates of the Hamiltonian, in a complete basis, then this matrix element would just be (Ei N⫺eB)2. However, as 兩 N,i 典 are only eigenstates of the Hamiltonian in a restricted basis, they will show a spread of energies when expanded in terms of the true continuum eigenstates of h. A measure of that spread is given by ⌫i N⫽ 冑 具 N,i 兩 共h⫺eB兲2 兩 N,i 典 ⫺共Ei N⫺eB兲2.共18兲 Let us use the fact that the THO states form a complete basis. Then, we have 具 N,i 兩 共h⫺eB兲2 兩 N,i 典 ⫽兺 n⫽0 ⬁ 具 N,i 兩 h⫺eB 兩 THO,n 典具 THO,n 兩 h⫺eB 兩 N,i 典 . 共19兲 Using Eqs. 共12兲and 共16兲 具 THO,n 兩 共h⫺eB兲 兩 N,i 典 ⫽1 2Nn 冕 dsexp共⫺s2兲dHn共s兲 ds dPi N⫺1共s兲 ds 冉 ds dx 冊 2 . 共20兲 This expression can be integrated by parts to give 具 THO,n 兩 共h⫺eB兲 兩 N,i 典 ⫽1 2Nn 冕 dsHn共s兲exp共⫺s2兲 冉 2s⫺d ds 冊 ⫻ 冠 dPi N⫺1共s兲 ds 冉 ds dx 冊 2 冡 .共21兲 Now, we can use the closure properties of the Hermite polynomials, to obtain 具 N,i 兩 共h⫺eB兲2 兩 N,i 典 ⫽ 冉 1 2 冊 2 冕 dsexp共⫺s2兲 冋 冉 2s⫺d ds 冊 ⫻ 冠 dPi N⫺1共s兲 ds 冉 ds dx 冊 2 冡 册 2 .共22兲 It is remarkable that the knowledge of the function x(s) is all we need to obtain wave functions, energies, and widths of the Hamiltonian eigenstates in the THO basis. C. Matrix elements of operators. Sum rules Let us consider the matrix elements of an arbitrary local operator O(x), which is a function of the coordinate x. The matrix element that connects the ground state 兩 N,0 典 to the continuum state 兩 N,i 典 is just 具 N,i 兩 O 兩 N,0 典 ⫽ ␲ 1/4 冕 dx Pi N⫺1关s共x兲兴O共x兲 兩 ␸ 0 THO共x兲 兩 2. 共23兲 This integral is more conveniently written in terms of the variable s 具 N,i 兩 O 兩 N,0 典 ⫽ 冑 ␲ 冕 ds Pi N⫺1共s兲O„x共s兲…exp共⫺s2兲, 共24兲 an expression that can be very simply evaluated using Gaussian quadratures. From this last formula we can define the following global magnitudes. 共i兲Total strength: It is given by CONTINUUM DISCRETIZATION IN A BASIS OF... PHYSICAL REVIEW A 63 052111 052111-3 ST共O;N兲⫽兺 i 兩 具 N,i 兩 O 兩 N,0 典 兩 2.共25兲 In the limit of very large number of states N, the set of states becomes a complete set, and one can use closure, so that ST共O兲⫽lim N→⬁ ST共O;N兲⫽ 具 N,0 兩 O2 兩 N,0 典 .共26兲 So, one should obtain, in the large Nlimit, ST共O兲⫽ ␲ ⫺1/2 冕 dsO„x共s兲…2exp共⫺s2兲 ⫽ 冕 dxO共x兲2 ␺ B共x兲2.共27兲 共ii兲Energy weighted sum rule: It is given by EW共O;N兲⫽兺 i共ei N⫺eB兲 兩 具 N,i 兩 O 兩 N,0 典 兩 2.共28兲 In the limit of very large number of states N, we can use the basis closure to express EW(O)⫽limN→⬁EW(O;N) in terms of a double commutator EW共O兲⫽1 2 具 N,0 兩 †O共x兲,关h⫺eB,O共x兲兴‡ 兩 N,0 典 .共29兲 The double commutator can be explicitly evaluated, to give †O共x兲,关h⫺eB,O共x兲兴‡⫽ 冉 dO共x兲 dx 冊 2 .共30兲 Thus, one obtains EW共O兲⫽ ␲ ⫺1/2 1 2 冕 ds 冉 dO共x兲 dx 冏 x⫽x(s) 冊 2 exp共⫺s2兲共31兲 ⫽1 2 冕 dx关dO共x兲/dx兴2 ␺ B共x兲2.共32兲 共iii兲Polarizability: It is given by P共O;N兲⫽兺 i⫽0共ei N⫺eB兲⫺1 兩 具 N,i 兩 O 兩 N,0 典 兩 2.共33兲 In the limit of a large number of states, P(O) ⫽limN→⬁P(O;N) converges to a constant value, that can be evaluated from the variation of matrix element of the Hamiltonian h⫺eBwith the ground state 兩 gs(t) 典 of a perturbed Hamiltonian h⫹tO(x)关30兴: P共O兲⫽1 2lim t→0 d2 dt2 具 gs共t兲 兩 h⫺eB 兩 gs共t兲 典 .共34兲 We can use the values of these global magnitudes to evaluate the convergence as the number of THO states included in the calculation is increased. With regard to the form of the operators O(x), we will consider two different cases. First, we take O(x)⫽x, a long-range operator, to describe effects of external fields, such as the Coulomb field. In this case, xrepresents the electric dipole operator. In second place, we consider a short-range operator O(x)⫽v(x), to describe possible effects of internal correlations, which would have a range similar to the potential. III. APPLICATION TO ANALYTIC