Modelling the Pauli Potential in the Pair Density Functional Theory

Modelling he Pauli Po en ial in he Pai Densi y Func ional Theo y C. Amo illiaand ´ A. Nagyb aDipa imen o di Chimica e Chimica Indus iale, Uni e si `a di Pisa, Via Riso gimen o 35, 56126 Pisa, I aly bDepa men o Theo e ical Physics, Uni e si y o Deb ecen, H–4010 Deb ecen, Hunga y Oc obe 22, 2008 Abs ac In he g ound s a e he pai densi y can be de e mined by sol ing a single auxilia y equa ion o a wo-pa icle p oblem. A no el me hod o de e mining he Pauli po en ial en e ing his equa ion is p esen ed and, s a ing om a eliable desc ip ion o he pai densi y, an analy ical exp ession is de i ed o a omic sys ems. Tes calcula ions a e p esen ed o Be and isoelec onic C2+ and O4+ ions. 1 In oduc ion Gene alized densi y unc ional heo ies ha e ecei ed a g owing impo ance in ecen yea s. Fo elec on sys ems, he in e es has been posed on he pai densi y as he undamen al a iable ins ead o he one pa icle densi y. I u ned ou ha he e exis a a ia ional p inciple o he pai densi y (analogous o he Hohenbe g-Kohn heo ems o he densi y unc ional heo y). I has been shown ha - ins ead o Kohn-Sham equa ions - in he pai densi y unc ional heo y [1–4] he g ound s a e 1 p oblem o an a bi a y sys em is educed o a wo-pa icle p oblem. The wo- pa icle equa ion is w i en [1–4] as −1 2∇2 1−1 2∇2 2+ ( 1) + ( 2) + N−1 + P( 1, 2)χ( 1, 2) =µχ( 1, 2),(1) whe e is he ex e nal po en ial, Nis he numbe o elec ons and he no a ion =| 1− 2|is used. The g ound-s a e eigen unc ion o his equa ion, ˜χ0say, co esponds o he pai densi y ampli ude and is ela ed o he pai densi y no he eal sys em as n=N(N−1) 2|˜χ0|2.(2) Eq. (1) con ains an unknown e m, P, o comple ely kine ic o igin. A e a densi y unc ional analogy Pis called Pauli po en ial. Eq. (1) is analogous o he densi y unc ional equa ion o he squa e oo o he densi y (which da es back o Thomas and Fe mi [5] and is analyzed by Le y, Pe dew and Sahni [6]). The pai densi y can be nume ically calcula ed ei he on he Ha ee-Fock le el o on highly co ela ed le el. The pai densi y can also be de e mined om Eq. (1) in a a he s aigh o wa d way i he Pauli po en ial is known. Howe e , he e a e no da a o he Pauli po en ial in he li e a u e, ye . Al hough ecen ly, he elec on-elec on cusp condi ion and asymp o ic beha iou o he Pauli po en ial ha e been de i ed [7], much wo k is called o o comple ely unde s and how such a po en ial could be modellized in analogy wi h he Kohn-Sham po en ial in o dina y densi y unc ional heo y (DFT). Conside ing he p esen knowledge, his kind o wo k appea s ex emely di icul . We belie e ha an impo an i s s ep, in o de o gain mo e insigh in his di ec ion, is he econs uc ion o he Pauli po en ial 2 om a eliable pai densi y o m o some ealis ic ac able elec on sys em. Fo hese cases, a six a iable unc ional o m o Pshould be, in p inciple, ob ained. This unc ion, a his poin , should be iewed as a sou ce o in o ma ion expecially wi h he aim o inding hose p ope ies ha can be ans e ed o o he sys ems o which he pai densi y is unknowm. Mo i a ed by he abo e conside a ion, in his pape we p esen model Pauli po en ial o he Be a om and isoelec onic a omic ions C2+ and O4+. The me hod, which is in ended o cap u e he main ea u es o P, is based on an