Solutions of the Radial Dirac Equation in a B

Solutions of the Radial Dirac Equation in a B-Polynomial Basis Muhammad I. Bhatti1 and Warren F. Perger2 1

Department of Physics and Geology, University of Texas Pan American, Edinburg, TX 78539, USA 2 Departments of Electrical Engineering and Physics, Michigan Technological University, Houghton, MI 49931-1295, USA

Abstract An algorithm is given for constructing accurate solutions to the radial Dirac equation in a B-polynomial basis set. The B-polynomial Galerkin method has been applied to produce the spectrum of the Dirac equation for the bound states of hydrogenic systems. Matrix formulation is used throughout the entire procedure and boundary conditions are applied to generate finite discrete eigenvalues, which include both negative and positive energies as well as corresponding states. Excellent agreement is found between previously existing accurate calculations. To check the quality of the spectrum, the resulting basis sets are used to evaluate the TRK sum rules. The procedure can be readily extended to produce the spectrum of complex systems. 1. Introduction When applied to complex atomic system, the relativistic Dirac equation presents a challenging problem when solving it for highly accurate solutions. Partly, the difficulty arises because the Dirac equation has complicated spectrum; it possesses an infinite number of bound states, a positive energy spectrum and a negative energy spectrum. This has been circumvented using B-spline methods by several authors such as Johnson and co-workers applying it to many-body perturbation theory (Johnson et al. 1987, Johnson et al. 1988) and Charlotte Fischer and co-workers (Fischer & Idrees 1989, Fischer & Parpia 1993, Qui & Fischer 1999) who applied it to Hartree-Fock and continuum problems. Recently Bhatti and co-workers (Bhatti et al. 2003) applied nth-degree B-spline techniques to calculate polarizabilities for non-relativistic systems. Also nth-degree B-polynomials were applied to solve second-order inhomogeneous differential equations in recent works (Bhatti & Bracken 2005, Bhatta & Bhatti 2005). This method is extended to solve the Dirac equations which improves the accuracy and efficiency over the traditional methods including the piecewise k-order B-spline method. This paper discusses a method for creating a finite basis set of the radial Dirac equation using polynomial functions of nth degree known as Bernstein-polynomials (Gelbaum 1995). The same method can be easily extended to calculate atomic properties in a model potential such as Hartree-Fock model potential. The method demonstrates

Solutions of the Radial Dirac Equation in a B-Polynomial Basis

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advantages over the previously applied methods for solving differential equations, in particular the Dirac equation (Johnson et al. 1987, Johnson et al. 1988, Fischer & Idrees 1989, Fischer & Parpia 1993, Qui & Fischer 1999, Bhatti et al. 2003). The Bpolynomials method will approximate any function to a desired accuracy depending on the size of the nth-degree polynomial functions. Finite basis sets for Dirac equation have been applied for evaluating many-body sums that appear in the many-body perturbation expressions (Johnson et al. 1988). The B-polynomial basis method is used to introduce a finite basis set for the Dirac equation on a finite interval [0, R]. The procedure also takes advantage of the continuous and unity partition property of B-polynomials over the entire interval [0, R]. All the B- polynomials vanish at the end points x=0 and x=R, except the first and last polynomials which are equal to 1 at the endpoints. This provides a greater flexibility in which to impose boundary conditions at the end points of the interval. It also ensures that the sum at any point x in the interval of all polynomial basis is equal to 1. In addition, the set of (n+1) B-polynomials of nth degree form a complete basis. We note that it is possible to confine a set of continuous basis functions for approximating solutions of the Dirac equation in an interval [0, R]. Usually, the confinement is attained by imposing the boundary conditions P (0) = Q (0) = 0 and P (R) = Q (R) = 0 for the large and small components of the Dirac equation, respectively. Finally the solution to the Dirac equation is presented as linear combination of these Bpolynomials. We will work with the B-polynomials representation of the wavefunctions. The radial Dirac equation can be written (Johnson et al. 1988): "

d V (r) c( dr − κr ) d −c( dr + κr ) −2mc2 + V (r)

#"

Pκ (r) Qκ (r)

#

"

= ε

Pκ (r) Qκ (r)

