Bachelor-Thess Many-Body Perturbaton Theory for the Nuclear Many-Body Problem Tll Tobas Böhlen Supervsor: Prof. Dr. Robert Roth Insttut für Kernphysk Technsche Unverstät Darmstadt Schlossgartenstr. 9, D-64289 Darmstadt, Germany Emal: DasBoeT@gmal.com August October 2006 Workng wthn the framework of the Untary Correlaton Operator Method (UCOM), ths work presents the dervaton of an exact-denomnator verson of the second-order many-body perturbaton theory correcton to the Hartree-Fock ground state energy. In addton, correctons to the Hartree- Fock sngle-partcle energes are nvestgated. 1
2
Contents Contents 1. Introducton 4 2. Calculatng Ground-State Energes wth UCOM-HF 4 2.1. Untary Correlaton Operator Method (UCOM)............... 5 2.2. Correlated Interacton V UCOM........................ 6 2.3. UCOM appled to a Hatree-Fock (HF) Scheme............... 9 3. Many-Body Perturbaton Theory (MBPT) 10 3.1. Raylegh-Schrödnger (RS) MBPT...................... 10 3.2. Brlloun-Wgner (BW) MBPT........................ 13 4. Improved MBPT Calculatons 14 4.1. Exact Denomnator (ED) of Second-Order MBPT............. 14 4.2. Calculatons done wth ED.......................... 15 4.3. Evaluaton and Interpretaton of the Results................. 15 5. Correctons for Sngle-Partcle (SP) Energes 19 5.1. Estmatng Sngle-Partcle Energes...................... 19 5.2. Calculatons and Results of SP Energy Correctons............. 22 6. Outlook 22 A. Slater rules 24 B. Bref Descrpton of BW MBPT 26 C. Implementaton-specfc Remarks 27 C.1. Changes to nsute................................ 27 C.2. Lst of Modfed (M) and Newly Created (C) Fles............. 27 D. Addtonal Graphs 28 3
1. Introducton 1. Introducton The descrpton of ground-state propertes of heaver nucle based on realstc nucleonnucleon (NN) nteractons, reproducng the expermental NN scatterng data, s stll an unsolved problem n the theory of nuclear structure. In state-of-the-art calculatons lght nucle can be descrbed n ab nto 1 schemes lke the no-core shell model or Green s functon Monte Carlo. However, these methods are currently computatonally too expensve to be appled to heaver nucle. Approxmatng many-body states n managable model spaces of smple antsymmetrc Slater determnants one encounters two propertes of realstc NN nteractons whch lead to dffcultes. Those are () a strong short-ranged (< 0.5 fm) repulson and () a strong tensor force. These nteracton components cause hgh momenta whch are assocated wth states not ncluded n a space spanned by a small number of Slater determnants. Huge Hlbert spaces would be needed to descrbe such states to a reasonable degree. We perform a untary transformaton of the bare realstc NN nteracton to make the many-body problem tractable wth model spaces and methods whch can be employed over the whole nuclear chart. Ths untary transformaton explctly descrbes the shortrange central and tensor correlatons nduced by the NN nteracton, and s the key dea of the Untary Correlaton Operator Method (UCOM). In our calculatons we start wth a Hamltonan transformed by UCOM. Frst, a standard Hartree-Fock (HF) scheme s appled. Further, we apply many-body perturbaton theory (MBPT) to the HF soluton n order to ntroduce resdual long-range correlatons. A summary of the essental steps s gven n Sec. 2 and 3. The am of ths work s to mprove the MBPT calculatons. To smplfy the formula of the MBPT correcton of the energy, an approxmaton was made n prevous calculatons: the second-order term was calculated wth a smplfed denomnator. In Sec. 4 we undo ths step and calculate a more precse energy correcton of MBPT. Ths secton also ncludes a dscusson of the devaton of the obtaned results from former ones. Furthermore, we employ MBPT as a tool to calculate correctons to the sngle-partcle (SP) energes n Sec. 5. 2. Calculatng Ground-State Energes wth UCOM-HF Snce bare realstc potentals, lke the Argonne V18 (AV18) potental, are unsutable for smple many-body model spaces that sngle Slater determnants provde, they must be modfed to yeld reasonable results. The Untary Correlated Operator Method (UCOM) provdes a novel approach to descrbe short-range central and tensor correlatons explctly by a untary transformaton. Snce the correlaton operators are gven explctly, UCOM allows for the dervaton of a bass-ndependent effectve nteracton operator V UCOM. To compute ground-state energes of nucle, the HF scheme s employed. In the present treatment of the nuclear many-body problem, we restrct ourselves to two-body nteractons, not consderng hgher orders of the cluster expanson of corre- 1 I.e., wthout ntroducng approxmatons to the nuclear many-body problem wth realstc nteractons. 4
2. Calculatng Ground-State Energes wth UCOM-HF lated operators (cf. Sec. 2.2). Ths approxmaton seems reasonable supposng that the range of the correlatons s short compared to the mean partcle dstances. A bref dscusson of UCOM s presented n ths secton. For more detals refer to [1, 2, 3, 4]. 2.1. Untary Correlaton Operator Method (UCOM) For computatonal reasons many-body states are preferentally approxmated by sngle antsymmetrc 2 Slater determnants Ψ a = A ( α 1 α 2... α A ), (2.1) where α = ϕ χ τ s a sngle-partcle (SP) state wth spatal component ϕ, spn component χ and sospn component τ. The UCOM transforms the crtcal central and tensor parts of the realstc nteracton by a correlaton operator C, conservng nherent operator symmetres and the NN phase shfts. Ths correlaton operator C can be wrtten as the product of two ndependent untary operators descrbng the tensor and the central correlatons, respectvely: C = C Ω C r = exp <j g Ω,j exp <j g r,j. (2.2) The hermtan two-body generators g r and g Ω are gven by g r = S,T 1 2 [s ST (r)q r + q r s ST (r)] Π ST (2.3) g Ω = T ϑ T (r)s 12 (r, q Ω )Π 1T, (2.4) where relatve momentum q, relatve dstance operator r, radal momentum q r, orbtal momentum q Ω, and s 12 (r, q Ω ) are defned by the followng relatons: q = 1 2 [p 1 p 2 ], r = x 1 x 2, q r = 1 [ r 2 r q + q r ] r (2.5) q Ω = q r r q r = 1 [l r r l] 2r2 (2.6) s 12 (r, q Ω ) = 3 2 [(σ 1 q Ω )(σ 2 r) (σ 2 r)(σ 1 q Ω )]. (2.7) x and p denote the partcle poston and momentum, l s the orbtal angular momentum, and σ are the spn operators. 2 The ndex a for antsymmetrc states s omtted n the followng for the sake of brevty. All states are assumed to be antsymmetrc. 5
