arXiv:1808.03629v2 [physics.atom-ph] 26 Oct 2018 lob nacddet hi olcientr nde- enhancement in largest nature the However, collective their ( [14]. may to nuclei Nuclear due quadrupoles formed enhanced magnetic [13]). be as enhanced also such mag- be moments and also EDM T,P-odd can (nuclear quadrupole [9] forces netic parity by nuclear mixed opposite be T,P-odd of can which the levels momentum nuclear angular close same the the with to due be may EDM. nucleon the of contribution the than larger times h olcieeetwr oehr[5 16]. [15, together and work effect effect level nuclear collective close the the both where deformation ocst h ula D n ci oetis moment Schiff and P,T-odd the EDM of nuclear contribution the the that to forces. [9] forces nuclear in nuclear P,T-odd shown the the was It by named) produced (and moment calculated Schiff [9–11] Refs. EDM. have moment Schiff [12]. the Ref. size for in nuclear considered formula finite been the the to with and corrections moment electrons Schiff atomic the relativistic of the Hamil- interaction the of describing nucleus, distribution the tonian inside The field electric moment [7–11]. Schiff added term with ex- screening potential multipole electrostatic electron the nuclear of the order the of third nside the pansion in field a appears electric It is moment produces nucleus. Schiff which electrons The multipole of moment. dia- vector Schiff interaction of nuclear by EDM the produced with [6]. is atoms atoms neutral magnetic in screened completely mecha- enhancement [3–5]. the small, e.g. for see very - looking are nisms are effects we super- Corresponding including therefore, cornered 2]. CP-violation already [1, of have symmetry models and popular theories many crucial unification provide (EDM) of atoms moments and tests dipole nuclei particles, electric elementary -violating of CP and moments T,P Schiff nuclear hanced ∼ ute nacmn ftencerShffmoment Schiff nuclear the of enhancement Further aclto fcletv ci moment Schiff collective of Calculation es 7 ]cluae h ci oetdet proton to due moment Schiff the calculated 8] [7, Refs. is EDM nuclear the theorem Schiff the to According en- and deformation octupole Introduction: nacdncerShffmmn n iervra ilto i violation reversal time and moment Schiff nuclear Enhanced 10 2 − 10 3 ASnumbers: axions. PACS for search to and forces nuclear T,P-violating and nadtoa nacmn ftm eesl()adprt (P parity ground and the (T) in appears reversal moments) time dipole electric of violating enhancement additional An fdaoi molecule diatomic of ie)hpesi uliwt h octupole the with nuclei in happens times) cuoedfrainrslsi togyehne collec enhanced strongly a in results deformation Octupole 225 , 223 1 colo hsc,Uiest fNwSuhWls yny205 Sydney Wales, South New of University Physics, of School RaOH 2 + oansGtnegUiesta an,509Miz Germ Mainz, 55099 Gutenberg-Universit¨at Mainz, Johannes orsodn xeiet a eue ots CP-violatio test to used be may experiments Corresponding . 229 h.Smlrehneet xs nmlclrions molecular in exist enhancements Similar ThO. esrmnsof Measurements . Dtd coe 9 2018) 29, October (Dated: ..Flambaum V.V. The . ∼ molecules 40 ci oeti endb h olwn xrsin[9]: expression following the by defined is moment Schiff W fti obe r ie by mixed are doublet this of h rjcinof projection the momentum where where t n iervra naine ned D n Schiff and polar EDM are Indeed, par- moment the invariance. by reversal time forbidden and are ity moment Schiff and EDM frame ln h ula spin nuclear the along hw pi h aoaoyfae[5 16]: [15, frame laboratory the in up shows ulu iha cuoedfrainas a small a