#

,

(1)

where energy ε is replaced by the ε = E − mc2 to make comparison with nonrelativistic 2 energies for the Coulomb case V (r) = − Ze . The negative energy states belong to r 2 ε < −2mc . The positive energy states have energy ε < 0 for low lying bound states and ε > 0 for the higher states with large principal quantum number n. The relativistic Coulomb problem for hydrogenic ions can be solved with energies accurate to machine precision using polynomial-Galerkin method (Fairweather 1978, Fletcher 1984). In this paper, we report the results of the Dirac equation also to machine precision. Since B-polynomial method does not depend on the nature of any interior sub-interval points, also known as knots, the matrix elements are integrated analytically exactly on the entire region. In the following sections, we explain the procedure for approximations, define the B-polynomials basis and present the results of the Dirac equation in the basis. 2. Description of the B-polynomials for approximating solutions Our main goal is to present solution of the Dirac equation on a closed interval [a, b] with continuous polynomials which require no particular knot sequence. The details of


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such B-polynomials are given in the Ref. (Bhatta & Bhatti 2005). In this section we briefly mention the B-polynomials of degree n which are defined as: Ã

Bi,n (r) =

n i

!

(r − a)i (b − r)n−i (b − a)n

0 ≤ i ≤ n,

(2)

for i = 0, 1, ..., n, where the binomial coefficient are given by Ã

n i

!

n! , i! (n − i)!

=

(3)

there are only (n+1) nth-degree polynomials. For convenience, we set Bi,n (r) = 0, if i < 0 or i > n. A simple code written in Mathematica or Maple is used to create all the non-vanishing B-polynomials of degree n supported over the interval. The first and the last polynomials are generally related to the boundary conditions of the problem currently under consideration. As an example, a set of 6 B-polynomials of degree 5 over an interval [-1, 1] is shown in Fig. 1. It is clear from Fig. 1 that all the B-polynomials are positive and the sum of all the polynomials is equal to 1 at any point x. It is also interesting to note that how these B-polynomials might add to approximate an arbitrary function f(x). It is also obvious from equation (2) that one can easily replace a and b points by the knot sequence to convert B-polynomials into piecewise B-polynomials over the subintervals. For approximating solutions of the Dirac equation on an interval [0, R], we let P (r) =

n P

i=0

pi Bi,n (r)and Q(r) =

n P

i=0

qi Bi,n (r) in equation (1). The Galerkin method

(Fairweather 1978, Fletcher 1984) requires that the residuals of the differential equations must be orthogonal to each of polynomial basis, which lead to the 2n × 2n generalized eigenvalues problem (Johnson et al. 1988), "

(V ) c[(D) − ( κr )] −c[(D) + ( κr )] −2mc2 (C) + (V )

#"

p q

#

"

−ε

p q

#

= 0,

(4)

where the n × n matrices (C), (D), (V), and (κ/r) are given by: ZR

(C)i,j =

Bi,n (r) Bj,n (r) dr,

(5)

0

ZR

(D)i,j =

Bi,n (r) 0

d Bj,n (r) dr, dr

(6)

ZR

(V )i,j =

Bi,n (r) V (r) Bj,n (r) dr,

(7)

0

ZR κ κ ( )i,j = Bi,n (r) Bj,n (r) dr, r r

(8)

0

One important observation to note is that the matrices of equations (5-8) are symmetric and analytic. The integrations are carried out using Mathematica code. The


