Momentum space approach to the relativistic atomic structure calculations

Momentum Space Approach to the Relativistic Atomic Structure Calculations REINALDO BARETTY* AND YASUYUKI ISHIKAWA Department of Chemistry and the Chemical Physics Program, University of Puerto Rico, Rio Piedras. Puerto Rico 00931 JOSE F. NIEVES Department of Physics, University of Puerto Rico, Rio Piedras, Puerto Rico 00931 Abstract A simple and straightforward derivation of the relativistic “no-pair” equation in momentum space is given for hydrogenic species. This is achieved by starting with a QED Hamiltonian in momentum representation and making a rigorous reduction into a system that contains a single electron but no positrons. The integral equation was solved for a series of hydrogenic systems using the Kwon-Tabakin-Lande technique. Numerical results were compared with those recently obtained by Hess, who employed the basis set expansion technique to solve the no-pair equation in configuration space. Introduction Currently the great majority of theoretical treatments of electronic behavior are based on the Schrodinger equation. The Schrodinger equation incorporates the major postulates of quantum mechanics, but it ignores effects explained by the other theory of nonclassical mechanics, the theory of relativity. The relativistic effects are minor when the electronic structures to be studied are those of molecules that contain only light atoms, atoms from, say, the first two rows of the fieriodic table. The small relativistic effects that are important for light atoms, spectroscopic fine structure effects, for example, may be incorporated with the perturbation theory [I]. This approach rapidly becomes unsatisfactory as heavy-atom species come under consideration. For heavy-atom species it is necessary to discard the Schrodinger equation [2]. The Dirac equation can be exactly solved only for a few of the one-electron problems to which it applies. Atomic and molecular wave functions are approximate, and the nature of the approximation must have certain qualities in order for it to be a useful one. An approximate method must be computationally This research was supported by a grant from the U.S. Army Research Office. *Permanent address: Department of Physics, Colegio Universitano de Humacao, Humacao, Puerto Rico 00661 International Journal of Quantum Chemistry: Quantum Chemistry Symposium 20, 109-1 17 (1986) CCC 0360-8832/86/200109-09$04.00 0 1986 by John Wiley & Sons, Inc. 110 BARETTY, ISHIKAWA, AND NIEVES feasible. At the same time it should be accurate, and whatever defects it may have should be consistent and well characterized. Finally, perhaps most importantly, an approximation should provide wave functions that are compact and easy to interpret chemically. These are properties that have caused the wide application of the HartreeFock self-consistent field method (HF SCF) to nonrelativistic problems. The corresponding technique widely used in relativistic theory is the Dirac-Fock self-consistent field method (DF SCF) [3,4]. However, all the relativistic atomic and molecular SCF calculations have been formally based on the familiar Dirac-Coulomb Hamiltonian ( H D C ) , which corresponds to an intuitively natural, but incorrect, extension of the Dirac oneelectron theory to many electrons. It has been known for some time that the H D C has no normalizable solutions [5]. This is because, as Sucher vividly describes [6], any normalizable product eigenfunctions of p:&lie embedded in a sea of nonnormalizable product states with the same total energy, in which one electron is in the positive-energy continuum and another in a negativeenergy continuum. The switching on of the electron-electron interaction terms will cause the normalizable product eigenfunctions to dissolve into the continuum (the so-called continuum dissolution (CD) problem). Furthermore, HW does not come from a rigorous application of quantum electrodynamics (QED). In essence, the problem with the H D c can be traced back to the fact that it does not incorporate the distinction between the electron and positron states of the underlying theory, QED. If one accepts the belief that the field-theoretic QED is the first principle of many-electron atoms and molecules, the problem is to construct an approximate scheme with which one