 It is demonstrated that atomic Hartree–Fock binding energies may be reproduced with great accuracy (within about four parts in a thousand) by a scaled model system in which the electrons are noninteracting, and are bound in a bare Coulomb potential. </jats:p

Bhaduri, R. K.

Brack, Matthias

English

University of Regensburg Publication Server

1684 Atomic ground state energies R. K . BHADURI A N D M . BRACK 1 Physics Department, McMaster University, Hamilton, Ont., Canada L8S4MI Received June 10, 1980 It is demonstrated that atomic Ha r t r ee -Fock binding energies may be reproduced with great accuracy (within about four parts in a thousand) by a scaled model system in which the electrons are noninteracting, and are bound in a bare Coulomb potential. O n demontre que les valeurs Hartree-Fock des energies de liaison atomique peuvent etre reproduites avec une grande precision (environ quatre partie sur mille) par un systeme de modeles a echelle dans lequel les electrons sont sans interaction et lies par un simple potentiel de Coulomb. . ~. „ [Traduit par le journal] C a n . J . Phys . , 58, 1684(1980) 1 In this article, we shal l present a very simple analyt ical expression for the total b ind ing energy o f a neutral atom. We are aware that this quantity is only o f peripheral interest i n modern atomic physics, since the major contr ibut ion to the total energy comes f rom electrons in the innermost shells. Never-theless, we report this work because our approach is novel, and the accuracy is better than a recently reported paper by Sucher (1) on the same subject. We demonstrate that the total ground state energy o f any atom may be reproduced by a model system in wh ich both the nuclear charge and the electron number are reduced by a scaling factor, and the electron-electron interactions switched off. The scaling factor is independent o f Z , and its value can be guessed f rom theory. Th is mode l is rather different f rom the convent ional effective charge approach (2, 3) i n wh ich only the nuclear charge, as seen by the electrons i n each shell, is modif ied. A neutral a tom wi th Z electrons is replaced by a model system o f Z 0 noninteract ing electrons bound in a bare C o u l o m b potential w i th nuclear charge Z 0 . F o r such a system the closed shell interpolat ion fo rmula for energy, in atomic units, is (4) [1] -E/Z02 = ( i Z 0 ) 1 / 3 ( l + 0 . 0 6 3 6 Z 0 - 2 / 3 ) - i We take Z ( ) = (3Z, where (3 = 0.831 for a l l atoms with Z > 8. Subst i tut ing this value o f Z 0 in eq. [1], our expression for the ground state energy o f an a tom wi th atomic number Z is [2] -EjZ1 = 0 . 7 4 3 2 Z 1 / 3 [1 + 0 . 0 7 1 9 Z " 2 / 3 ] - 0.3453 In Table 1, the energies o f some atoms as predicted by eq. [2] are compared wi th the corresponding (nonrelativistic) Ha r t r e e -Fock results. O u r results 'Permanent address: Physics Department, University of Re-gensburg, 8400 Regensburg, W. Germany. T A B L E 1. Compar i son of the Hart ree -Fock energies EHF with those predicted by eq. [ 2 ] for some neutral atoms. The energies are in atomic units, and divided by Z2 z -EHF/Z2 -E/Z2, eq. [ 2 ] 8 1 . 1 6 9 1 . 1 6 8 1 0 1 . 2 8 6 1 . 2 8 1 1 3 1 . 4 3 1 1 . 4 2 5 1 5 1 . 5 1 4 1 . 5 0 9 1 8 1 . 6 2 6 1 . 6 2 3 2 0 1 . 6 9 2 1 . 6 9 2 2 5 1 . 8 4 0 1 . 8 4 6 3 1 2 . 0 0 1 2 . 0 0 6 3 6 2 . 1 2 4 2 . 1 2 5 4 2 2 . 2 5 4 2 . 2 5 3 4 9 2 . 3 9 1 2 . 3 8 9 5 4 2 . 4 8 0 2 . 4 7 8 6 5 2 . 6 5 7 2 . 6 5 6 7 0 2 . 7 3 3 2 . 7 3 1 7 5 2 . 8 0 6 2 . 8 0 2 8 0 2 . 8 7 6 2 . 8 6 9 8 7 2 . 9 6 9 2 . 9 6 0 9 0 3 . 0 0 7 2 . 9 9 7 9 5 3 . 0 6 9 3 . 