An Alternative Scheme for Calculating the Unrestricted Hartree-Fock Equation: Application to the Boron and Neon Atoms.

Transcription

1 An Alternatve Scheme for Calculatng the Unrestrcted Hartree-Fock Equaton: Applcaton to the Boron and Neon Atoms Mtyasu MIYASITA 1,2, Katsuhko HIGUCHI 1, Masahko HIGUCHI 3 1 Graduate School of Advanced Scence of Matter, Hroshma Unversty, Hgash-Hroshma , Japan 2 CCSE, Japan Atomc Energy Agency, Hgash-Ueno, Tato-ku, Tokyo , Japan 3 Department of Physcs, Faculty of Scence, Shnshu Unversty, Matsumoto , Japan (June 15, 2009) Abstract We present an alternatve scheme for calculatng the unrestrcted Hartree-Fock equaton. The scheme s based on the varatonal method utlzng the sophstcated bass functons that nclude no adustable parameters. The valdty and accuracy of the present scheme are confrmed by actual calculatons of the boron and neon atoms. It s shown that the present scheme not only gves the reasonably lower total energy but also conserves the vral relaton wth enough accuracy. Key words: unrestrcted Hartree-Fock equaton, atomc structure, boron atom, neon atom

2 1 Introducton The sngle-partcle wave functon and spectra for the atomc system are fundamental for understandng the electronc and magnetc propertes of solds. For nstance, hoppng and Coulomb ntegrals ncluded n the model Hamltonan[1,2] and LDA+U method[3,4], etc, are defned usng the atomc wave functons. Also n the tght-bndng method,[5] the matrx elements of the Hamltonan are constructed from the atomc wave functons and spectra. The electronc and magnetc propertes that are related to the well-localzed electrons of solds are necessarly related to the atomc wave functons and spectra. Thus, there s no doubt that approprate descrpton for the atomc system wll lead to better understandng of solds. Concernng the theoretcal framework for calculatng the atomc structures, the most standard one s the Hartree-Fock (HF) equaton.[6,7] The HF equaton does not contan the correlaton effects at all, but such correlaton effects are small n the atomc system compared to the solds. The rato of the correlaton energy to the exchange energy s roughly 1/10 as large as that of solds.[8,9,10] Therefore, the HF equaton s a good startng pont for descrbng the atomc structures. There exst two knds of HF equatons,.e., restrcted Hartree-Fock (RHF) equaton and unrestrcted Hartree-Fock (UHF) equaton.[11] The former has the mert of conservng the total spn moment rgorously, but has the demert concernng the matrx type of the Lagrange multplers. The Lagrange multplers generally appear n the sngle-partcle equaton n a form of the matrx elements by mposng the orthonormalty on the orbtals. Such the matrx cannot always be transformed nto the dagonal form n the case of the RHF equaton.[12] Ths means that the RHF equaton cannot always be transformed nto the canoncal form. These nondagonal elements of the Lagrange multplers are small but can not be neglected n some case.[12,13] If the nondagonal elements of the Lagrange multplers are neglected for the practcal am, one can utlze the so-called term-dependent HF equaton, whch s derved by takng the functonal dervatves of the total energy of a sngle term wth respect to the radal functons.[12,14] It s shown by Slater that ths scheme s equvalent to the approxmaton of takng the sphercal average of the potentals.[14,15] Extended verson of ths approxmaton has been also developed by Slater.[13] It starts wth the total energy that s averaged wth respect to all the allowed terms to some electron confguraton. The resultant RHF equaton, whch may be called the term-averaged HF equaton, has been numercally solved by Fscher et al.[16] The results have been regarded as a standard reference data of the RHF equaton. In addton to the above methods, there s an effectve method to solve the RHF equaton,.e., the Hartree-Fock-Roothaan (HF-R) method.[17,18,19] In the HF-R method, the solutons are expanded wth the emprcal bass functons such as the Gaussan-type orbtal (GTO) and Slater-type orbtal (STO). The problem of solvng the RHF equaton results n the generalzed egenvalue problem, whch s a strong advantage from the vewpont of the numercal