can calculate the physical properties of the many-electron systems. There are several alternative approaches to this problem, e.g., a Hamiltonian approach [5,6], a mean-field approach based on the QED Lagrangian [7], and a many-particle generalization of the Bethe-Salpeter equation. Although the renormalization algorithm of the perturbation series involved is not yet established in an unambiguous manner, the Hamiltonian approach is probably the most attractive of all because it translates the idea that atoms and molecules are weakly bound inhomogeneous many-electron systems in which the pair production processes are absent and the particle number is conserved. In fact Brown and Ravenhall [5], and Sucher [6] used a field-theoretic description of QED to derive a legitimate N-electron configuration-space Hamiltonian that no longer suffers from the CD problem. Their configuration-space Hamiltonian differs from the H D c by the presence of suitable projection operators (e.g., free-particle [5,6], external field [6], and HF projection operators [6]) to force the exclusion of positron states. However, construction of these projection operators is technically very difficult in a position-space representation. This is because all these projection operators are nonlocal in such a representation [6]. In momentum space, however, the free-particle projection operators are simple algebraic operators. Thus, they can easily be implemented if one works in momentum representation. In 111 MOMENTUM SPACE APPROACH fact, Hardekopf and Sucher [8] have derived a relativistic “no pair” equation in momentum space using the free-particle projection operators by reducing the N-electron Hamiltonian into Pauli form with the use of a (pseudo) unitary Foldy-Wouthuysen transformation. The obvious route to take in obtaining the correct N-electron Hamiltonian in momentum space is to start with QED in momentum space and make a rigorous reduction to a system that contains N-electrons (and no positrons). This we have done for one- and two-electron atoms in order to make a start on formal and computational studies of relativistic wave equations in momentum space [9]. The purpose of the present study is two-fold: (1) to review our field-theoretic derivation of the relativistic “no pair” equation in momentum space for hydrogenic systems, and (2) to compare our numerical results with those obtained by Hess using the basis set expansion technique. No-pair Equation in Momentum Space Brown and Ravenhall IS] and Sucher [61 pointed out that the HDc yields no stable bound-state solutions. They returned to QED to derive the legitimate N-electron configuration-space Hamiltonian that does not suffer from the CD problem. Their Hamiltonian can be written as H+ = N N i= 1 i= I c h ~ i +) A + (2 vext(i) + vee) A+ (1) Here hD(i) is the free-particle Dirac Hamiltonian (a.u.) and is given by hD(i) = c a p + pc2 * Vext(i) is the external Coulomb potential for the ith electron and is given by -Z/ri, where Z is the nuclear charge. A + is the projection operator onto the space spanned by products of the positive energy eigenstate of hD: A + = A + ( l ) * - - A+(N) where A + (i) is the so-called Casimir positive-energy projection operator defined by A+(i) = (Eo(i) + hD(i)/2E0(i) (2) with Eo(i) = c(p2(i) + C*)I’~ The operator A+(i) is nonlocal in configuration space. Therefore, it is very difficult to handle in such a representation. In momentum space, however, the A+(i) reduces to a simple algebraic operator. In fact, Hardekopf and Sucher [8] obtained, for one- and two-electron systems, the so-called no-pair externalfield equations in momentum space by subjecting (1) to a pseudounitary Foldy-Wouthuysen transformation. Their no-pair integral equation takes the Schrodinger-Pauli form that operates on 2-spinors. 