2. Calculatng Ground-State Energes wth UCOM-HF The central correlator C r shfts a par of partcles apart by a dstance-dependent amount n the radal drecton, untl they are out of the range of the repulsve core. s ST (r) s the shft as a functon of the nter-partcle dstance for each spn-sospn channel. The tensor correlator C Ω does not act n the S = 0 channels. The mpact of the tensor correlator C Ω n the S = 1 channels s a perpendcular shft to the relatve orentaton r. ϑ T (r) descrbes strength and range of the tensor correlatons. It s only a functon of the nter-partcle dstance. Both s ST (r) and ϑ T (r) vansh for large dstances, and the correlators reduce to dentty operators. The optmal correlaton functons are determned varatonally for each (S,T)-channel by mnmzng the energy of the two-nucleon system. The correlaton volume I ϑ = dr r 2 ϑ(r) (2.8) mposes an addtonal constrant on the range of the tensor correlaton. More detals can be obtaned from [4] and [5]. As an example, Fg. 1 panel (a) depcts a radal two-body wave functon as t typcally results from a sngle Slater determnant. Note the hgh, unrealstc probablty at small nterpartcle dstances (r < 1 fm). Ths flaw s elmnated by applcaton of the central correlator 3 c r to the state, n panel (c). Panel (e) shows the consequences of the applcaton of the tensor correlator c Ω to a radal s-wave two-body wave functon. In panels (b), and (d) of Fg. 1 the central and tensor correlaton functons are dsplayed schematcally. Fg. 2 vsualzes how the troublesome hgher off-dagonal matrx elements n relatve momentum-space (of the form q(ls)jt vucom q (L S)JT, where q s the relatve two-body momentum) are moved closer to the zero-plane by the untary transformaton. The low-momentum matrx elements become strongly attractve. The ncluson of the tensor correlator does not change the matrx elements of the spn-snglet channel, but suppresses off-dagonal matrx elements n the spn-trplet channel. Thus, the untary transformatons predagonalzes the nteracton, decouplng low and hgh momenta. 2.2. Correlated Interacton V UCOM Now we wll apply the correlaton operator C to an uncorrelated Hamltonan and calculate correlated matrx elements of the effectve correlated potental V UCOM. To dstngush correlated and uncorrelated operators and states, correlated operators and states are denoted by a tlde,.e. Ψ = C Ψ for a correlated many-body state and à = C AC for a correlated operator. A transformaton of an arbtrary operator A leads to a correlated operator wth contrbutons to all partcle numbers. We formulate a cluster expanson [1] of the correlated operator à = Ã[1] + Ã[2] + Ã[3] +..., (2.9) 3 Small letters denote two-body operators. 6
2. Calculatng Ground-State Energes wth UCOM-HF [arb. unts] 0.15 0.1 0.05 (a) r φ 0 L=0 0.2 (b). 0 0.15 [arb. unts] 0.1 0.05. 0 0.15 [arb. unts] 0.1 0.05 (c) (e) r c r φ 0 L=0 r c Ωc r φ 0 L=0 L=2 [fm]. 0.15 0.1 0.05 0 0.08 0.06 0.04 0.02. R +(r) r ϑ(r) (d) 0 0 1 2 3 4 5 r [fm]. 0 0 1 2 3 4 5 r [fm] Fgure 1: Applcaton of the central and tensor correlators resultng n a transformed s-wave two-body tral state, as onbtaned from the Argonne V18 potental. Panels (a), (c) and (e) depct the uncorrelated, the central correlated, and the central and tensor correlated radal wave functon, respectvely. The panels (b) and (d) show the correspondng central and tensor correlaton functons, where R ± (r) r ± s(r) for slowly varyng shft functons s(r). I ϑ = 0.09 fm 3. where the rreducble n-body contrbutons are denoted by Ã[n]. When startng wth a k-body operator rreducble contrbutons wth n < k vansh. Consder an uncorrelated Hamltonan H consstng of an one-body knetc energy T = p2 /(2m N) and a two-body nteracton V = <j v j. As long as the range of the correlaton functons s(r) and ϑ(r) s small compared to mean partcle dstances, the hgher partcle number contrbutons of the cluster expanson of the correlated Hamltonan H are neglgble. Ths leads us to the followng two-body approxmaton 4 : H C2 = T [1] + T [2] + Ṽ[2] T + V UCOM, (2.10) where T [1] turns out to be the smple uncorrelated knetc energy. The two-body terms are subsumed n a correlated effectve nteracton V UCOM = <j v UCOM,j. V UCOM can be consdered a realstc potental on ts own. The fact that an explct form of the correlated operator exsts makes t applcable n methods whch do not allow the use of partal wave matrx elements. To apply the correlated nteracton V UCOM n a HF scheme formulated n the sphercal harmonc-oscllator bass we have to calculate ts two-body matrx elements. We employ jj-coupled harmonc-oscllator states n 1 l 1 j 1, n 2 l 2 j 2 ; JT for ths purpose. It turns out 4 The two-body approxmaton s denoted by the superscrpt C2. 7
2. Calculatng Ground-State Energes wth UCOM-HF central correlatons central & tensor correlatons 3 S 1 3 D1 3 S1 1 S0 bare Fgure 2: Relatve momentum-space matrx elements q(ls)jt vucom q 0 (L0 S)JT of the bare AV18 potental (left-hand column), the central-correlated AV18 potental (center column), and the fully correlated AV18 potental. The dfferent rows correspond to dfferent partal waves, as ndcated on the left-hand sde. The red dots mark the plane of vanshng matrx elements. The momenta are gven n fm 1 and the matrx elements n MeV. that t s computatonally more effectve to map the tensor correlator CΩ back onto the two-body states whereas the central correlator Cr acts onto the operators. Skppng the extensve dervaton [5, 7, 9], the non-normalzed antsymmetrc jj-coupled matrx elements read as follows: 8