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matrices are finally written out to a computer disk later to be read in by a FORTRAN code for solving generalized eigensystem, eigenvalues and eigenvectors, using FORTRAN EISPACK library (Garbow et al. 1977). Internal Mathematica diagonalization routines were also used for comparison and verification. As an example, let us consider the case of a Coulomb field with charge Z=1. We choose the interval [0, R] with interval length b=R= 30 atomic units (a.u.) and calculate the spectrum of the Dirac equation for κ = −1. The wavefunctions are expressed as a linear combination of B-polynomials with n = 23 (degree) polynomials. Imposing the boundary condition that the wavefunction must tend to zero at the extrema of the interval effectively reduces the number of B-polynomials from 24 to 22. The EISPACK (RSG) (Garbow et al. 1977) is called to compute the eigenvalue problem in equation (4). The results are provided in Table 1. It is obvious from the Table, 22 of these eigenvalues (the positive-energy group) lie below −2mc2 while the remaining 22 eigenvalues (positive-energy group) grow from small negative values to large positive values but considerably smaller numbers as compared to those from the B-spline spectrum (Johnson et al. 1988). The negative and positive energy spectra are shown in column 3 and 4, respectively. In Table 2, we report the eigenvalues of the Dirac equation using the present Bpolynomial method and compared with the existing eigenvalues computed using the k-order B-spline method. The present method is also applied to the Z=2 case considered by Johnson et al. (Johnson et al. 1988) and C. Fischer et al (Fischer & Parpia 1993) both included B-spline methods based on different type of logarithmic grids. Our calculations are not based on any particular type grid; instead we integrate analytically, and therefore exactly, to evaluate the matrix elements, equations (5-8), which are expressed in terms of continuous B-polynomials over the entire length (R) of the interval. The RSG (Garbow et al. 1977) routine is called to compute all the eigenvalues of the symmetric generalized eigenvalues equation (4). The results of the Dirac spectrum are reported in Table 2, where they are compared against other available data. The present method provides fast, efficient, and accurate results to machine accuracy of a PC computer with double precision. Several tests can be carried out to check the quality of this spectrum. We choose to test our basis by calculating the Thomas-Reiche-Kuhn (TRK) relativistic sum rule. The rule is formulated in the relativistic case in terms of the two partial waves and κ = −l − 1 associated with orbital angular momentum, l. The TRK rule is given by (Johnson et al. 1988): n l X ωi0 | hκ = −1, i = 0 | r |κ = l, ii |2 2l + 1 i=0

+

n l+1 X ωi0 | hκ = −1, i = 0 | r |κ = −l − 1, ii |2 = 0, 2l + 1 i=0

(9)


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where ωi0 = εi,κ − ε0,−1 . The nonrelativistic TRK sum rule is given by: n X

ωi0 | hl = 0, i = 0 | r |l, ii |2 =

i=0

l(l + 1) + 1 . 2

(10)

Table 3 represents the results of the TRK sum rules using the basis sets from Bpolynomials. The basis set with n=22 polynomials is generated in a coulomb potential with Z = 1. Accuracy better than one part in 107 is observed in the cancellation of positive- and negative-energy sums. It is noticed that the sum over positive-energy states in the relativistic TRK sum rule gives close to non relativistic expression equation (10). The differences in the sum over positive-energy states from the nonrelativistic limit are also shown in the Table 3. 3. Results and Discussions To summarize, we have demonstrated a powerful method to solve relativistic radial Dirac equation using B-polynomial-method. It also shown that the quality of the spectrum is superior when calculating the completeness property and TRK sum rules. Using the matrix formalism, the symmetric generalized eigensystem equation (4) is solved to provide 2n eigenvalues and 2n orthonormal eigenfunctions. The present method is for the first time applied to solve the Dirac equation and has been determined to be faster and more straightforward to implement. The current procedure may be extended to calculate other atomic properties such as energies, many-body perturbation sums, and transition amplitudes as explained in the Ref. (Perger et al. 2001). References Bhatta D D & Bhatti M I 2005 Applied Mathematics and Computation . in press. Bhatti M I & Bracken P 2005. submitted. Bhatti M I, Coleman K D & Perger W F 2003 Phys. Rev. A. 68(4), 44503–1–4. Fairweather G 1978 Finite element Galerkin methods for differential equations Dekker, New York. Fischer C F & Idrees M 1989 Computers in Physics 23, 53–58. Fischer C F & Parpia F A 1993 Phys. Lett. A 179, 198–204. Fletcher W A 1984 Computational Galerkin Methods Springer New York, NY. Garbow B S, Boyle J M, Dongarra J J & Moler C B 1977 Matrix Eigensystem Routine-EISPACK Guide Extension Springer, Berlin. Gelbaum B R 1995 Modern Real and Complex Analysis John Wiley and Sons New York. Johnson W R, Blundell S A & Sapirstein J 1988 Phys. Rev. A. 37(2), 307–315. Johnson W R, Idrees M & Sapirstein J 1987 Phys. Rev. A. 35, 3218–3226. Perger W F, Xia M, Flurchick K & Bhatti M I 2001 Computers in Science and Engineering 3(1), 38–47. Qui Y & Fischer C F 1999 J. Comput. Phys. 156, 257.