3 calculatons. The HF-R method usually deals wth the canoncal RHF equaton. In ths sense, the HF-R method also gves an approxmaton of the RHF equaton. Next let us sketch the features of the latter one (UHF equaton). The UHF method has the mert that the equaton can take the canoncal form rgorously, but t has the demerts that the total spn moment s not always guaranteed to be conserved. Due to such the demert, the UHF method has not been actvely treated, as compared to the RHF method.[11,20-24] However, the UHF equaton s obvously more natural and reasonable than the RHF equaton because the spatal wave functons for the up-spn and down-spn states are generally dfferent from each other. If we deal wth the UHF equaton approprately, the total energy would be lower than that of the RHF method. In ths paper, we present an alternatve scheme for calculatng the UHF equaton, whch s based on the varatonal method. As the expanson bass, we choose the solutons of the Xα method wth the sphercally-averaged potentals[25]. Ths choce of the bass functons reduces the computatonal task n terms of the number of the expanson bass compared to the HF-R method. Furthermore, as shown n the subsequent sectons, the present scheme not only mproves the total energy but also conserves the vral relaton wth enough accuracy. Of course, the present scheme ncludes the effects of the nonsphercal dstrbuton of electrons. Ths effect can not be neglgble for the atomc structures, as already ndcated by Slater[26] and ours[27,28], and unambguously shown by our precedng works.[29,30] Organzaton of ths paper s as follows. In Sec. 2, we explan the outlne of the present scheme. The results of the test calculaton for the boron and neon atoms, and correspondng dscussons are gven n Sec. 3. Fnally, n Sec. 4, we summarze the results and gve some comments on t. 2 Varatonal Method In ths secton, we present a varatonal method for calculatng the UHF equaton by means of sophstcated bass functons. Let us start wth the UHF sngle-partcle equaton: Ze r 2m r r r 2 * occ. occ. 2 r 2 r r r r r r, e dr r e dr r (1)

4 where denotes the up-spn or down-spn. The basc dea of our scheme s almost the same as that of our recent work[29]. Namely, the soluton of eq. (1) s expandng wth the set of known functons as bass functons. It should be noted that the varatonal method always ncludes the arbtrarness of choosng bass functons and the choce affects calculaton results drectly. As the bass functons, we adopt the egenfuncton for the Hamltonan defned by 2 2 ˆ 2 Ze H X, ex H 0 : VS r VS, r 2m r, (2) where X, ex V r and V r H S S, stand for the sphercal part of Hartree and exchange potental of the Xα method[25], respectvely. Utlzng the numercal methods such as the Herman-Skllman method[31], the egenfunctons and egenvalues of the eq. (2) can be easly obtaned, whch are denoted as r, p and nl Y lm respect to such the egenfunctons: nl, respectvely. We shall expand the soluton of eq. (1) wth ( ) C r p ( r) Y (, ). (3) nlm nl lm nlm Substtutng eqs. (2) and (3) nto eq. (1), and wrtng dstnctly the sphercal part and the rest one (whch s denoted by V H NS r ) of the Hartree potental, we get r Hˆ V V r C p ( r) Y (, ) H X, ex 0 NS S, nlm nl lm nlm r C p ( r) Y (, ) e d C p ( r) Y (, ). * occ. nlm nl lm 2 nlm r r nlm nl lm r r nlm (4) Multplyng p ( r) Y (, ) on both sdes of eq. (4) and ntegratng over the whole space, we a * * nala lama have