112 BARETTY, ISHIKAWA, A N D NIEVES In the present work, we give an alternative, but simple and straightforward, derivation of the nonpair external field equation in momentum space for oneelectron systems. An advantage to our approach is that the method can be extensible in a straightforward manner to many-electron systems. The procedure is to start with the field-theoretic QED Hamiltonian in momentum space [lo], writing the Dirac matter field in terms of creation and annihilation operators for electrons and positrons, and to delete ulf the terms that contain positron creation or annihilation operators. The resulting expression is a no-pair equation in momentum space, which incorporates the Dirac “hole theory” [lo]. It is free of the CD problem because, in field-theory language, a crucial separation between electrons and positrons is made from the outset. In this section, a simple and straightforward derivation of the no-pair equation in momentum space is given for the hydrogenic systems. In Coulomb gauge, the matter Hamiltonian H , of QED is given by [lo] H,,, = H, + Hex, + H c (3) where and Here HD is the Hamiltonian of the free Dirac field in the Schrodinger picture. Agxt is the external static potential, jo(x) the charge density, and j*(x) the in terms of the plane-wave eigenfuncelectromagnetic current. Expand $,(x) tions, u(p,s)eip’x(s = 1, 2, 3, and 4) of the free-particle Dirac Hamiltonian [9], where 113 MOMENTUM SPACE APPROACH Here E, is the positive energy of the free-particle Dirac Hamiltonian, hD and ~ ; = (A) for s = 1, 3 and w(s) = (9) for s = is given by E, = (c2p2 + c ~ ) ] ' w(s) 2, 4. In c number theory, the are the Fourier coefficients. In field theory, = correspond to annihilation and creation operators of and a:,. = electrons in the state p,s, respectively, whereas = 3,4 corre= 3,4 and spond to creation and annihilation operators, respectively, of positrons with momentum - p and reverse spin [ 101. The H,. acting on the space of one-electron states gives identically zero. Then, substituting (4) in (3) and deleting all the terms that involve apIs= 3,4 and = 3,4r we obtain the Hamiltonian for one-electron systems; with ' ~ .Hamiltonian H is bounded below. In c where A@) = [(E,, + c * ) / ~ E , . ] ~The number theory, the procedure corresponds to the exclusion of negative-energy states by the use of the projection operator. Taking the electron state as a superposition of one of the Kramers degenerate pairs, or in coordinate space w4) yields the following integral equation for the eigenvalue equation H14) = the Fourier coefficients 4(p): x w+(I){ 1 (p' x p) c2p. p' + + (E,,ic2+uc2)(EP + c2) (E,, + c2)(E,+ c') * } ~ ( 1 ) (6) By deleting all the terms that contain positron creation or annihilationoperators, terms that correspond to creation or destruction of virtual electron-positron pairs are effectively eliminated; this conserves the difference between the num- 114 BARETTY, ISHIKAWA, A N D NIEVES ber of electrons and positrons in the system. In the present case, the procedure gives rise to a “no-pair’’ external-field equation in momentum space for oneelectron systems. As the nonrelativistic limit (c + co) is approached, A@) -+ 1 and Ep+ c2 + p2/2. Thus, Eq. (6) reduces to the Schrodinger equation in momentum space. where the nonrelativistic binding energy, En, = W - c2. For the central Coulomb potential, we may assume momentum-space eigenfunctions of the form [lo], +(PI = G@)Xfrm ( O N ) where xKm (O,d = 2 C u = -c 112 YY-“ is the sphercial harmonic. +c2 are Pauli 2-spinors and are given as &;j = (A) and c$;;i”= (9) C(11/2j; m - a,a)are the Clebsh-Gordan coefficients. This separation of angular variables leads us to the integral equation for G @ ) . For the ground lSIRstate, it takes the form, where with Here Qo(z)and Ql(z) are Legendre functions of the second kind. They are given by 1 z-1 Qo(z) = --In2 z + l M O M E N T U M SPACE APPROACH 115 with = ($2 + pf2)/2pp’ Equation (7) is identical to the no-pair external-field equation recently obtained by Hardekopf and Sucher in a different approach [B]. Computational Method We have applied the Kwon-Tabakin-Lande (K-T-L) technique [ 1 I] to the “no-pair” integral Eq. (7). By casting the integral equations in momentum space into a discrete matrix eigenvalue problem with a carefully chosen grid of momentum points, Kwon and Tabakin [ 1 I] solved the eigenvalye problem for bound-state energies of hadronic atoms. They employed the so-called Lande subtraction technique to remove the logarithmic singularity of the Coulomb potential (7a,b,c)at p = p ‘ . Since the K-T-L scheme is described in detail in Ref. 10, we do not repeat it here. The grid of momentum points used in their hadronic-atom calculations [ 1 I] is given by r where C, and C,, are parameters that control the distribution of momentum points. The Xiare the roots of