ONE-DIMENSIONAL HAMILTONIANS A. The Poeschl-Teller Hamiltonian The Poeschl-Teller potential 关28兴is widely used in molecular physics, for example, to model bending vibrations, and conveys a considerable attention in other fields 关6兴.Itis written as v共x兲⫽⫺D1 cosh2共x兲,共35兲 where ⫺Dis the value of the potential in its minimum. The variable x⫽ ␣ r, where ris the relative coordinate and ␣ is the inverse of the range of the potential. The depth of the potential Dcan be written as D⫽1 2j共j⫹1兲,共36兲 in terms of a new parameter j关31兴which is a positive real number. The bound eigenstates of the Poeschl-Teller Hamiltonian can be written in terms of jas ⌿jv共x兲⫽NjvPj (j⫺v)共z兲,共37兲 where vis an integer taking values from 0 to the integer part of j,Njv⫽ 冑 (j⫺v)v!/(2j⫺v)! is a normalization constant, z⫽tanh(x), and Ps (p)(z) are the associated Legendre functions when sis integer. In the present paper we consider j ⫽1, the only true bound state v⫽0, has an energy of eB⫽ ⫺1 2and its wave function is written as ␺ B PT共x兲⫽1 冑 2 cosh共x兲.共38兲 In this case there is another state for v⫽1, which is not normalizable, corresponding to a resonance in the continuum at zero energy. The use of an integer value of jis assumed, in the present paper for simplicity, but it is not mandatory. The relationship between xand sstems from the Eq. 共5兲 1 冑 2 cosh共x兲 ⫽ 冑 ds dx ␲ ⫺1/4 exp共⫺s2/2兲.共39兲 Direct integration gives both the dependence of son xand vice versa: erf(s)⫽tanh(x). The s(x) function is presented in Fig. 1. In this case the derivative of the function s(x) can be written in terms of the svariable as F. PE ´REZ-BERNAL et al. PHYSICAL REVIEW A 63 052111 052111-4 ds dx⫽ 冑 ␲ 2exp共s2兲关1⫺erf2共s兲兴,共40兲 facilitating the calculations. This result allows us to write the THO basis in the scoordinate space as ␸ n THO共x兲⫽1 冑 2n⫹1n!Hn关s共x兲兴 冑 1⫺erf2关s共x兲兴.共41兲 In Fig. 2 we plot the first five states of the THO basis for j ⫽1. In this case the Hamiltonian matrix can be easily computed from Eq. 共12兲and its diagonalization provides us with eigenvalues and eigenfunctions. According to Eq. 共6兲,we obtain one negative eigenvalue at the precise energy eB⫽ ⫺1 2, and in addition, N⫺1 positive eigenvalues corresponding to the continuum discretization. The resulting energies for increasing values of the Nparameter are depicted in Fig. 3, with Nranging from 2 to 20. The appearance of symmetry doublets is due to the symmetric form of the potential, which provides wave functions with well-defined parity. As the dimension of the THO basis increases, new energy levels appear. On the one hand, many of them lie close to the zero energy, increasing the level density in the region. On the other hand, new levels explore higher energies. The wave functions obtained are orthonormal 共see Fig. 4 where we present the N⫽5 case兲and, as expected, with increasing energy they extend to higher xranges while the nodes accumulate in the vicinity of the origin. They have the desired asymptotic behavior and their nature allows for a straightforward use in calculations. Once the eigenvalues and eigenfunctions of the Hamiltonian are obtained, we proceed to check convergence and closure of the truncated basis, calculating the total strength, energy weighted sum rule, and polarizability for a typical long-range operator (x) and a short-range one 关the potential v(x)]. The results obtained are presented in Tables I and II, respectively. In Table I we include only even Nvalues. The odd N ⫹1 cases give identical result since the negative parity states are the only ones connected to the ground state due to the antisymmetric nature of the xoperator. The operator v(x)is FIG. 1. Function