has also deformation EDM octupole intrinsic an with nucleus A hsmxn oaie ula axis nuclear spin polarises mixing This ulo nua oetmhsadulto ls oppo- close states of rotational doublet parity a site has momentum angular nucleon la hredensity charge clear h lcrnsreigadcnan ula ensquared radius mean nuclear charge contains and screening electron the where uduoedeformation quadrupole rm h ci moment Schiff the frame cuoemoment octupole 1,16]: [15, ulu iha cuoedfrainadnon-zero and deformation octupole an with nucleus A fanceshsa cuoedeformation octupole an has nucleus a If h iigcecetis: coefficient mixing The . I 1 1 , S R tt n ntemetastable the in and state Σ , Z r < 2 n < intr stencerrdu.Hwvr ntelaboratory the in However, radius. nuclear the is stencercharge. nuclear the is ieShffmmn in moment Schiff tive S n z ≈ = I > > iltn ffcs(uha T,P- as (such effects violating ) ≡ r < ( 5 2 = 10 D | e √ I R 3 ± [ I 35 2 u oadffrnebtenteproton the between difference a to due r < α T α ρ O S nt h ula xs.Tesae of states The axis). nuclear the to on /Z > > ( I ee etr hc utb directed be must which vectors -even O = r intr I ρ +1 = 2 = ,Asrlaand Australia 2, 2 z ) h eodtr rgntsfrom originates term second The . intr r r I I < n h nrni ci moment Schiff intrinsic the and , n ..i a olcienature collective a has it i.e. , √ > 229 d α 1 hc is which β hoispredicting theories n n ula EDM nuclear and β any E 2 3 2 n I − 2 − ( ThOH r ntefie-oy(rotating) fixed-body the in , + S | | 1 + Ω | ≈ I I 3 intr r h oet ftenu- the of moments the are W 229 − 5 ± Z ,T P, 229 > 20 | E > S I h-containing - Th r < + hnucleus. Th + − intr π |− ±| spootoa othe to proportional is T , − ihtesm angular same the with 3 √ 3 > odpseudovector. -odd 229 2 ∆ iltn interaction violating n 35 . . >< 1 ThF ln h nuclear the along Ω eZR state > r + ,weeΩis Ω where ), > 3 d β ] = , 2 β β < e 3 3 , n a and r (1) (2) (4) (3) > , 2 and neutron distributions which results in the laboratory as the static octupole deformation. This means that a I frame nuclear EDM d =2α I+1 D [15, 16]. similar enhancement of the Schiff moment may be due A similar Ω- doublet mixing mechanism produces huge to the dynamical octupole effect [23–25] in nuclei where enhancement of electron EDM de and T,P-odd interac- <β3 >= 0. tions in polar molecules, such as ThO. Interaction of de Unfortunately, the nuclei with the octupole deforma- with molecular electric field produces the mixing coef- tion and non-zero spin have a short lifetime. Several ficient α resulting in the orientation of large intrinsic experimental groups have considered experiments with 225 223 molecular EDM D eaB along the molecular angular Ra and Rn. The only published EDM measure- momentum J, and we∼ obtain d =2α J D αea [26], ment [27] has been done for 225Ra which has 15 days J+1 ∼ B where aB is the Bohr radius. As a result, the T, P - vio- half-life. In spite of the Schiff moment enhancement the 225 lating molecular EDM d exceeds electron EDM de by 10 Ra EDM measurement has not reached yet the sensi- orders of magnitude. tivity to the T,P-odd interaction Eq. (6) comparable to In the papers [15, 16] the numerical calculations of the the Hg EDM experiment [17]. The experiments continue, Schiff moments and estimates of atomic EDM produced however, the instability of 225Ra and a relatively small by electrostatic interaction between electrons and these number of atoms available may be a problem. moments have been done for 223Ra, 225Ra, 223Rn, 221Fr, To have a breakthrough in the sensitivity we need a 223Fr, 225Ac and 229Pa. The Schiff moment of 225Ra ex- more