the Legendre polynomial. Since the parameters for hadronic atoms are not suitable to electronic systems, we have reoptimized them using 96 momentum points. C, = 2, and C , = 1OI6 were found to give six-digit accuracy in binding energies. All the calculations were performed with 96 momentum points. Results and Discussion Results of calculations on the binding energies of 1SlIzhydrogenic atoms are presented in Table I. Hess [I21 has recently applied the Gaussian basis expansion technique to the no-pair equation in the configuration representation [ 131. His results on hydrogenic atoms are reproduced in the second columns for comparison. Hess’ results were obtained by using 62 Gaussian-type functions and are estimated to have %digit accuracy. Results obtained with the K-T-L technique differ from those given by Hess in the sixth digit and are consistently overestimating the relativistic energy by about 2.0 x lo-’ a.u./P. This is attributable to numerical errors introduced by use of the limited number of momentum discretization points. However, in view of the fact that a large Gaussian basis set is required to obtain higher accuracy and that rather elaborate 116 BARETTY, ISHIKAWA, AND NIEVES TABLE1. Comparison of scaled ISlR energies" of one-electron atoms. 2 K-T-Lh Hess' 8 14 27 - ,5004868 - ,5015752 - .5063721 - .5304658 - .5521210 - .50048485 - ,5015733 - ,50637023 - 33046413 - ,55211933 55 69 Dirac eq.J .50042674 - .50131151 - .50494907 - .52194169 - s3648497 - 'Scaled binding energies are given in a d z 2 . bCalculated with the K-T-L technique using 96 momentum points. 'A basis set of 62 Gaussian-type functions used. "Binding energies givcn by Dirac equation. transformations are necessary to solve the no-pair equation in configuration space [ 121, solution of the no-pair equation directly in momentum space offers a simpler alternative to relativistic atomic structure calculations. In momentum space, relativistic kinematics [ 14,151 as well as the non-local potentials that arise in QED effects can be handled more directly and in a simpler manner. Our numerical results confirm the findings of Hess [ 121 and Hardekopf and Sucher [8] that the binding energies obtained with the no-pair equation are consistently larger than the binding energies obtained with the Dirac equation. This is due to the specific choice of expansion of the Dirac matter field used in Eq. (4). The expansion in terms of the eigensolutions of the free-particle Dirac Hamiltonian is the usual practice in field theory [ 101, and conceptually it describes bound electrons in terms of free electrons. Mathematically it is the most convenient form because the projection operator (2) reduces to an algebraic operator in momentum representation. However, its drawback is that the bound-state solutions of the no-pair Eq. (7) span only the subspace of the positiveenergy Dirac plane-wave solutions, whereas the expansion of the boatldspace eigensolutions of the Dirac external-field Hamiltonian in terms of the plane-waves necessarily contain the negative-energy plane-wave components as well as the positive-energy ones. Bibliography [ I ] E. R. Davidson, Y. Ishikawa, and G. L. Malli. Chem. Phys. Lett. 84, 226 (1981). [2] P. Pyykko, Adv. Quantum Chem. 11, 353 (1978). [3] I. P. Grant, Adv. Phys. 19, 747 (1971). [4] J. P. Desclaux. Comput. Phys. Commun. 9, 31 (1975). [5] G. E. Brown and D. G. Ravenhall, Proc. R. SOC.London A208, 552 (1951). 161 J. Sucher, Phys. Rev. A22, 348 (1980); Int. J. Quantum Chem. 25, 3 (1984). MOMENTUM SPACE APPROACH 117 [7] K. Dietz, 0. Lechtenfeld, and G . Weymans, J. Phys. B15, 4301 (1982). [8] G. Hardekopf and J. Sucher, Phys. Rev. A N , 703 (1984). [9] R. Baretty, Ph.D. thesis, University of Puerto Rico, 1985 (unpublished). [ 101 J. J. Sakurai, Advanced Quantum Mechanics (Addison-Wesley, Reading, Mass., 1967). [ I l l Y. R. Kwon and F. Tabakin, Phys. Rev. C18, 932 (1978). [I21 B. A. Hess, Phys. Rev. A32, 756 (1985). [I31 For a mixed momentum-configuration space approach to the no-pair equation, see J. Almlof, K. Faegri, Jr., and H. H. Grelland, Chem. Phys. Lett. 114,53 (1985). [I41 A. M. Simas and V. H. Smith, Jr., J. Chem. Phys. 81, 5219 (1984). [I51 A. Farazdel, W. M. Westgate, A. M . Simas, R. P. Sagar, and V. H.Smith, Jr., Int. J. Quantum Chem. Symp. 19, 61 (1985). Received May 1 , 1986