5 nlm a 0 0 occ. n al a nl Cnlm l 0 al mamon ala anl V nalama anlm E nalama anlm 2 (5) where O p ( r) p ( r) r dr, (6) a * 2 nala anl nala nl NS S, ex a V * ( ) *, H X, n ( ),, alama anlm pn al r Y a lam V r V r p a nl r Ylm dr (7) r p ( r) Y, * 2 * * nl lm a En ( ),, alama anlm e pn al r Y a lam a r drdr r r (8) and r s the result of the prevous teraton. Equaton (5) s ust the generalzed egenvalue problem. If the matrx elements, O, nala anl V, nalama anlm nalama anlm E and the energy spectra of the sphercal approxmaton, egenfunctons, C nlm 0, are gven, then we can obtan the egenvalues,. It should be noted that the egenvalues snce two matrces ncluded n eq. (5) are hermtan. be analytcally calculated by usng the Wgner 3-symbols.[32], and are guaranteed to be real The angular ntegraton n eqs. (7) and (8) can The potentals eq. (7) should be determned n a self-consstent way[6]. The correspondng bass functons n eq. (3) are modfed for each teraton snce the functon p ( r) nl s the radal part of soluton for the Hamltonan (2). Ths s the strkng feature of the present scheme, whch enables us to solve the UHF equaton more rapdly wth not so many of bass functons. The teraton s contnued untl self-consstency for the potentals s acheved. 3 Results and Dscussons In order confrm the effectveness of our scheme, actual calculatons are appled to the boron and neon atoms. The number of bass functons of eq. (3) s 23, whch have the followng quantum

6 numbers: ( nlm) (100), (200), (211), (210), (211), (300), (311), (310), (311), (322), (321), (320), (32 1), (32 2), (400), (411), (410), (411), (422), (421), (420), (42 1), (42 2). As mentoned above, we use the sphercal approxmated Xα method for preparng the bass functons. The parameter α s determned by requrng that the vral theorem holds. The most optmzed values are for boron and for neon, respectvely. (See Fgure 1) Let us gve the results of the test calculaton for the boron and neon atoms. Fgure 2 shows the energy spectra of the present scheme, together wth those of conventonal ones[16,19]. The dfferences between ours and conventonal ones are not small, especally for the outer states. Table 1 shows the total energy of the present and conventonal schemes. Present scheme gves the lower total energy than that of the conventonal ones. Ths s not only because the UHF has no restrcton on the wavefuncton such that the spatal parts are dentcal for the up- and down-spn states, but also-because the choce of the bass functons n more approprate than the conventonal HF scheme. The latter reason s confrmed by the fact that the total energy of the neon atom that has the same spatal wavefuncton for the up- and down-spn states due to no spn-polarzaton becomes lower than the conventonal RHF scheme. Furthermore, we would lke to stress that the vral relaton s conserved wth enough accuracy n the present scheme. 4 Concludng Remarks In ths paper, we present an alternatve scheme for calculatng the UHF equaton. The valdty and accuracy of ths scheme have been confrmed by actual calculatons for the boron and neon atoms. The present scheme can lower the total energy reasonably, whle conservng the vral relaton. The present scheme clearly shows the necessty of modfyng the sngle-partcle pcture of atomc systems. Furthermore, we can say that the present scheme may open the possblty of mprovng the estmaton of the varous parameters concernng the sold state physcs, whch are based on the atomc sngle-partcle wave functons and spectra. For the future work, we have to consder the relatvstc and/or correlaton effects n order to mprove the sngle-partcle pcture beyond the HF approxmaton. These are completely neglected n the present calculatons. Especally concernng the relatvstc effect, t seems to be ndspensable for descrbng the electrc structures of the heaver atoms.

7 Acknowledgements Ths work was partally supported by Grant-n-Ad for Scentfc Research (No ) and for Scentfc Research n Prorty Areas (No ) of The Mnstry of Educaton, Culture, Sports, Scence, and Technology, Japan.