0 5 8 are generally more accurate than those o f Sucher (1)' considerably so in the region 8 < Z < 50. Earlier, Scott (5) and F o l d y (6) had given simple expressions for atomic b ind ing energies wh ich are also much less accurate than our eq. [2]. Scott derived his formula by adding to the leading T h o m a s - F e r m i term corrections due to the exchange contr ibut ion , and a term to compensate for the diverging Thomas -Fe rm i density near the or ig in . O u r approach in deducing eq. [2] w i l l be entirely different. The atomic Hami l t on i an wi th Z electrons may b e written as \ i i r i I \i<j rij i 1 0008-4204/80/121684-02$01.00/0 ©1980 National Research Counci l o f Canada/Conseil national de recherches du Canada B H A D U R I A N D B R A C K 1685 where the sum is over a l l electrons. We choose = (3Z so that the expectation value o f the last term is zero, i.e., t4] \Z{Z - 1 K / V 1 ) = Z 2 ( l - P X r , " 1 ) or (i - P) = Knj'l>Krrl> f°r Z » I. A s before, we have assumed that P is ^dependent o f Z . In the T h o m a s - F e r m i approx-^ a t i o n , which is exact for Z oo, p = 6/7, while Hartree-Fock calculations over a wide range o f l v a l u e s yield a value o f P close to 5/6, thus sub-stantiating our c la im (1). Us ing the v i r ia l theorem, w e may now write the Har t r ee -Fock energy (for * » l ) a s IS] £ H F = - iZ 2 P <r r 1 >2 F ! n the above equation, the expectation value o f rt~1 ' s for a Z-electron system with H F wave functions. °ur mode l equation [2], on the other hand, is for a sVstem o f Z 0 noninteract ing electrons in a bare nuclear charge Z 0 : E= -iZ02<rrl>%T w here the expectation value is now taken wi th hydro-p i c wave functions. F o r [5] and [6] to be equal, we ^ust have ^his relation should ho ld for any Z » 1, but we are U n ab l e to prove it for the general case. However, it l s easy to verify it in the T h o m a s - F e r m i l imit oo). In this case (3) l 8 J <^ 1 >z H F -<^ - 1 >I F = 4 ( ^ r ) 1 / 3 x 1.588Z 1/ 3 P] <rrl>%rt = 2 ( 3 / 2 ) 1 / 3 ( p Z ) 1 / 3 Subst i tut ing these i n eq. [7], we find p = 0.833 « 5/6. This is the same value o f P obtained f rom eq. [4] in H F calculations, and explains why the H F b ind ing energies are reproduced by the noninteract ing scaled model . In fact, it is easy to verify eq. [7] numerical ly by calculat ing ( r , - 1 ) ^ using Table 2.3 in the book by Fischer (7). F o r the range o f Z values that we have considered, we find that the r ight-hand side o f eq. [7] tends to overestimate O , - 1 ) " * 7 by about 1 to 1.5% for P = 0.831. Th is was the scaling factor used to fit the H F energies by eq. [2]. If we had used eqs. [5] and [8], wi th the T F value o f p = 6/7, we would have obtained the coefficient of Z 1 / 3 to be 0.7687, which is correct in the l imi t o f Z -> oo. O n the other hand, by using the empir ica l value o f p ^ 5/6, we have shown that the energy may be written in the form o f eq. [6], which is the content o f our model . O u r arguments, however, do not amount to a ' ' de r i va t i on^ o f eq. [2], nor explain its extreme accuracy. Acknowledgements This work was supported by the Na tu ra l Sciences and Engineering Research C o u n c i l o f Canada . The authors thank Y u k i N o g a m i and Peter Sutherland for useful discussions. 1. J . S U C H E R . J . Phys. B , 11, 1515(1978). 2. J . C . S L A T E R . Quantum theory of matter. McGraw Hi l l , New York , N Y . 1951. p. 475. 3. R. O . M U E L L E R , A . R . P. R A U , and L . S P R U C H . Phys. Rev. A , 8, 1186(1973). 4. N . H . M A R C H . Self-consistent field in atoms. Pergamon Press, Oxford, England. 1975. pp. 49, 100. 5. J . M . C . S C O T T . Philos. Mag. 43, 859(1952). 6. L . L . F O L D Y . Phys. Rev. 83, 397 (1951). 7. C . F . F I S C H E R . The Ha r t r ee -Fock method for atoms. John Wiley, New York , N Y . 1977. p. 27. 

Atomic Ground-State Energies

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