2. Calculatng Ground-State Energes wth UCOM-HF n1 l 1 j 1, n 2 l 2 j 2 ; JT v UCOM n 1l 1j 1, n 2l 2j 2; JT = = [j 1 ] [j 2 ] [j 1 ] [j 2 ] l 1 l 2 L 1 1 2 2 S L,L,S N,Λ ν,λ ν,λ j j 1 j 2 J l 1 l 2 L { } { } 1 1 Λ λ L Λ λ L 2 2 S j 1 j 2 J S J j S J j NΛ, νλ n 1 l 1, n 2 l 2 ; L NΛ, ν λ n 1l 1, n 2l 2; L [j] [S] [L] [ L ] ( 1) L+L { 1 ( 1) λ+s+t } ν(λs)jt vucom ν (λ S)jT. (2.11) n, n = 0, 1, 2,... are the radal oscllator quantum numbers, [j] 2j + 1, and {...} are the 6j and 9j symbols. Moreover,...... are the harmonc oscllator brackets [12, 13]. 2.3. UCOM appled to a Hatree-Fock (HF) Scheme The standard HF scheme s a varatonal approach to the nuclear many-body problem. It assumes an A-body state as n Eq. 2.1. The α are the varatonal degrees of freedom used to mnmze the expectaton value of the Hamltonan. The varatonal soluton of the many-body states then leads to the HF equatons [11]. The Hamltonan conssts n our case of the effectve realstc potental V UCOM and the ntrnsc knetc energy T nt : H nt = T nt + V UCOM. (2.12) Center-of-mass contrbutons are excluded from the energy,.e. the center-of-mass knetc energy T cm s subtracted from the knetc energy to yeld the ntrnsc knetc energy: T nt = T T cm = 2 1 A q 2 j. (2.13) A It depends only on the relatve two-body momentum q j. Ths leads to a Hamltonan H nt whch s Galle-nvarant and a quas two-body operator. The effect of center-ofmass exctatons to T nt s not excluded, however, t s expected to yeld only a small contrbuton. The jj-coupled Hartree-Fock SP states a expressed n the harmonc oscllator bass read a = ν(l 1 2 )jm; 1 2 m t νljmmt = n m N <j C n (νljmmt) nljmmt, (2.14) 9
3. Many-Body Perturbaton Theory (MBPT) E/A [MeV]. 0-2 -4-6 -8 He4 O16 O24 S34 Ca40 Ca48 N48 N56 N78 Sr88 Zr90 Fgure 3: Bndng energes resultng from a HF calculaton wth a correlated AV18 (I ϑ = 0.09 fm 3 ), ( ) n reference to the expermental values ( ). HF calculatons were done for e Max = 12, l Max = 10. wth radal harmonc-oscllator quantum number n, orbtal angular momentum l, total angular momentum j wth projecton m, and sospn projecton m t. The regular teratve HF scheme enables us to calculate the HF energes [4]. To do so we use a truncated harmonc oscllator bass wth e = 2n + l e Max and optonally l l Max. Usng ths HF framework wth a correlated AV18 potental we obtan the bndng energes shown n Fg. 3. Although a szable (nearly constant!) dfference to expermental results remans, we are able to obtan bound states, whch would be mpossble wth a bare nteracton. The prmal cause of the devaton to expermental results are the bndng energy contrbutons from long-range correlatons, whch are not covered by the UCOM-HF scheme. Assessng these neglected correlatons s the subject of the next secton. 3. Many-Body Perturbaton Theory (MBPT) To account for the effect of the resdual long-range correlatons the HF soluton s corrected by MBPT [10]. More precsely, we add the frst non-vanshng order of MBPT, whch s the second-order, to the HF energes. 3.1. Raylegh-Schrödnger (RS) MBPT Raylegh-Schrödnger (RS) MBPT s the smplest and most frequently used form of MBPT. The orgnal Hamltonan H s dvded nto an unperturbed Hamltonan H 0 and a perturbatve Hamltonan W, related by H = H 0 + λw, wth 0 λ 1. λ s a formal expanson parameter and wll be set later to unty. Next, the exact egenfunctons and egenvalues of the correspondng Schrödnger equaton are expanded n a seres n λ, E = E (0) + λe (1) + λ 2 E (2) +... (3.1) Φ = Φ (0) +λ Φ (1) +λ 2 (2) Φ +.... (3.2) 10
3. Many-Body Perturbaton Theory (MBPT) We assume the unperturbed wavefuncton to be normalzed, Φ (0) (0) Φ = 1, and choose a benefcal ntermedate normalzaton, Φ Φ (0) = 1, for Φ. In consequence, we fnd Φ Φ (n) = 0 for n = 1, 2, 3,.... Eqs. 3.1 and 3.2 are related by the Schrödnger equaton. Comparng λ-coeffcents one easly obtans followng set of relatons: H 0 Φ (0) (0) = E Φ (0) (3.3) H 0 Φ (1) +W Φ (0) (0) = E (1) (0) Φ +E (0) Φ (3.4) H 0 Φ (2) +W Φ (1) (0) = E (2) (1) Φ +E (1) (2) Φ +E (0) Φ (3.5) H 0 Φ (3) +W Φ (2) (0) = E (3) (1) Φ +E (2) Φ + + E (2) (1) (3) Φ +E (0) Φ. (3.6) Upon combnng these equatons they lead to the followng expressons for the nth-order energes: E (0) = Φ (0) H 0 Φ (0) (3.7) E (1) = Φ (0) W Φ (0) (3.8) E (2) = Φ (0) W Φ (1) (3.9) E (3) = Φ (0) W Φ (2) (3.10) E (4) =... Solvng agan the set of Eqs. 3.3 3.6, ths tme for (k) Φ one can determne the kthorder energy wth Eqs. 3.7 3.10. We wrte (1) Φ nsertng the unty operator n form of the H 0 egenstates n Φ (0) n : (1) Φ = n nφ (1) = n nφ (1), (3.11) n where the last representaton s obtaned due to the ntermedate normalzaton. n (1) Φ s zero for = n. Multplyng Eq. 3.4 by n, one fnds (E (0) E n (0) ) n (1) Φ = n W Φ (0), (3.12) consderng that the zeroth-order wave functons are orthogonal. Now we substtute the expanson Eq. 3.11 nto Eq. 3.9: E (2) = Φ (0) W Φ (1) (0) = Φ W n n Φ (1), (3.13) n and fnally, upon combnng ths equaton wth Eq. 3.12 the expresson for the secondorder energy W n 2 E (2) = n n E (0) E n (0) (3.14) 11
3. Many-Body Perturbaton Theory (MBPT) can be wrtten n terms of the solutons of the ntal unperturbed egenproblem. Lkewse, expressons for hgher order energes and states can be obtaned. It s also possble to calculate hgher order energes teratvely. In our case the egenbass n conssts of the egenstates of the HF soluton. The unperturbed Hamltonan s assocated wth the dagonal matrx elements of H nt. In second quantzaton ts two-body part reads H 0 = αβ Hnt αβ a α a β a β a α, (3.15) αβ where a s the creaton operator and a the extncton operator. The perturbaton W conssts of the off-dagonal contrbutons of the correlated Hamltonan αβ H nt δγ wth α δ, β γ, α γ, or β δ. To descrbe excted Slater determnants, we ntroduce the followng conventons: the latn letters a, b, c,... are used to denote occuped one-partcle states below the Ferm energy ɛ F (hole states). The letters r, s, t,... mark states above Ferm energy ɛ F (partcle states). Further, latn subscrpts followng a many-body state or energy denote removed partcles, whereas superscrpts ndcate newly occuped states. For nstance, the manybody state Ψ rs ab denotes a state where orgnally all levels beneath the Ferm energy were occuped. The ndces followng Ψ tell us that the partcles of the levels a, b were removed whereas the levels r, s were newly occuped. Smlarly, Eab rs s shorthand for Ψ rs Hnt Ψ rs ab ab. Orgnally, the sum n Eq. 3.14 runs over all possble exctatons n. Yet, we can exclude certan excatons types when applyng t to our framework. Sngle-exctaton HF Slater determnants are absent n the sum because of Brlloun s theorem 5, whle trple and hgher order determnants do not appear because the Hamltonan contans at most two-partcle nteractons. Hence, the second-order perturbaton theory deals wth a correcton nduced by the subspace of 2p2h-states 6 ( Ψ rs ab ). We fnd for the ground state energy correcton followng Eq. 3.14: E (2) 0 E (2) = a b<a r s<r Ψ 0 Hnt Ψ rs ab E (0) E rs ab 2. (3.16) Usng the Slater rules (App. A) we rewrte the numerator, E (2) = 1 ab H nt rs 2 4 E (0) Eab rs. (3.17) a,b r,s To make ths approach computatonally less demandng an addtonal approxmaton was used n the prevous calculatons [4]: E (0) Eab rs = Ψ rs ab E (0) H nt Ψ rs = ab ɛa + ɛ b ɛ r ɛ s, (3.18) 5 Brlloun s theorem states that the HF equatons can be rewrtten as a decouplng condton between ground-state and one-partcle-one-hole (1p1h) exctatons. 6 Shorthand for two-partce-two-hole-states. 12