ansa z on he o m o he pai densi y ampli ude. We gene alize he me hod o Amo illi e al. [8]. The o iginal me hod was used o ob ain he exac Hamil onian o an analy ic g ound-s a e wa e unc ion o He-like ions. He e, a gene aliza ion is p esen ed o p oducing he Pauli po en ial om a model pai densi y ampli ude. The pape is o ganized as ollows: In sec ion 2 he pai densi y unc ional he- o y is e iewed. In sec ion 3 a model pai densi y ampli ude and he co esponding po en ial is p esen ed. Sec ion 4 desc ibes nume ical examples: he Be and some isoelec onic a omic ions. The las sec ion is de o ed o discussion. 2 The pai densi y unc ional heo y Fi s , he pai -densi y unc ional heo y [1–3] is summa ized. Conside he many elec on Hamil onian H, ˆ H=ˆ T+ˆ Vee + N X i=1 ( i),(3) 3 whe e ˆ T= N X i=1 (−1 2∇2 i) (4) is he kine ic ene gy ope a o , ˆ Vee = N X i<j 1 | i− j|(5) is he elec on-elec on epulsion ene gy ope a o and ( ) is a local ex e nal po en- ial. Fo con enience we conside an e en numbe o pa icles. The second-o de ed educed densi y ma ix is de ined as n2(x1,x2;x0 1,x0 2) = N(N−1) 2ZΨ(x1,x2,x3..., xN)Ψ∗(x0 1,x0 2,x3..., xN)dx3...dxN,(6) whe e xis ands o he spa ial and he spin coo dina es: i, σiand he in eg al sym- bol when e e ed o spin deno es summa ion. The diagonal o he spin-independen second-o de ed densi y ma ix n( 1, 2) = X σ1,σ2 n2( 1, σ1, 2, σ2) (7) also called pai densi y is he key quan i y. I is con enien o in oduce new posi ion a iables qJ= ( j, j0) = (qJ1, qJ2, qJ3, qJ4, qJ5, qJ6),(8) i.e. he pai s will be deno ed by capi al indices while he pa icles in each pai will be iden i ied by he co esponding unp imed and p imed le e s. Wi h he abo e no a ion he numbe o a iables is he same as ha o he ini ial sys em. Each pa icle is associa ed wi h a single pai , i.e. he numbe o indices Jis N/2. The ’in e nal’ po en ial o he pa icles in pai Jis gi en by ˜ (qj) = ˜ ( j, j0) = 1 | j− j0|.(9) 4 while he in e ac ion be ween pai s Iand Jis WIJ =W(qI,qJ) = W( i, i0; j, j0) = 1 | i− j|+1 | i− j0|+1 | i0− j|+1 | i0− j0|.(10) The ene gy o he pai s due o he ex e nal po en ial is ˆ U= M X I=1 u(qJ) = N X I=1 ( ( j) + ( j0)) .(11) De ining he ope a o ˆ L ep esen ing he in e nal ene gy o pai s ˆ L= M X I=1 (−1 2∇2 I+ ˜ (qI)) ,(12) he ini ial Hamil onian can be exp essed as ˆ H=ˆ L+ˆ W+ˆ U , (13) whe e ˆ W=1 2 M X I6=J WIJ (14) is he in e ac ion ene gy be ween di e en pai s (M=N/2). ˆ His he same as he ini ial Hamil onian, bu now i is w i en in e ms o pai s o pa icles, wi h ˆ L+ˆ U ep esen ing he Hamil onian o independen (nonin e ac ing wi h each o he ) pai s and ˆ W ep esen ing he in e pai in e ac ion. The Laplacian in he kine ic ene gy ope a o can also be w i en as ∇2 I=∇2 qI=∇2 i+∇2 i0= 6 X α=1 ∂2 ∂q2 Iα .(15) The ene gy o he independen pai s has he o m Q[n] = min Ψ→nhΨ|ˆ L+ˆ W|Ψi.(16) 5 The sea ch o he minimum is o e all an isymme ic wa e unc ions Ψ which yield he gi en n. Then he g ound s a e ene gy can be w i en as E= min n1 N−1Zu( 1, 2)n( 1, 2)d 1d 2+Q[n].(17) The ac o 1/(N−1) comes om he no maliza ion o n. The densi y o pai I n(qI) = n( i, i0) = X σi,σi0 n2( i, σi, i0, σi0) (18) is he pai densi y in he o iginal space. The Hohenbe g-Kohn heo ems [10] ha e been gene alized o he pai densi y [11,12] o he o iginal space. The g ound s a e inequali y is 1 N−1Zn(q)u(q) + Q[n]≥E0.