8 Reference [1] N. F. Mott, Proc. Roy. Soc. (London) A62, 416 (1949). [2] P. W. Anderson, Phys. Rev. 115, 2 (1959). [3] V. I. Ansmov, J. Zaanen and O. K. Andersen, Phys. Rev. B 44, 943 (1991). [4] A. I. Lechtensten, V. I. Ansmov and J. Zaanen, Phys. Rev. B 52, R5468 (1995). [5] W. A. Harrson, Electronc Structure and the Propertes of Solds The Physcs of the Chemcal Bonds (W. H. Freeman and Co., San Francsco, 1980). [6] D. R. Hartree, Proc. Cambrdge Phl. Soc., 24, 111 (1928). [7] V. Fock, Z. Physk, 61, 126 (1930); 62, 795 (1930). [8] Ths estmaton s based on the data of the atomc systems (Ref. [9]) and the homogeneous electron lqud model (Ref. [10]). Ths s the reason why the HF approxmaton works comparably well for the atomc system though t fals to descrbe the electronc structures of solds. The falure of the HF approxmaton s predomnant especally near the Ferm level, whch s well-known the Hartree-Fock problem. [9] A. D. Becke, J. Chem. Phys. 84, 4524 (1986). [10] G. F. Gulan and G. Vgnale, Quantum Theory of the Electron Lqud (Cambrdge, New York, 2005) Chap. 17. [11] A. Szabo and N. S. Ostlund, Modern quantum chemstry (Dover, New York, 1996) Chap. 3. [12] J. C. Slater, Quantum theory of atomc structure Vol. II (McGraw-Hll, New York, 1960) Sec. 17. [13] In order to avod ths dffculty, several approaches have been developed so far. For nstance, see, C. C. J. Roothaan, Rev. Mod. Phys. 32, 179 (1960), and Ref. [12]. [14] S. E. Koonn and D. C. Meredth, Computatonal physcs (Westvew Press, Colorado, 1990) Chap. 3. [15] J. C. Slater, Quantum theory of atomc structure Vol. I (McGraw-Hll, New York, 1960) Sec. 9. [16] C. F. Fscher, The Hartree-Fock method for atoms (John Wley and Sons, New York, 1977). [17] C. C. Roothaan, Rev. Mod. Phys. 23, 69 (1951). [18] C. C. Roothaan and P. S. Bagus, Methods n Compt. Phys. 2, 47 (1963). [19] E. Clement and C. Roett, Atomc data and nuclear data tables, 14, 177 (1974). [20] Usually the UHF method does not conserve the total spn moment, so that the modfed scheme such as the proected UHF method has been devsed. However, unfortunately, the proected UHF method s complcated and needs the heavy computatonal task. For nstance, see, Refs. [21], [22], [23] and [24]. [21] L. M. Sachs, Phys. Rev. 117, 1504 (1960). [22] W. A. Goddard, III, Phys. Rev. 157, 93 (1967).

9 [23] R. P. Hurst, J. D. Gray, G. H. Brgman and F. A. Matsen, Mol. Phys. 1, 189 (1958). [24] W. A. Goddard, III, Phys. Rev. 169, 120 (1968). [25] J. C. Slater, Phys. Rev., 81, 385 (1951). [26] J. C. Slater, The calculaton of molecular orbtals (John Wley and Sons, New York, 1979) Chap. 2. [27] M. Hguch and A. Hasegawa, J. Phys. Soc. Jpn. 66, 149 (1997). [28] M. Hguch and A. Hasegawa, J. Phys. Soc. Jpn. 67, 2037 (1998). [29] M. Myasta, K. Hguch, A. Narta and M. Hguch, Mater. Trans. 49, 1893 (2008). [30] M. Myasta, K. Hguch and M. Hguch, submtted to Mater. Trans. [31] F. Herman and S. Skllman: Atomc Structure Calculatons (Prentce-Hall Inc., New Jersey, 1963). [32] A. Messah: Quantum Mechancs (Dover Publcatons, NY, 1999).

10 Fgure captons Fg. 1. Parameter alpha and vral rato for boron and neon atoms. born and neon atoms, respectvely. Open and black crcle denotes Fg. 2. Energy spectra for the boron and neon atoms. The frst and thrd columns show the results for the conventonal HF method whch s calculated by E. Clement and C. Roett or C. F. Fscher[16,19]. The second and forth columns are the results for the present scheme. The up- and down-arrows denote the occuped states, and open crcle the unoccuped states. All values are gven n Rydberg Unt.

11 Tables 1

12 Fgure 1

13 Fgure 2