E/A [MeV]. -6-6.5-7 -7.5-8 -8.5-9 -9.5 3. Many-Body Perturbaton Theory (MBPT) He4 O16 O24 S34 Ca40 Ca48 N48 N56 N78 Sr88 Zr90 Fgure 4: Bndng energes resultng from HF+MBPT calculatons ( ), where UCOM was appled to the Hamltonan ntally, n reference to the expermental values ( ). HF calculatons were done wth e Max = 12, l Max = 10 (AV 18, I ϑ = 0.09 fm 3 ). where ɛ x are the HF SP energes of the correspondng states. Employng the expressons gven above, the ground state energes for selected closedshell nucle were calculated n Ref. [4]. These calculatons use a correlated AV18 potental, a correlaton volume I ϑ = 0.09 fm and 13 major oscllator shells (e Max = 12, l Max = 10). It s found that the resultng energes reproduce the expermental data to a good extent throughout the nuclear chart from 4 He to 208 Pb and even far off the valley of β-stablty (Fg. 4). In order to estmate hgher order perturbatve effects, the thrd order was also calculated but yelded only a small devaton from second-order results, ndcatng a possble onset of convergence of the perturbaton seres. The suprsngly good results for the bndng energes n two-body approxmaton are accredted to a net cancellaton of three-body effects [14]. However, theoretcal results obtaned from no-core shell model 7 calculatons ndcate that some addtonal bndng energy can stll be ganed from long-range correlatons. 3.2. Brlloun-Wgner (BW) MBPT Another varant of MBPT s the Brlloun-Wgner (BW) scheme. Ths method yelds often better results, at the same order of MBPT than Raylegh-Schröder (RS) MBPT. Moreover, t s often an easy way to avod the somewhat demandng case of degenerated RS MBPT, gven that the degeneraces are resolved n the aspred order of MBPT. Contrary to the RS MBPT, the BW MBPT already uses the fnal energy n the defnton of the energy contrbuton and so requres an teratve scheme for the calculaton. For a compact descrpton, see App. B. 7 The no-core shell model provdes the exact soluton of the many-body problem for a gven nteracton,.e. long-range correlatons are fully ncluded (Ref. [5]). 13
4. Improved MBPT Calculatons 4. Improved MBPT Calculatons In the last secton MBPT was used to assess the mpact of resdual long-ranged correlatons not treated by the HF method. To be more accurate n our calculaton, we wll undo the approxmaton made n Eq. 3.18 and calculate MBPT wth an exact denomnator (ED). Afterwards we evaluate the changes n the obtaned results. BW MBPT s not further consdered. Frst, as t s computatonally very demandng. (It requres an teratve scheme, and net energy cancellatons, whch reduce the denomnator-sum n RS MBPT extremely, do not occur n the BW varant.) Furtheron, the devaton from results obtaned from RS MBPT calculatons s expected to be small. As the followng methods were newly mplemented we wll address possble complcatons n the course of the dscusson, and wll also make some programmcode-specfc comments. 4.1. Exact Denomnator (ED) of Second-Order MBPT Let us go back to Eq. 3.17 and manpulate the denomnator: E (0) Eab rs = Ψ 0 Hnt Ψ0 Ψ rs ab Hnt Ψ rs ab. (4.1) Applyng the Slater rules and consderng the cancellaton of matrx elements not ncludng ether a,b or r,s one obtans: Ψ0 Hnt Ψ0 Ψ rs ab Hnt Ψ rs ab = = 1 de H 2 nt de de H nt de = d,e d,e d a ad Hnt ad + bd Hnt bd (4.2) = d rd d r Hnt rd d sd Hnt sd, d where prme ndcates that n the summaton over the occuped SP states the followng replacement s made: a r and b s. Ths smplfed denomnator s then used for further calculatons. Frst, the ED MBPT was mplemented n the standard calculaton method (HFM) whch conssts n addng explctly the energy contrbutons of all SP levels. It turned out that calculatons done wth ths method were extremely tme-consumng and only the ground state energes of lghter nucle could be obtaned n a reasonable tme. To make the routne more effcent, we restrcted the calculatons to closed-shell nucle, benefttng from averagng over the contrbutons due to dfferent m-values (HFC) and thereby reducng the sums n Eq. 3.17. 8 For closed shells the HFC method s exact 8 The calculaton tme for 90 Zr wth e Max = 12, l Max = 10 s 10 d. d 14
4. Improved MBPT Calculatons Nucleus 4 He 16 O 24 O 34 S 40 Ca 48 Ca 48 N 56 N 78 N 88 Sr 90 Zr a HO (fm) 1.3 1.5 1.7 1.6 1.6 1.7 1.7 1.7 1.8 1.8 1.8 Table 1: Optmal Oscllator Length for selected closed-shell Nucle. Nucleus 4 He 16 O 24 O 34 S 40 Ca 48 Ca 48 N 56 N 78 N 88 Sr 90 Zr E(MeV) 4.24 5.84 2.75 5.69 6.08 3.26 2.43 3.69 4.72 5.08 5.07 Table 2: Devaton of the results of ED MBPT from AD MBPT for selected closed-shell nucle, calculated wth HFC and e Max = 12, l Max = 10. wthn the framework of HF calculatons, but t consttutes an approxmaton for the energy contrbutons from shells contanng the two-hole states and the shells contanng the two excted nucleons above Ferm energy. 4.2. Calculatons done wth ED The calculatons made wth the exact-denomnator (ED) method were restrcted to some selected representatve closed-shell nucle used n prevous calculatons: 4 He, 16 O, 24 O, 34 S, 40 Ca, 48 Ca, 48 N, 56 N, 78 N, 88 Sr, 90 Zr. A correlated AV18 potental and a correlaton volume of I ϑ = 0.09 fm 3 were used for the calculatons n both the HFM and the HFC method. In a frst HF run, the optmal oscllator lengths a HO were determned for each nucleus (cf. Tab. 1). Employng the HFM method, two runs were done: The frst one wth a bass of 11 major oscllator shells (e Max = 10). Ths confguraton allowed us only to calculate some lghter nucle. The second one wth a smaller bass of e Max = 8 and therefore a shorter calculaton tme. Wth the HFC method we performed calculatons for each e Max = 6, 8, 10, and e Max = 12 wth the addtonal restrcton l Max = 10. Smlar calculatons wth the approxmated-denomnator (AD) method were done for comparson. 