(19) whe e E0and n0a e he g ound-s a e ene gy and he diagonal o he spin indepen- den second-o de densi y ma ix, espec i ely. In he pai densi y unc ional heo y he adiaba ic connec ion is de ined by he pa ame ized Hamil onian ˆ Hα=ˆ L+αˆ W+ˆ Uα,(20) whe e ˆ Uα=PIuα I(q) is gi en by he condi ion ha he pai densi y n(q), o he o iginal space keeps being independen o α. Fo α= 0 he ’non-in e ac ing Hamil onian’ ˆ Hα=0 =ˆ L+ˆ Uα=0 =X I=1 hα=0 I(21) is ob ained. In his auxilia y sys em he in e ac ion be ween he pai s is ze o and he auxilia y equa ions ha e he o m ˆ H0Ψ0=E0Ψ0.(22) 6 The wa e unc ion in his auxilia y sys em can be w i en as a symme ized exp ession o an isymme ic wo-pa icle unc ions χI: Φ0(x1, ..., xN) = ˆ S(χ1(x1,x2)...χM(xN−1,xN)) .(23) ˆ S=1 N!X P ˆ P(24) is he symme ize ope a o . Pis he pe mu a ion ope a o and he sum is o e all pe mu a ion o he elec on pai s. This wa e unc ion is an isymme ic wi h espec o he exchange o he a iables o a single pai and symme ic wi h espec o he exchange o he pai s. The disad an age o he p esen no a ion is ha i does no allow ansposi ion o a iables belonging o wo di e en pai s. In he g ound s a e n(q) = NN−1 2X σ |χ0(x1,x2)|2=NN−1 2|˜χ0(q)|2,(25) whe e he wo-pa icle unc ion ˜χ0sa is y he eigen alue equa ion h0(q)˜χ0(q) = −1 2∇2 q+ e (q)˜χ0(q) = ε0˜χ0(q),(26) We men ion in passing ha i is possible o w i e he pai densi y in e ms o geminals. The p esen e sion o he pai densi y e sion o heo y has he ad an age ha he calcula ion o nis always educed o he solu ion o a wo-pa icle equa ion ha is he N-body p oblem can be educed o a wo-body p oblem. I has been p o ed [1] ha he auxilia y po en ial is uniquely de e mined by he diagonal o m o he spin independen second-o de densi y ma ix and he e ec i e po en ial is o he o m e (q) = ( 1) + ( 2) + N−1 12 + p,(27) 7 whe e p= (N−1)δTP δn (28) and TP=T−T0(29) is he di e ence o he kine ic ene gies o he eal sys em (T=hΨ|ˆ T|Ψi) and he auxilia y sys em T0= M X I=1 Zχ∗ I(x1,x2)[−1 2∇2 q]χI(x1,x2).(30) By a densi y unc ional analogy he unc ional TP[n] is called Pauli ene gy. The o al ene gy has he o m E[n] = T0[n] + TP[n] + Zn(q) 12 dq+1 N−1Zn(q)u(q)dq.(31) The disad an age o he p esen ea men is ha i is ha d o cap u e he e mionic s uc u e o an elec onic sys em wi h a single e ec i e po en ial. Howe e , we ha e always a wo-pa icle p oblem o sol e independen ly o he numbe o elec ons. I is wo h o make e o s o ind adequa e app oxima ion o he Pauli po en ial in o de o u ilize his bene i . The p esen s udy is a s ep in his di ec ion. The auxilia y equa ions can also be de i ed by cons ained sea ch [1,13]. The wo-pa icle equa ion (26) was la e de i ed [14] in a di e en way which is no es ic ed o e en numbe o elec ons. 8 3 A model pai densi y ampli ude and he co e- sponding po en ial As i was shown in he i s pape [1] he Pauli po en ial is uniquely de e mined by he pai densi y. Tha is, om he knowledge o n, Pcan be gi en by in e ing Eq. (1) P( 1, 2) = −Kloc( 1, 2)−w( 1, 2),(32) whe e Kloc( 1, 2) = −1 2˜χ0( 1, 2)h∇2 1+∇2 2i˜χ0( 1, 2) (33) and w( 1, 2) = ( 1) + ( 2) + N−1 −µ. (34) We ha e ecen ly p o ed [7] ha he Pauli po en ial asymp o ically beha es as P→(N−2) 1 1 +1 2 −1 (35) when 1→ ∞, 2→ ∞ and → ∞. The elec on-elec on cusp condi ion has he o m: P=2−N (36) as →0. Wi h he aim o econs uc Pin some unc ional o m, we s a ou om a model unno malized pai densi y ampli ude in he o m χ=χHF (λ 1, λ 2)(1 + g( )) ,(37) 9 a λ E Tp< −2> < −1> < > < 2> < 3> 4 0.9835 –14.660(4) 0.878(3) 9.319 4.279 15.537 54.485 232.96 5 0.9847 –14.669(4) 0.864(3) 9.541 4.320 15.469 54.107 230.88 6 0.9860 –14.669(4) 0.855(3) 9.701 4.349 15.418 53.815 229.23 HF(a)–14.573 1.005 10.536 4.489 15.120 51.956 218.11 co (b)–14.667 — 9.536 4.337 15.272 52.854 222.48 (a)In he Tpcolumn is epo ed he di e ence Tex −T(1) wand momen s a e om e s. [16,18]. (b)Momen s om [18]. Table 2: To al ene gy (E), Pauli kine ic ene gy (Tp) and some momen s < k> o Be a om o di e en choices o he co ela ion unc ion pa ame e aand he scaling cons an λcalcula ed in his wo k and compa ison wi h HF and co ela ed li e a u e da a. Da a a e in a omic uni s. a λ E Tp< −2> < −1> < > < 2> < 3> 5 1.00090 –36.538(9) 3.253(8) 25.37 7.540 8.013 14.031 29.466 6 1.00063 –36.538(9) 3.226(8) 25.35 7.577 7.997 13.991 29.369 7 1.00055 –36.539(9) 3.204(8) 25.94 7.604 7.985 13.961 29.296 HF(a)–36.408 3.475 27.06 7.716 7.945 13.863 29.06 co (b)–36.534 — 25.50 7.548 8.118 14.502 31.14 (a)In he Tpcolumn is epo ed he di e ence Tex −T(1) wand momen s a e om e s. [16,18]. (b)Momen s om [16]. Table 3: To al ene gy (E), Pauli kine ic ene gy (Tp) and some momen s < k> o C2+ a omic ion o di e en choices o he co ela ion unc ion pa ame e aand he scaling cons an λcalcula ed in his wo k and compa ison wi h HF and co ela ed li e a u e da a. Da a a e in a omic uni s. ampli ude as hey esul om a i ing o he same accu a e unc ion. The comple e de ini ion o ˜χ0depends a his poin by he pa ame e aen e ing he co ela ion unc ion g( ) and he scaling ac o λ. We made di e en choices o such pa ame e s o all he h ee cases and he inal esul s a e collec ed in Tabs. 2,3,4. The main p oblem encoun e ed in he calcula ion o he o al ene gy by Mon e Ca lo me hod has been ela ed o he high a iance o he unc ion o be a e aged 16 a λ E Tp< −2> < −1> < > < 2> < 3> 7 1.00005 -68.40(1) 6.91(1) 49.11 10.712 5.492 6.534 9.280 8 1.00003 -68.40(1) 6.91(1) 49.45 10.741 5.486 6.522 9.260 9 1.00003 -68.41(1) 6.90(1) 49.73 10.763 5.481 6.513 9.244 HF(a)-68.257 7.374 51.8 10.887 5.455 6.469 9.167 co (b)-68.411 — 49.17 10.694 5.570 6.769 9.843 (a)In he Tpcolumn is epo ed he di e ence Tex −T(1) wand momen s a e om e s. [16,18]. (b)Momen s om [16]. Table 4: To al ene gy (E), Pauli kine ic ene gy (Tp) and some momen s < k> o O4+ a omic ion o di e en choices o he co ela ion unc ion pa ame e aand he scaling cons an λcalcula ed in his wo k and compa ison wi h HF and co ela ed li e a u e da a. Da a a e in a omic uni s. which is de ined in Eq. (59). This equi es a long simula ion o achie e an ene gy mean alue wi h an accu acy o he o de o some mHa ee. The same occu s o Tp. Looking a he esul s o Tabs. 2,3,4, i is e iden ha he op imal alues o he pa ame e s aand λmus be ound by sea ching o a comp omise be ween he need o ge ing eliable alues o he momen s < k>and he bes ene gy. The esul s show also ha a conside able ac ion o co ela ion ene gy has been aken in o accoun . I is also impo an o no ice ha he a iance becomes la ge when he nuclea cha ge inc eases bu also ha ou app oxima ion, mainly based on sho ange co ela ion, should wo ks be e in such cases. I is also in e es ing o look a he alues o Tp. F om he bounds on he gene alized Weizs¨acke - ype kine ic ene gy in oduced by Aye s [15] i ollows ha 0≤Tp≤Tex −T(1) w.