stable nucleus and a larger number of atoms. An ex- ceeds the Schiff moment of 199Hg (where the most accu- cellent candidate is 229Th nucleus which lives 7917 years rate measurements of the Schiff moment have been per- and is very well studied in numerous experiments and formed [17]) 200 times. Even larger enhancement of the calculations (this nucleus is the only candidate for the 225Ra Schiff moment has been obtained in Ref. [18]. For nuclear clock which is expected to have a precision sig- other nuclei the enhancement factors relative to Hg are nificantly better than atomic clocks [28, 29], has strongly between 30 and 700. Atomic calculations of EDM in- enhanced effects of ”new physics” [30, 31] and may be duced by the Schiff moment in Hg, Xe, Rn, Ra and Pu used for a nuclear laser [32]). 229Th is produced in macro- atoms have been performed in Refs. [9, 19–22] and in- scopic quantities by the decay of 233U (see e.g. [33]), and clude additional atomic enhancement mechanisms. its principal use is for the production of the medical iso- It is useful to make an analytical estimate of the Schiff topes 225Ac and 213Bi. moment. According to Ref. [16] the T,P-violating matrix According to Ref. [34] the 229Th nucleus has the element is approximately equal to octupole deformation with the parameters β3=0.115, β2=0.240, I =5/2 and the interval between the opposite + β3η + < I− W I > eV. (5) parity levels E(5/2−) E(5/2 )=133.3 KeV. The analyt- | | ≈ A1/3 ical formula in Eq. (7)− allows us to scale the value of the 225 Here η is the dimensionless strength constant of the nu- Schiff moment from the numerical calculations for Ra which has β3=0.099, β2=0.129, I =1/2 and interval be- clear T, P - violating potential W : + tween the opposite parity levels E(1/2−) E(1/2 )=55.2 G η KeV [16]. Then Eq. (7) gives: − W = (σ )ρ, (6) √2 2m ∇ S(229Th) = 2 S(225Ra), (8) where G is the Fermi constant, m is the nucleon mass 225 8 3 Using S( Ra) = 300 10− eη fm [16] we obtain and ρ is the nuclear number density. Eqs. (2,3,4,5) give 229 8 × 3 the analytical estimate for the Schiff moment: S( Th) = 600 10− eη fm . Within the meson× exchange theory the π-meson ex- 4 I 2 2/3 KeV 3 change gives the dominating contribution to the T,P- S 1. 10− β2β3 ZA e η fm , (7) ≈ · I +1 E E+ violating nuclear forces [9]. According to Ref. [36] the − − neutron and proton constants in the T,P-odd potential 3 225 6 This estimate gives S = 280 e η fm for Ra, very close (6) may be presented as η η 5 10 ( 0.2gg¯0 + n ≈− p ≈ × − to the result of the numerical calculation in Ref. [16] gg¯1 +0.4gg¯2). In Refs. [15, 16] we have not separated the S = 300 e η fm3. proton and neutron contributions. Majority of the nucle- The values of the Schiff moments for the nuclei with ons are neutrons, so it make sense to take η = ηn. How- octupole deformation listed above vary from 45 to 1000 ever, the proton interaction constant has an opposite sign 8 3 199 129 203 10− eη fm [16]. For spherical nuclei Hg, Xe, Tl and may cancel a part of the neutron contribution, so we and 205Tl, where the Schiff moment measurements have multiply the interaction constant by ((N Z)/N)=0.36 − been performed, the calculations [9–11] give the Schiff and use η = 0.36ηn. This way we can obtain a rough 8 3 225 3 moment S 1 10− eη fm . estimate: S( Ra) = ( 2.2gg¯0 + 11gg¯1 + 4gg¯2) e fm , ∼ × 229 − 3 The Schiff moment in Eq. (7) is proportional to the S( Th) = ( 4.4gg¯0 + 22gg¯1 +8gg¯2) e fm . 