4.3. Evaluaton and Interpretaton of the Results The newly calculated ED MBPT energy correctons gve mostly a lttle lower bndng energes than the AD MBPT calculatons. Fg. 5 depcts the bndng energes calculated wth the old AD MBPT vs. the ones calculated wth HFC ED MBPT. Fg. 6 also dsplays values for the bndng energes obtaned by treatng the longranged correlatons by means of random phase approxmaton (RPA) [8]. The values of the ED varant are usually closer to the RPA results. The mean devaton between the AD method and the ED method s Ē = 4.4 MeV (cf. Tab. 2). Employng a realstc NN potental for the descrpton of nuclear ground states, sets a theoretcal framework for the calculatons whch does not descrbe perfectly the physcal crcumstances hence, even an exact soluton of the many-body problem wthn ths nteracton mght yeld a devaton from the expermental results. Two bndng energes 15
4. Improved MBPT Calculatons E/A [MeV]. -6-6.5-7 -7.5-8 -8.5-9 -9.5 He4 O16 O24 S34 Ca40 Ca48 N48 N56 N78 Sr88 Zr90 Fgure 5: Bndng energes for representatve closed-shell nucle, resultng from HFscheme+AD MBPT calculatons ( ) vs. HF-scheme+ED MBPT calculatons ( ). Furthermore, expermental values ( ) are dsplayed. HF and MBPT calculatons were done wth the HFC code and e Max = 12 wth restrcton l Max = 10. E/A [MeV]. -6-6.5-7 -7.5-8 -8.5-9 -9.5 He4 O16 O24 S34 Ca40 Ca48 N48 N56 N78 Sr88 Zr90 Fgure 6: Bndng energes for some representatve closed-shell nucle: AD MBPT ( ), ED MBPT ( ), RPA ( ), expermental values ( ), and exact values wthn theoretcal framework ( ) of 4 He and 16 O orgnatng from the no-core shell model. MBPT and HF calculatons were done wth the HFC code and e Max = 12 wth restrcton l Max = 10. 16
4. Improved MBPT Calculatons E/A [MeV]. -5-6 -7-8 -9 He4 O16 O24 S34 Ca40 Ca48 N48 N56 N78 Sr88 Zr90 Fgure 7: Comparson of the two MBPT calculatons methods, HFM ( ) and HFC ( ). MBPT calculatons were done wth e Max = 8. calculated by applyng the no-core shell model wth a realstc NN potental are dsplayed n Fg. 6. These energy values are exact wthn our framework gven by V UCOM. As a result of the method-ntrnsc devaton dscussed above, expermental values 9 can only gve us a messure for the overall qualty of the many-body approxmaton, but they make t dffcult to dfferentate between the qualty of dstnct approxmaton methods wthn the accuracy level reached by these approxmatons. Exact values wthn our theoretcal framework can presently only be calculated for low-mass nucle. Those values hnt at somewhat lower bndng energes than the expermental bndng energes (Ref. [5, 6]). There s a devaton of the ground state energes obtaned from the HFC method to those obtaned from the exact HFM method, but Fg. 7 shows that the approxmaton s qute good. Ths s expected, snce only a few SP energes are affected by the approxmaton, and thus they should contrbute wth a small devaton to the fnal energy. Calculatons of the same type wth a dfferent number of major oscllator shells dffer consderably. The calculatons wth e Max = 12 and l Max = 10 are the most accurate as they employ the largest bases. An nvestgaton of the energy gan by ncludng more shells, shown n Fg. 8, suggests that convergence has not yet been reached and consderable changes may occur f even more shells are ncluded. To estmate the contrbutons orgnatng from the ncluson of more shells, whch s computatonally not practcable, we perform an exponental ft of the form E (0+2) (e Max ) = a exp( b e Max ) + c, (4.3) wth the ft parameters a,b, and c. Some representatve fts are dsplayed n Fg. 9. For e Max we obtan the ground state energes dsplayed n Fg. 8. For nucle up to 56 N the predcted values seem reasonable. For heaver nucle the extrapolaton seems to overestmate the bndng energes. The exponental ft provdes only a rough estmate of convergence behavor, and s probably an unappropate choce for 78 N, 88 Sr and 90 Zr. Bases used to determne the ft parameters are less relable for heaver nucle, and the energy gan for larger orbtal bases mght no longer follow a suffcently unform 9 Expermental values are always marked by bars n the graphs. 17
4. Improved MBPT Calculatons -4 E/A [MeV] -6-8 -10. -12 He4 O16 O24 S34 Ca40 Ca48 N48 N56 N78 Sr88 Zr90 Fgure 8: Bndng energes from HFC ED MBPT calculatons wth e Max = 6, 8, 10, and e Max = 12 wth l Max = 10 (n ths order: ( )( )( )( )). Extrapolated values obtaned from an exponental ft ( ). -5-5 -6 O16-6 Ca40-7 -7-8 -8-9 -9-10 -10-11 -11. -12. -12 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10-5 -5-6 N56-6 Zr90-7 -7-8 -8-9 -9-10 -10-11 -11. -12. -12 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 Fgure 9: Convergence behavor of bndng energes. The graph dsplays the some exemplatory fts ( 16 O, 40 Ca, 56 N, 90 Zr). The data ponts are calculated by HF+MBPT HFC method wth e Max = 6, 8, 10, and e Max = 12 wth addtonal restrcton l Max = 10. E/A [MeV] E/A [MeV] E/A [MeV] E/A [MeV] 18