(63) Looking a ou esul s, his inequali y is sa is ied in he ange o a alues consid- 17 e ed he e. De ia ions om his beha io mus lead o conside a ions ela ed o N- ep esen abili y. Finally, i is wo hwhile o look a he plo s o he e ec i e po en ial de i ed by he app oxima e pai densi y ampli ude. This has been done o he con ibu ions V1( 1, 2) and V2( ) while V3( 1, 2, ) canno be easily shown being dependen on h ee independen a iables. Fo his pu pose, and only o Be, we plo V1( 1, 2) in Fig.1 and V2( ) in Fig.2. F om Fig.1, i is clea ha V1is domina ed by he ex e nal nuclea po en ial when 1o 2 ends o 0 while is abou cons an o bo h la ge 1 and 2, being −µ he limi in his case. The ipples o he wo dimensional su ace o Fig.1 a e ins ead a consequence o he exchange in e ac ion and de e mine he shell s uc u e o he one pa icle densi y o Be a om. Finally, V2, shown in Fig.2, is always epulsi e. Fo small , i beha es as he elec on-elec on in e ac ion po en ial while i goes o ze o mo e apidly o la ge . The ipples o V1, he long ange beha io o V2and he con ibu ion V3a e special ea u es o P. 5 Conclusions In his wo k, we ha e illus a ed a me hod o econs uc he Pauli po en ial o pai densi y unc ional heo y o ou elec on a omic ions. The po en ial is de i ed by in e ing he e ec i e wo elec on equa ion in ol ing he pai densi y ampli ude assuming ha he pai densi y i sel can be w i en in an analy ical ac able o m. Cusp and asymp o ic condi ions ha e been sa is ied and app op ia e adjus able pa- ame e s ha e been used in o de o ep oduce, wi hin a easonable accu acy, he o al ene gy and some lowe momen o he in acule densi y. Some in e es ing ea- 18 u es o he Pauli po en ial ha e been ound o he sys ems ea ed he e. These ea u es a e con ained in he exp essions (40), (44) and (45) o V1,V2and V3.V1and V2include also he nuclea and he elec on-elec on elec os a ic po en ial ene gies. Some illus a ions a e gi en also in Figu es 1 and 2. We would like o emphasize ha he p esen me hod is no es ic ed o ou - elec on sy ems. In he he pai densi y heo y one has o sol e an e ec i e wo elec on equa ion independen ly on he numbe o elec ons. Tha is, he no el me hod in oduced he e o in e he e ec i e wo elec on equa ion can always be applied i he he pai densi y (o he he pai densi y ampli ude) is a ailable. Fo he u u e, i will be in e es ing o analyze in de ails each indi idual e m in o de o ind a gene aliza ion o he abo e exp essions o all polyelec onic sys ems in a o m which does no equi e he in e sion o he e ec i e wo elec on equa ion wo ked he e. Abou V2, we would like o e e b ie ly o he ’a e age-pai -densi y heo y’ o Go i-Gio gi and Sa in. In his heo y he sphe ically and sys em-a e aged pai densi y ( ) is de e mined by simple adial equa ions conjec u ed by Go i-Gio gi and Sa in [21]: h−∇2 +we ( )iφi( ) = iφi( ),(64) he solu ions o which gi e ( ) as X i θi|φi( )|2= ( ),(65) ha is, ( ) is gi en by a weigh ed sum o he squa e o some o hogonal ’e ec i e’ geminals φiwi h weigh ing ac o s o ’occupancy’ θi. The po en ial we ( ) in Eq. 19 (64) was app oxima ed as we ( ) = w(0) e ( ) + wc e ( ),(66) whe e w(0) e ( ) = ∇2 1/2 KS 1/2 KS (67) and wc e ( ) = 1 + 2 2 3 s −3 2 s!