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1 0.8 0.6 0.4 0.2

1

0.5

0

0.5

1

Figure 1. The set of 6 B-polynomials of degree 5 are shown in the interval [-1, 1]. The quantities are dimensionless.

Table 1. Eigenvalues are given of the symmetric generalized eigensystem equation (4). The B-polynomials are used to approximate the solutions of the radial Dirac equation with κ = −1 in a Coulomb potential with Z=1. The interval length b=R=30 a.u. and 22 polynomials (degree 23) are used in the approximation. State 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

B-poly neg: E +mc2 -37581.37127060313 -37564.19164570285 -37562.54422353429 -37561.938472607006 -37560.68858734584 -37559.52743600588 -37559.147874070935 -37559.02925616232 -37558.894938358455 -37558.67873874412 -37558.545418969006 -37558.43377570914 -37558.324345570465 -37558.22338535256 -37558.132848733614 -37558.05226496846 -37557.981415907074 -37557.920069316766 -37557.86785074069 -37557.82415342494 -37557.787775740595 -37540.9785212177

B-poly pos: E −mc2 -0.5000066565939798 -0.12500207823741233 -0.05545660073296264 -0.02548920877956791 0.010532600612851712 0.059647840223736835 0.12092534389074307 0.1937630336672984 0.277787933743422 0.37269378533258635 0.4781853892701458 0.595763210908932 0.7200710802950812 0.8385153800943299 1.0263492739509505 1.3208651539529874 1.4116713871599007 1.7044089811382868 3.3979716957364694 4.814156281058436 4.952798746483542 14.30726042627141

Coulomb-Dirac equation -0.5000066565953603 -0.12500208018900594 -0.05555629517766647 -0.03125033803007682 -0.020000181059003808 -0.01388899675293942 -0.01020415094535565 -0.007812547130015446 -0.006172872985189315 -0.005000024626497179 -0.00413225004740525 -0.0034722366654023062 -0.0029585912998300046 -0.002551029592723353 -0.0022222297156986315 -0.0019531311954779085 -0.0017301089901593514 -0.001543214253615588 -0.0013850452814949676 -0.0012500032062234823 -0.001133789621235337 -0.0010330602672183886


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Table 2. Computed eigenvalues using B-polynomials and B-splines are compared with the exact results of the Coulomb-field Dirac equation for various Z values. The B-polynomial results used 26 polynomials of degree 27 and an interval of 40 a.u., whereas the B-spline results both used 40 B-splines of order=7 and an interval of 40 a.u. The B-spline exponential grid was that used in ref. (Johnson et al. 1987) and the r4 grid of ref. (Fischer & Parpia 1993). Z B-polynomials method B-splines, exp grid B-splines, r4 grid Exact

1 -0.5000066565652576 -0.5000066564787851 -0.5000066565884519 -0.5000066565953603

2 -2.00009871358548 -2.000106511490881 -2.000106514004648 -2.000106514067738

Table 3. TRK sum rules equations (9-10) are calculated and shown for various l values using Dirac basis sets in the Coulomb-field. We use 22 B-polynomials, with rmax=30au. The basis set is created for κ = −1 and Z=1. l 0 1 2 3 4 5 6 7 8 9

Sum over positive- and negative-energy states 4.143 × 10−9 −6.306 × 10−9 1.015 × 10−8 −2.154 × 10−9 −5.053 × 10−8 −1.388 × 10−7 4.720 × 10−8 3.611 × 10−8 −2.533 × 10−7 −3.639 × 10−7

Sum over positiveenergy states only 0.499934 1.49993 3.49936 6.49708 10.4914 15.4802 21.4607 28.4299 36.3842 45.3197

Nonrelativistic limit l(l+1)+1 2

0.5 1.5 3.5 6.5 10.5 15.5 21.5 28.5 36.5 45.5