5. Correctons for Sngle-Partcle (SP) Energes scheme 10. Moreover, t seems that second-order MBPT tends to over-estmate bndng energes [17], and ths effect s lkely compensated by hgher order MBPT contrbutons. In concluson, we fnd that the exact-denomnator (ED) method yelds n general better results. Stll, the correcton to the approxmated-denomnator (AD) method can be consdered a fne tunng compared to the typcal devatons between dfferent approxmaton methods, e.g. RPA. 5. Correctons for Sngle-Partcle (SP) Energes In ths secton correctons to the HF sngle-partcle (SP) energes are calculated. Agan we employ MBPT as a tool to estmate the mpact of long-range correlatons. 5.1. Estmatng Sngle-Partcle Energes To calculate the SP energy of a gven level we take the energy dfferences of two systems, one wth N-partcles and the other wth (N 1)-partcles. We frst suppose a N-partcle system n the many-body state N Ψ 0 wth energy N E. Now we subtract the energy N 1 E c of a (N 1)-partcle system n state N 1 Ψ c, where nucleon c s mssng. The unperturbed Hamltonan s chosen to be the HF Hamltonan and energy correctons up to second order are consdered ( N E N E (0+1+2), N 1 E c N 1 E (0+1+2) c ): ε c = N E (0+1+2) N 1 E (0+1+2) c = = ( N E (0) + N E (1) + N E (2) ) ( N 1 E (0) c + N 1 E (1) c + N 1 E (2) c ) = = N E HF N 1 E (0) c N 1 E (1) c + N E (2) N 1 E (2) c, (5.1) where the HF energy N E HF ncludes the zeroth and the frst order of MBPT energes. We have to stress that we do not do standard MBPT for the (N 1)-partcle system, snce an approxmaton s made n representng the zeroth-order wave functon of the (N 1)-partcle system by the N-partcle HF ground state Slater determnant, removng the partcle of SP level c. In general, the HF SP levels of the N-partcle system are not equal, but merely smlar, to the orbtals of the (N 1)-partcle system. Yet ths smplfctaon leads to a tremendous reducton of the sums later on. Applyng ths ansatz to our many-body Hamltonan the energy contrbutons organzed by perturbaton order are gven by 10 The fttng ponts of the last three nucle le nearly on a straght lne. Thus, a mnmal devaton (esp. of the last one) affects consderably the over-all curvature of the exponental ft. 19
5. Correctons for Sngle-Partcle (SP) Energes N E (0) N 1 E (0) + N E (1) N 1 E (1) = ε c (5.2) N E (2) = N Ψ 0 Hnt N 2 N Ψ N 0 Hnt N Ψ 0 N Hnt N = = 1 N Ψ 0 Hnt N Ψ rs 2 ab 4 a,b,r,s N E (0) N E (0)rs = 1 ab Hnt rs 2 (5.3) 4 Z ab a,b,r,s N 1 Ψ c H nt N 2 N 1 Ψ c H N 1 nt Ψ c N H nt N (5.4) N 1 E (2) = N wth da H nt da + db H nt db Z = d a r,b s d d a dr H nt dr a r,b s d r ds H nt ds. (5.5) ε c denotes the HF SP energy. N runs over all possble exctatons of the N-partcle system or the (N 1)-partcle system respectvely. The latn letters denote occuped and unoccuped states as stated n Sec. 3.1. Eq. 5.3 s just the second order MBPT term as n Sec. 3. The frst-order contrbutons n Eq. 5.2 are the same for the N-partcle and the (N 1)-partcle system as a consequence of Koopman s theorem [10]. Thus to fnd the lowest contrbutng order correctons, we subtract second-order energes of both partcle systems. In preparaton for the subtracton, we splt the second-order contrbutons of the (N 1)-partcle system (Eq. 5.4) by exctaton type: : These nclude all exctatons a r. Despte of Brl- A) Sngle exctatons N 1 Ψca r loun s theorem these exctatons do contrbute to the energy correcton, snce they allow the levels of the (N 1)-partcle system 11 to relax, and therefore make them closer to the optmum SP levels. The resultng contrbuton reads: N 1 E (2) A = N 1 Ψ c H nt N 1 Ψca r 2 1 a,r 2 d,e de H nt de 1 a r, 2 d,e de H nt de = = ac (5.6) Hnt cr 2 d da H nt da a r, d dr H nt dr. a c,r All of the above sum ndces whch cannot be equal c are marked wth a star. 11 These levels are (n the appled approxmaton) the HF levels of the N-partcle system. 20
5. Correctons for Sngle-Partcle (SP) Energes B) Double exctatons of type N 1 Ψcab rs : Include all exctatons a r and b s. Ths s the usual type of exctatons whch occur n MBPT only that a, b c. Thus we fnd N 1 E (2) B = 1 ab Hnt rs 2 4 1 a,b,r,s 2 d,e de Hnt de 1 a r,b s, = 2 d,e de Hnt de = 1 ab (5.7) Hnt rs 2, 4 X a,b,r,s where X Z wth the constrant that the sums do not run over c. C) Double exctatons of type N 1 Ψcab cr : Includes all exctatons wth a c and b r. Ths type of exctatons has to be consdered n the (N 1)-partcle system snce exctatons nto the now unoccuped c-level occur. The sum wrtes wth N 1 E (2) C = 1 2 = 1 2 = 1 2 Y = N 1 Ψ c Hnt N 1 Ψcab cr 2 N 1 Ψ c Hnt N 1 Ψ c N 1 Ψcab cr Hnt Ψ = cr cab ab H nt cr 2 de H nt de = a,b,r 1 a,b,r a,b,r 2 d,e de H nt de 1 2 ab Hnt cr 2 Y, a c,b r, d,e da Hnt da + db Hnt db d a c,b r, d d a a c,b r, dc Hnt dc d r (5.8) dr Hnt dr. (5.9) When we approxmate the denomnators of the above terms, analogous as n Eq. 3.18, by dfferences n HF SP energes, the sums are tremendously reduced by cancellaton and yeld the followng result: ε c = ε c + 1 2 a,b,r ab H nt cr 2 ε c + ε r ε a ε b + 1 2 a,r,s rs H nt ca 2 ε c + ε a ε r ε s (5.10) Note that the summatons over a and b nclude the c-level as t s ntroduced to the fnal sum by the contrbuton of the sngle exctaton terms. Ths result, deduced by the smplest form of MBPT, can also be obtaned by the most elementary approxmaton of the Green s Functon formalsm [10]. 21
6. Outlook 5.2. Calculatons and Results of SP Energy Correctons Some of the closed-shell nucle were calculated wth the AD SP energy correctons gven by Eq. 5.10. The calculatons were mplemented wth the HFM method wth the same settngs as the calculatons n the last secton. Unphyscally large correcton-terms to the HF SP energes resulted by the AD calculatons, mxng the energetcal order of SP levels. Montorng the ndvdual contrbutons to the sums, large contrbutons (> 4 MeV) of only a few terms of the second term of Eq. 5.10 were observed. The large contrbutons usually had a small denomnator (< 0.02 MeV). Ths could hnt at some near-degeneraces evoked by nadequate approxmatons. Comparng the denomnators of frst term and second term of Eq. 5.10 t s easly seen that n the second term the total denomnator energy s at least equal to the Ferm gap. Ths does not hold true for the frst term snce ths denomnator nvolves the dfference of a 2p state wth a 1p1h state. Near-degeneraces can occur. A hybrd verson, a mxture of ED and AD sums, of the energy correctons was nvestgated. 