θ( s− ).(68) θ( s− ) is he Hea iside s ep unc ion and s=4π 3%−1/3 ,(69) whe e %is he a e age elec on densi y. The co ela ion po en ial wc e ( ) o iginally p oposed by O e hause [22], has been used o sol e Eq. (64) o he uni o m elec on gas [21, 23]. I leads o an accu a e desc ip ion o he sho - ange pa o . Ou po en ial V2( 1, 2) (44), using he exp ession (51) o g, has he o m V2( 1, 2) = 1 (1 + a )2(1 + (a+ 1/2) ).(70) We immedia elly no ice ha he dominan e m in (70) o small is 1/ . I is he same as he i s e m in he O e hause po en ial, which is also he dominan pa o he O e hause po en ial o small . Thus he po en ial V2( 1, 2) has some esemblance o he O e hause po en ial. The 1/ e m in he O e hause po en ial comes om he cusp condi ion on ( ) [21]. The dominan e m in (70) o small has he same o igin. We also men ion in passing ha i was de i ed ia a double adiaba ic connec- ion by one o he p esen au ho s [4] ha he squa e oo o he sphe ically and 20 sys em a e aged pai densi y is he solu ion o a simple adial equa ion, ha is, con- a y o he heo y o Go i-Gio gi and Sa in, i is possible o ob ain ( ) h ough a solu ion o a single equa ion. I a single geminal is used he e mionic cha ac e should be e lec ed in he po en ial which is consequen ly mo e complica ed. I mo e han one geminals a e used he sphe ically and sys em a e aged pai densi y has a mo e complica ed o m bu he po en ial can be mo e easily app oxima ed. The numbe o geminals N(N−1)/2 depends on he numbe o elec ons. The e o e a single geminal app oach migh gain an impo an ole as he numbe o elec ons inc eases. In he densi y unc ional heo y he e has been a g owing in e es in de e - mining he exac exchange, exchange-co ela ion and Kohn-Sham po en ials in he knowledge o he densi y. Se e al me hods ha e been wo ked ou [24–29]. The ex- ac po en ials a e e y use ul, o example o check he accu acy o app oxima e me hods. An analogous p oblem in he pai densi y unc ional heo y is o ob ain he Pauli po en ial in he knowledge o he pai densi y as he e he elec on-elec on i e ac ion is exac ly ea ed, bu he kine ic ene gy unc ional is unknown. The p ob- lem he e is mo e complica ed in he sense ha a wo-pa icle po en ial Pshould be calcula ed, ins ead o a one-body exchange-co ela ion po en ial o he densi y unc- ional heo y. On he o he hand, i is also simple as only a single equa ion has o be in e ed ins ead o se e al Kohn-Sham egua ions in he densi y unc ional heo y. The accu a e o m o he Pauli po en ial ob ained by he p esen me hod can be used la e o ind app oxima e exp essions o i . One has o be, howe e , ex emely ca e ul in he cons uc ion because o he N- ep esen abili y p oblem [11,15,30–43]. Dal Ri e al. [19] de i ed densi y ma ices om Jas ow- ype ial wa e unc- 21 ions. The pai densi y used in his wo k can be conside ed as he lowes o de app oxima ion o he gene al, N- ep esen able pai densi y p esen ed by Dal Ri e al. Consequen ly, ou pai densi y is, a leas app oxima ely , N- ep esen able. Re e ences [1] ´ A. Nagy, Phys. Re . A 66, 022505 (2002). [2] ´ A. 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