s(x) for the Poeschl-Teller Hamiltonian characterized by j⫽1. FIG. 2. THO basis for the Poeschl-Teller Hamiltonian, from n ⫽0to4. FIG. 3. Eigenvalues of the Poeschl-Teller Hamiltonian in the THO basis with dimensions ranging from N⫽2 to 20. FIG. 4. Eigenstates (n⫽0 to 4) of the Poeschl-Teller Hamiltonian diagonalized in the N⫽5 THO basis. CONTINUUM DISCRETIZATION IN A BASIS OF... PHYSICAL REVIEW A 63 052111 052111-5 symmetric and thus the situation is the opposite. Consequently, only odd Nvalues are shown in Table II. In the long-range operator case the convergence is very fast for the three observables computed. For N⫽4共three states in the continuum, only two with the right parity兲we obtain the exact values within a 2/1000 relative error. For the potential operator, the convergence is also fast although we need N⫽17 共eight states with the right parity兲in the worst of the cases, to reach the same relative error as before. It is remarkable that the polarizability associated to the potential operator converges very rapidly. This indicates that perturbative corrections to the energy of the bound state, due to changes of the potentials, will be obtained accurately in this basis. In both cases we should stress the fast convergence obtained, which points out that the discretization performed is able to simulate correctly the continuum effects with the inclusion of few states in the THO basis. B. The Morse Hamiltonian The Morse potential 关29兴is a commonplace to model anharmonic vibrations in diatomic molecules 关32兴and it is becoming of wide use in polyatomic structure calculations through the local mode picture 关33,34兴. The form of the potential is v共x兲⫽D 兵 关1⫺exp共⫺x兲兴2⫺1 其 ,共42兲 where x⫽ ␣ r, with rthe relative coordinate and ␣ the inverse of the potential range, and Dis the potential depth in the minimum (x⫽0). Dcan be written in terms of a parameter j关35兴, which is a positive real number, as D⫽1 2 冉 j⫹1 2 冊 2 .共43兲 The bound wave functions for the Morse potential are written as ⌿jv共x兲⫽Njvexp共⫺z/2兲zj⫺vLv 2j⫺2v共z兲,共44兲 where vis an integer number taking values from 0 up to the integer part of j,Njv⫽ 冑 (2j⫺2v)v!/(2j⫺v)! is a normalization constant, z⫽(2j⫹1)exp(⫺x) is the Morse variable, and Ls (p)(z) are the generalized Laguerre polynomials of degree sand order p. As in the previous case we take j⫽1. The only true bound state in this case, v⫽0, has energy eB⫽ ⫺1 2and its wave function is ␺ B M共x兲⫽3 exp共⫺x兲exp关⫺3 exp共⫺x兲/2兴.共45兲 For v⫽1 there is another state that is not normalizable and corresponds to a resonance in the continuum at zero energy. Direct integration in Eq. 共7兲provides the relation between xand s: 关1⫹erf共s兲兴/2⫽关1⫹3 exp共⫺x兲兴exp关⫺3 exp共⫺x兲兴. 共46兲 Numerically solving this equation we get the s(x) function plotted in Fig. 5. The behavior of s(x) reflects the potential TABLE I. Convergence of the total strength (ST), energy weighted sum rule (EW), and polarizability 共P兲of the operator xas a function of the THO basis dimension for the Poeschl-Teller Hamiltonian. Nis the total number of basis states. In this case, because of the parity selection rule, only odd parity states are connected to the ground state through the xoperator. NST(x,N)EW(x,N)P(x,N) 2 0.815 77 0.527 09 1.262 54 4 0.822 45 0.500 34 1.420 50 6 0.822 467 0.499 99 1.423 44 8 0.500 00 1.423 49 10 0.500 00 1.423 50 Exact Value 0.8224 67 0.500 00 1.423 50 TABLE II. Convergence of the ST,EWsum rule, and Pof the operator v(x) as a function of the THO basis dimension for the Poeschl-Teller Hamiltonian. Nis the total number of basis states. In this case, because of the parity selection rule, only even parity states are connected to the