12 The troublesome second term of Eq. 5.10, whch contaned the large sum terms, was substtuted by the ED terms. Nevertheless, the results of ths mplementaton yelded no mprovement to the AD verson. Some of the SP energes stll were extremely large. The double exctatons of type C ( N 1 Ψcab cr ) have bg contrbutons to SP energes. Omttng these exctatons n the calculatons correcton energes of a reasonable range of about 2 to 8 MeV were obtaned for selected closed-shell nucle. It seems that energy correctons to the HF SP energes are not tractable by smple MBPT. 6. Outlook The bndng energes obtaned by a transformed realstc potental by applyng, frst, Hartree-Fock (HF) and then many-body perturbaton theory (MBPT) were mproved n Sec. 4. Wthn the range of devaton of dstnct approxmaton methods ths correcton s small and systematcal errors due to the restrcton to two-body nteractons are expected to yeld greater naccuraces. So far, bndng energes are already n a good agreement wth the experment. In contrast, there s less consstency concernng the charge rad and densty dstrbutons wth current approxmatons. Presently nvestgaton n the nfluence of tree-body nteracton n ths framework s done [14]. Consderng the effects of three-body forces one expects a better consstency of the results, especally for observable lke the charge rad. There s also nvestgaton n advanced HF versons to mprove the descrpton of long-range correlatons. The HF Bogolubov scheme [16] employs a quas-partcle approxmaton, whereas the Brückner scheme [15] unfes HF and MBPT before teratng t n a smlar way to the standard HF teratons. 12 The hybrd method, and not an completely ED method, was chosen to reduce calculaton tme. 22
6. Outlook Regardng the sngle-partcle (SP) energes t seems that more sophstcated approxmaton methods are needed. A promsng ansatz s to treat ths problem wth Green s Functon (GF) theory gong beyond the smplest reducton of the Dyson equaton 13. 13 F. Dyson ntroduced an effectve energy-dependent (self-energy) potental. Moreover, he showed that the exact Green s Functon can be wrtten n an ntegral equaton the so-called Dyson equaton. Ths expresson can be expanded as a perturbaton seres n matrx representaton. Applyng GF theory wth the Dyson equaton results, when consderng only effects up to second order, n Eq. 5.10, whch s n Sec. 5 deduced by MBPT. 23
A. Slater rules A. Slater rules In ths secton we present a set of rules whch allows us to easly evaluate matrx elements Ψ1 O Ψ2 (A.1) wth states consstng of A-partcle Slater determnants 14 Ψ and operators that are a sum of one- or two-partcle operators O 1 = O 2 = A =1 A T =1 j> A V j A V j. <j (A.2) (A.3) Those operators depend on the poston, momentum, spn, and sospn of the nvolved partcles. The rules for evaluatng the matrx elements depend on whether O contans one- or two-body operators and, furthermore, to whch degree the two determnants Ψ1 and Ψ 2 dffer. For one-body operators two nontrval cases exst, for two-body operators there are three. We dstngush the followng cases: 1. Ψ1 and Ψ2 are dentcal: Ψ1 = Ψ 2 Ψ, 2. 3. Ψ1 and Ψ2 dffer by exactly one spn orbtal: Ψ1 Ψ, Ψ2 Ψ a a, Ψ1 and Ψ2 dffer by exactly two spn orbtals: Ψ1 Ψ, Ψ2 Ψ a b a b. When two determnants dffer by more SP states than the order x of the x-body operator, the matrx element s automatcally zero, snce the operator cannot couple those states. To apply the Slater rules, the determnants have to be put n maxmum concdence by approprate permutatons of the SP states 15. We wll present an exemplary devaton of one of the rules followed by a complete summary of the rules (Tab. 3 and 4). An A-nucleon Slater determnant contanng the SP states, j,..., k }{{} A 14 As before, all consdered states are supposed to be antsymmetrc. 15 E.g., by permutatng the SP states of Ψ2 untl all equal SP states appearng n Ψ1 and Ψ2 have parwse the same poston. 24
A. Slater rules s defned as A! { } Ψ = (A!) 1/2 ( 1) pn P n j... k, n=1 (A.4) where A! s the number of possble permutatons of the nucleons. P n s the permutaton operator that generates the nth perturbaton of the many-body state. p n s the number of transpostons requred to obtan ths permutaton. Takng two determnants n maxmum concdence, we have A! A! ΨΨj = (A!) 1 ( 1) p ( 1) pj j P { a b } Pj { a b }. (A.5) The SP states are assumed to form an orthonormal set. Thus the expresson above s only non-zero f the prmed SP states are dentcal to the unprmed ones and two many-body states Ψ and Ψj are orthogonal f they do not contan the dentcal SP states. Snce the SP states are normalzed we fnd Ψ Ψ = 1. Let us now derve explctly case 1 for a one-partcle operator O 1. The matrx elements of T 1, T 2, T 3,... are the same when the partcles are ndstngushable, so A Ψ O1 Ψ = Ψ T Ψ = A Ψ T1 Ψ = = A A! =1 A! A! ( 1) p ( 1) pj j P { a b } T1 P j { a b }, (A.6) where we choose the operator for the frst partcle T 1 by conventon. In case 1 the prmed SP states are the same as the unprmed. For the combnaton of the partcles 2, 3,..., A we obtan zero unless the partcles are n the same level n the th permutaton as n the jth permutaton. Gven that all partcles occupy the same levels the frst partcle also has to be n the same level n both of the permutatons. Thus, the permutatons have to be the same ( = j) to gve a contrbuton: A! } } Ψ O1 Ψ = ((A 1)!) 1 P { a b T1 P { a b. (A.7) The combnaton of the partcles 2, 3, 4,... always yelds 1 snce the SP states are normalzed. In the sum over A! permutatons, partcle 1 occupes each SP state (A 1)! tmes because there are (A 1)! ways to arrange the other partcles among the remanng (A 1) SP states. Ths leads us to Ψ T1 Ψ = a T1 a a T a (A.8) a a 25
B. Bref Descrpton of BW MBPT Case 1: Case 2: Case 3: O 1 = A =1 T Ψ O1 Ψ = a a T a Ψ O1 Ψ r a = a T r Ψ O1 Ψ rs ab = 0 Table 3: Matrx elements between determnants for one-partcle operators. Case 1: Case 2: Case 3: O 2 = A <j V j Ψ O2 Ψ = 1 2 a,b ab V ab Ψ O2 Ψ r a = ab V rb Ψ O2 Ψ rs ab b = ab V rs Table 4: Matrx elements between determnants for two-partcle operators. as case 1 of the Slater rules for an one-partcle operator. B. Bref Descrpton of BW MBPT We ntroduce projecton operators to separate the many-body Hlbert space nto two subspaces: the model space wth projector P and the resdual space wth projector Q wth P + Q = 1 and PQ = QP = 0. We splt the Hamltonan accordng to H = H 0 + W, wth PH 0 P = QH 0 Q = H 0. Our am s to fnd an effectve Hamltonan H eff whch s restrcted to the model space, PH eff P = H eff = PH 0 P + PW eff P, such that H eff P E n = En P E n for at least a few egenvalues. Thus the effectve Hamltonan actng n model space should have the same egenenerges E n as the orgnal Hamltonan H, and egenstates that are projectons of the exact egenstates. The effectve resdual perturbaton W eff can be defned recursvely and depends on the (yet unknown) egenvalue E n : W eff (E n ) = W + WQ(E n H 0 ) 1 QW eff (E n ), (B.1) To employ the BW MBPT for our purposes, we choose H 0 to be the dagonal matrx elements of H n the HF bass, and W = H H 0 are therefore the off-dagonal matrx elements. The HF state E n (0) constructs our model space P = E n (0) (0) E n, thus Q = n n n E n (0) (0) E n and E n (0) H 0 E n (0) (0) = E n. The corrected energy s gven by E n = E n (0) H0 E (0) n + E (0) n Weff E (0) n = E (0) n + E corr,n. (B.2) 26