ground state through the v(x) operator. NST(v,N)EW(v,N)P(v,N) 3 0.511 992 0.061 675 0.073 9799 5 0.527 628 0.108 383 0.074 0682 7 0.531 714 0.132 586 0.074 0737 9 0.532 854 0.143 771 0.074 0741 11 0.533 187 0.148 694 13 0.533 287 0.150 812 15 0.533 318 0.151 713 17 0.533 328 0.152 096 Exact Value 0.533 333 0.152 381 0.074 0741 FIG. 5. Function s(x) for the Morse Hamiltonian characterized by j⫽1. F. PE ´REZ-BERNAL et al. PHYSICAL REVIEW A 63 052111 052111-6 asymmetry. Once the s(x) function is computed we can define the THO basis following Eq. 共5兲. The result for N⫽5is depicted in Fig. 6. The Hamiltonian diagonalization in the THO basis provides with eigenvalues and eigenfunctions. We plot in Fig. 7 the energies obtained increasing the dimension of the basis from N⫽2 to 20. The bound-state energy lies at its exact value, eB⫽⫺ 1 2, while the behavior of the positive eigenvalues is similar to the preceding case, excluding the appearance of parity doublets. Note the different scaling in Figs. 3 and 7, which shows the different behavior of the Poeschl-Teller and Morse potentials. The eigenfunctions 共see Fig. 8兲form an orthonormal set. They are not symmetric, as expected, but as in the previous case, they both increase the number of nodes in the region around the origin and explore higher 兩 x 兩 values as nincreases. Positive values of xare explored much more rapidly as a function of nthan the negative ones. With the obtained eigenvalues and eigenfunctions, we again check the convergence and closure of the truncated basis calculating the total strength, energy weighted sum rule, and polarizability for the xoperator and the potential v. The results obtained are presented in Tables III and IV. In this case we cannot make any symmetry simplification. In Table III the results for the xoperator are presented, showing a very fast convergence for all the computed observables. For N⫽5共four states in the continuum兲we obtain around 1/1000 maximum relative error. For the potential operator 共see Table IV兲with N⫽6 the maximum relative error is around 1/1000. Also, in this case, the convergence of the polarizability associated to the potential operator is very fast. As in the Poeschl-Teller potential we should stress the fast convergence obtained, even faster in this case. That supports the evidence for considering the truncated THO basis as a suitable tool for continuum discretization. IV. SUMMARY, CONCLUSIONS, AND OUTLOOK In this paper a THO basis has been introduced to produce appropriate normalizable states for discretizing the continuum. This is a fundamental problem in quantum mechanics and is especially relevant when treating weakly bound systems. The THO basis used in this paper is obtained by a local scale transformation 共LST兲that converts the ground state of the system into the harmonic-oscillator 共HO兲ground state. Thus the only previous requirement to apply this forFIG. 6. Basis of THO for the Morse Hamiltonian, from n⫽0 to 4. FIG. 7. Eigenvalues of the Morse Hamiltonian in the THO basis with dimensions ranging from N⫽2 to 20. FIG. 8. Eigenstates (n⫽0 to 4) of the Morse Hamiltonian diagonalized in the N⫽5 THO basis. TABLE III. Convergence of the ST,EWsum rule, and Pof the operator xas a function of the THO basis dimension for the Morse Hamiltonian. Nis the total number of basis states. NST(x,N)EW(x,N)P(x,N) 2 1.082 07 0.593 44 0.658 937 3 1.100 99 0.500 68 0.894 608 4 1.101 67 0.500 04 0.950 650 5 1.101 68 0.500 00 0.960 174 6 0.961 193 7 0.961 235 8 0.961 235 Exact Value 1.101 68 0.500 00 0.961 237 CONTINUUM DISCRETIZATION IN A BASIS OF... PHYSICAL REVIEW A 63 052111 052111-7 malism is to know 共either