C. Implementaton-specfc Remarks E corr,n can be expanded as a seres: E (0) W m 2 n (0) E W m m W k k W E (0) E n = E n (0) + m E n E m (0) + m,k (E n E m (0) ) (E n E (0) k ) +.... (B.3) As E n appears n both sdes of the defnton, not permttng an algebrac soluton, ths equaton requres a self-consstent method lke an teratve scheme to be solved. C. Implementaton-specfc Remarks Ths secton contans nformaton and remarks concernng the mplementaton of the new routnes nto the nsute project. C.1. Changes to nsute n A new opton check was ntroduced when parsng the command lne of the fles calcenergypt_hfm.c and calcenergypt_hfc.c. The added varable CMode specfes the Calculaton Mode : 0: for the approxmated-denomnator (AD) method, 1: for the exact-denomnator (ED) method. Both versons, HFM and HFC, calculate the denomnator utlzng the D2VTCenerges 16. Choosng a dfferent value for the bnary sel-scheme does not (!) alter the calculaton energes utlzed to calculate the denomnator, e.g. D2VC 17, D2V 18. C.2. Lst of Modfed (M) and Newly Created (C) Fles PTE_Calc_HFC.h (M): updated functon declaraton PTE_Calc_HFC.c (M): added the m-averaged ED method PTE_Calc_HFM.h (M): updated functon declaraton PTE_Calc_HFM.c (M): added the standard ED method calcenergypt_hfm.c (M): ntroduced the calculaton mode CMode calcenergypt_hfc.c (M): ntroduced the calculaton mode CMode PAR_Base.h (M): ntroduced calculaton mode varable n PAR structure n 16 Second order MBPT energes (D2) whch are calculated wth a transformed Hamltonan ncludng NN-nteractons (V), knetc energy (T) and Coulomb nteractons (C). 17 Hamltonan ncludes only V and C. 18 Hamltonan ncludes only V. 27
D. Addtonal Graphs PAR_Base.c (M): ntroduced calculaton mode to functons HFC_Base.h (M): added HFC.eeee to HFC_Struct HFC_Base.c (M): ntalzed HFC.eeee HFM_Base.h (M): added HFM.eeee to HFM_Struct HFM_Base.c (M): ntalzed HFM.eeee COE_* (C): calculaton routne for SP energy correctons calcorbten_hfm (C): calculaton routne for SP energy correctons makefle (M): entres for complng calcorbten_hfm lbhf\makefle (M): entres for COE_* D. Addtonal Graphs In ths secton we present two graphs depctng exemplatory results of the SP energy correctons. 28
D. Addtonal Graphs 0 1p1 2 1p1 2 1p3 2 1p3 2 0 1p1 2 1p1 2 1p3 2-20 -40-60 1s1 2 1s1 2-10 -20 1p3 2 E [MeV] E [MeV] -80-30 1s1 2 1s1 2-100.. -40 Protons Neutrons Protons Neutrons Fgure 10: SP energes of protons and neutrons for 16 O (HF and HF+Correcton) are depcted. Left-hand-sde wth all terms of Eq. 5.10 and rght-hand-sde wth double exctatons of type C ( N 1 Ψ cr cab ) omtted. (rght-hand-sde). 29
D. Addtonal Graphs 100 20 75 1f7 2 1f7 2 50 0 2s1 2 1d3 2 1d5 2 2s1 2 25 0-25 1f7 2 2s1 2 1d3 2 1d5 2 1p3 21p1 2 1f7 2 2s1 2 1d5 21d3 2-20 -40 1d5 21d3 2 1p3 21p1 2 1p1 2 1p3 2 E [MeV] E [MeV] -50 1p1 2 1p3 2 1s1 2-60 1s1 2-75. 1s1 2 1s1 2. Protons Neutrons Protons Neutrons Fgure 11: SP energes of protons and neutrons for 56 N (HF and HF+Correcton) are depcted. Left-hand-sde wth all terms of Eq. 5.10 and rght-hand-sde wth double exctatons of type C ( N 1 Ψ cr cab ) omtted. (rght-hand-sde). 30
References References [1] H. Feldmeer, T. Neff, R. Roth, J. Schnack, Nucl. Phys. A632, 1998 [2] H. Feldmeer, T. Neff, Nucl. Phys. A713, 2003 [3] H. Feldmeer, T. Neff, R. Roth, H. Hergert, Nucl. Phys. A745, 2004 [4] R. Roth, P. Papakonstantnou, N. Paar, H. Hergert, Hartree-Fock and Many- Body Perturbaton Theory wth Correlated Realstc NN-Interactons, Phys Rev. C73 044312, 2006 [5] R. Roth, P. Papakonstantnou, T. Neff, H. Hergert, H. Feldmeer Phys. Rev. C72, 2005 [6] R. Roth, P. Navrátl, n preparaton, 2006 [7] I. Talm, Helv. Phys. Acta 25, 185, 1952 [8] C. Barber, N. Paar, R. Roth, P. Papakonstantnou, Correlaton energes n the random phase approxmaton usng realstc nteractons, Phys. Rev., 2006 [9] M. Moshnsky, Nucl. Phys. 13, 104, 1959 [10] A. Szabo, N.S. Ostlund, Modern Quantum Chemstry, Mneola, New York, Dover Edton, 1996 [11] P.Rng and P. Schuck, The Nuclear Many-Body Problem, Sprnger Verlag, New York, 1980 [12] G. P. Kamuntavcous, R. K. Kalunauskas, B. R. Barrett, S. Mckevcus and D. Germanas, Nucl. Phys. A695, 191, 2001 [13] B. Buck and A. C. Merchant, Nucl. Phys. A600, 387, 1996 [14] A. Zapp, Dploma Thess n prep., IKP TU Darmstadt, Germany, 2006 [15] P. Hedfeld, Dploma Thess n preparaton, IKP TU Darmstadt, Germany, 2006 [16] H. Hergert, Doctor Thess n preparaton, IKP TU Darmstadt, Germany, 2006 [17] R. Roth, Prvate communcaton [18] H. Feldmeer, Notes about BW many-body perturbaton theory [19] W. Noltng, Quantenmechank - Methoden und Anwendungen, 5. Edton, Sprnger, 2004 [20] Horst Stöcker, Taschenbuch der Physk, 4. Aufl., Verlag Harr Deutsch, 2000 31
References Acknowledgement My thanks goes to the supervsng Prof. Dr. Robert Roth for the flexble and anmatng way of managng my Bachelor-Thess, for the relaxed dscussons about the topc, and for the open and pensve gaze, furrowng hs brow, upon my two knocks on the doorframe before enterng for some short questons more. I thank Heko Hergert for the helpfull nstructons on the topc and hs very accurate and constructve way of correctng. He used up at least one red-nk pen on my prntouts. I d lke to thank my roommate Pascal Büscher for good off-physcal dstractons and the theoretcal nuclear physcs groups THQ, NHQ and TNP++ for the harmonous workng atmosphere on the fourth floor never ate so much cake. Fnally, I want to thank my parents for ther support and trust n me, for the ntrest n my studes and the never-endng curous questons. I wouldn t be wrtng these lnes wthout them. 32