analytically or numerically兲the ground state of the system. This defines the LST and allows to generate all the states in the THO basis by transforming the HO wave functions. The states in the THO basis are discrete, normalizable, and have exponentially decreasing asymptotic behavior. Although the basis is infinite, it is possible to get good approximations to the exact results when calculating observables of interest by truncating the basis to few states in the continuum region. In the calculations presented in this paper, truncating just to 7 or 8 states in the continuum gives the exact results within around one per mil relative error in the worst of the cases. In this paper we have presented the formalism for onedimensional potentials and we have chosen the case of just one bound state. However we have performed calculations for several bound states and the same kind of results are obtained. The THO basis converge very rapidly to the exact energies of the bound eigenstates while states in the continuum lie close to zero energy, increasing the level density in that region, and few of them explore higher-energy regions. The formalism presented here can be of use whenever bound states close to the dissociation limit are concerned or in the cases in which the coupling between bound and continuum states are important. It can be used for structure calculation to evaluate strength functions into the continuum and to perform scattering calculations taking into account the breakup effects. The use of THO wave functions in practical calculations involves increasing the number Nof states considered in the calculation until convergence is achieved. In this sense, the THO basis has an advantage over the use of a box to calculate continuum effects, because, in this case, one has to deal with two continuum parameters, which are the radius of the box, and the maximum energy of the states considered. Thus, it is much harder to demonstrate convergence when there are two parameters to vary, instead of just one discrete parameter. The CDCC calculations have similar convergence problems. The bins describing continuum discretization are such that when expressed in terms of the coordinates, they vanish at distances of the order of 1/⌬k. A practical CDCC calculation requires to fix the maximum kvalue considered and the interval ⌬kof the bins, so that the number of bins is given by N⫽kmax /⌬k. Here, also to demonstrate convergence, one has to deal with two parameters. In addition, the CDCC method requires to solve the Schro ¨dinger equation for all the energies in the continuum. In our case, there is only one discrete parameter to check convergence and it is only required to solve the Schro ¨dinger equation for the ground state. The THO basis also has some similarities with the Sturmian basis. In both cases the basis is discrete and normalizable, and the wave functions have the same asymptotic behavior as the ground state. However, the Sturmian basis requires solving the Schro ¨dinger equation for increasing values of the potential depth, obtaining in this way wave functions with the same energy, but more nodes. These wave functions are not orthogonal, as they correspond to different Hamiltonians. Besides, the Sturmian basis gives an accurate description of the interior of the potential, but it converges very slowly to describe large separations. Thus, the THO basis has the advantage that one has to solve only the Schro ¨- dinger equation for the ground state, the wave functions are orthogonal, and the description of distances beyond the potential range seems to be satisfactory. ACKNOWLEDGMENTS This work was supported in part by the Spanish DGICYT under Project Nos. PB98-1111 and FPA2000-1592-C03-02. We acknowledge useful discussions with F. Iachello, P. H. Vaccaro, A. Frank, R. Lemus, and R. C. Johnson. 关1兴M.V. Stoitsov, P. Ring, D. Vretenar, and G.A. Lalazissis, Phys. Rev. C 58, 2086 共1998兲. 关2兴M.V. Stoitsov, W. Nazarewicz, and S. Pittel, Phys. Rev. C 58, 2092 共1998兲. 关3兴M.V. Stoitsov, J. Dobaczewski, P. Ring, and S. Pittel, Phys. Rev. C 61, 034311 共2000兲. 关4兴N. Sandulescu, Nguyen Van Giai, and R.J. Liotta, Phys. Rev. C61, 061301 共2000兲. 关5兴S. Mizutori, J. Dobaczewski, G.A. Lalazissis, W. Nazarewicz, and P.-G. Reinhard, Phys. Rev. C 61, 044326 共2000兲. 关6兴K. Bennaceur, J. Dobaczewski, and M. Ploszajczak, Phys. Rev. C 60, 034308 共1999兲. 关7兴I. Cacelli, R. Moccia, and A. Rizzo, Phys. Rev. A 57, 1895 共1998兲. 关8兴T.N. Rescigno, A.E. Orel, and C.W. McCurdy, Phys. Rev. A 55, 342 共1997兲. 关9兴A.K. Kazansky, J. Phys. B 29, 4709 共1996兲. 关10兴J.M. Vogels, B.J. Verhaar, and R.H. Blok, Phys. Rev. A 57, 4049 共1998兲. 关11兴M.B. Campbell, T.J. Bensky, and R.R. Jones, Phys. Rev. A 57, 4616 共1998兲. 关12兴B.I. Schneider, Phys. Rev. A 55, 3417 共1997兲. TABLE IV. Convergence of the ST,EWsum rule, and Pof the operator v(x) as a function of the THO basis dimension for the Morse Hamiltonian. Nis the total number of basis states. NST(v,N)EW(v,N)P(v,N) 2 0.575 235 0.120 856 0.013 4195 3 0.669 899 0.178 806 0.093 6744 4 0.740 342 0.689 425 0.093 7489 5 0.749 840 0.919 480 0.093 7500 6 0.749 996 0.938 555 7 0.750 000 0.937 510 8 0.937 500 Exact Value 0.750 000 0.937 500 0.093 7500 F. PE ´REZ-BERNAL et al. PHYSICAL REVIEW A 63 052111 052111-8 关13兴A.K. Kazansky, J. Phys. B 30, 1401 共1997兲. 关14兴A.C. Mueller et al.,Nuclear Physics in Europe: Highlights and Opportunities, edited by J. Vervier et al., NuPECC Report, 1997, p. 31. 关15兴E.P. Wigner and L. Eisenbud, Phys. Rev. 72,29共1947兲; Atomic and Molecular Processes: An R-Matrix Approach, edited by P.G. Burke and K.A. Berrington 共IOP, Bristol, 1993兲. 关16兴M. Rotenberg, Adv. At. Mol. Phys. 6, 233 共1970兲. 关17兴F. Antonsen, Phys. Rev. A 60, 812 共1999兲. 关18兴R. Szmytkowski and B. Zywicka-Mozejko, Phys. Rev. A 62, 022104 共2000兲. 关19兴O.I. Tolstikhin, V.N. Ostrovsky, and H. Nakamura, Phys. Rev. A58, 2077 共1998兲. 关20兴R.G. Lovas, R.J. Liotta, A. Insolia, K. Varga, and D.S. Delion. Phys. Rep. 294, 265 共1998兲. 关21兴N. Austern, Y. Iseri, M. Kamimura, M. Kawai, G. Rawitsher, and M. Yahiro, Phys. Rep. 154, 125 共1987兲. 关22兴M. Moshinsky and Y. Smirnov, The Harmonic Oscillator in Modern Physics 共Harwood Academic, Amsterdam, 1996兲. 关23兴I.Zh. Petkov and M.V. Stoitsov, C. R. Acad. Bulg. Sci. 34, 1651 共1981兲. 关24兴I.Zh. Petkov and M.V. Stoitsov, Theor. Math. Phys. 55, 584 共1983兲. 关25兴I.Zh. Petkov and M.V. Stoitsov, Yad. Fiz. 37, 1167 共1983兲 关Sov. J. Nucl. Phys. 37, 692 共1983兲兴. 关26兴M.V. Stoitsov and I.Zh. Petkov, Ann. Phys. 共N.Y.兲184, 121 共1988兲. 关27兴I.Zh. Petkov and M.V. Stoitsov, Nuclear Density Functional Theory, Oxford Studies in Physics 共Clarendon, Oxford, 1991兲. 关28兴G. Po ¨schl and E. Teller, Z. Phys. 83, 143 共1933兲. 关29兴P.M. Morse, Phys. Rev. 34,57共1929兲. 关30兴O. Bohigas, A.M. Lane, and J. Martorell, Phys. Rep. 51, 267 共1979兲. 关31兴Y. Alhassid, F. Gursey, and F. Iachello, Ann. Phys. 共N.Y.兲 148, 346 共1983兲. 关32兴G. Herzberg, Molecular Spectra and Molecular Structure, Vol 1: Spectra od Diatomic Molecule 共Van Nostrand Reinhold, NY, 1966兲. 关33兴M.S. Child and L. Halonen, Adv. Chem. Phys. 57,1共1984兲. 关34兴J.K. Watson, Chem. Phys. 190, 291 共1995兲. 关35兴F. Iachello and R.D. Levine, Algebraic Theory of Molecules 共Oxford University, Oxford, 1995兲. CONTINUUM DISCRETIZATION IN A BASIS OF... PHYSICAL REVIEW A 63 052111 052111-9