2 − squared octupole deformation parameter β3 which is A more accurate job has been done in Ref. [18] about (0.1)2. According to Ref. [23], in nuclei with a where they presented the Schiff moment as S(225Ra) = 3 soft octupole vibration mode the squared dynamical oc- (a0gg¯0 + a1gg¯1 + a2gg¯2)e fm . To estimate the er- tupole deformation < β2 > (0.1)2, i.e. it is the same ror the authors of Ref. [18] have done the calcula- 3 ∼ 3 tions using 4 different models of the strong interaction. [39], the latter is enhanced in 229Th by the same oc- They obtained the following 4 sets of the coefficients: tupole mechaninism. Indeed, the axion dark matter field a0 = 1.5, 1.0, 4.7, 3.0; a1 = 6.0, 7.0, 21.5, 16.9; a(t) = a0cos(m t) (m is the axion mass) generates os- − − − − a a a2 = 4.0, 3.9, 11.0, 8.8. Taking the average val- cillating nuclear forces which are similar to the T,P-odd ues of− the coefficients− − and− using Eq. (8) we obtain: nuclear forces producing the Schiff moments. To obtain

225 3 the result for the oscillating Schiff moments and EDM it S( Ra) = ( 2.6gg¯0 + 12.9gg¯1 6.9gg¯2) e fm , (9) is sufficient to replace the constant θ¯ by a(t)/fa, where 229 − − 3 S( Th) = ( 5.1gg¯0 + 25.7gg¯1 13.9gg¯2) e fm . (10) fa is the axion decay constant [38, 39]. Search for the − − effects produced by the oscillating axion-induced Schiff We will use these expressions as our final values for the moments in solid state materials is in progress [40]. A Ra and Th Schiff moments. We can express the results promising direction here may be to use 229ThO molecule in terms of the more fundamental parameters such as the placed in a matrix of Xe (or other) atoms. A proposal ¯ ˜ QCD θ-term constant θ and the quark chromo-EDMs du to use paramagnetic molecules in the matrix of rare-gas ˜ ¯ and dd using the relations gg¯0 = 0.37θ [37] and gg¯0 = atoms for the electron EDM search has been described 15 ˜ ˜ −15 ˜ ˜ 0.8 10 (du + dd)/cm, gg¯1 =4 10 (du dd)/cm [1]: in Ref. [41]. · · − 225 ¯ 3 Promising objects for the Th Schiff moment measure- S( Ra) = 1.0 θe fm , (11) + 229 ¯ 3 ment may be ThO molecule and ThOH molecular ion. S( Th) = 2.0 θe fm , (12) Both molecules have zero electron angular momentum S(225Ra) = 104(0.50 d˜ 0.54 d˜ ) e fm2, (13) in the ground state and very close opposite parity levels u − d 229 4 2 which enhance T,P-violating EDM. S( Th) = 10 (1.0 d˜u 1.1 d˜d) e fm . (14) − Use of polar diatomic molecules for the measurement of Note that the contributions of θ¯ and d˜u,d should not be the nuclear Schiff moment was suggested by Sandars [7, 8] added to avoid double counting since d˜u,d may be induced because electric field inside polarised molecule exceeds by θ¯. external electric field ǫ by several orders of magnitude Molecular enhancement. Atomic EDM da pro- and has the same direction. The molecular polarisation duced by the Schiff moment S very rapidly increases with is P Dǫ/(E E+), where D eaB is the intrinsic ∼ − − ∼ the nuclear charge Z [3, 4, 9, 16]: electric dipole moment of the polar molecule. Therefore, to have a significant polarization degree P the interval 2 aB 2 2γ da Z ( ) − S, (15) between the opposite parity molecular rotational levels ∝ 2ZR (E E+) should be sufficiently small. Indeed, the ro- − − where R is the nuclear radius, aB is the Bohr radius, tational interval in molecules is 3-5 orders of magnitude γ = p1 (Zα)2. Th and Ra have close nuclear charges, smaller than a typical interval between the opposite par- Z = 88− and 90, and similar electronic structure up to ity levels in atoms. last filled 7s2 subshell. Two extra 6d2 electrons in Th We may interpret the molecular enhancement in a dif- have high angular momenta, do not penetrate the nucleus ferent way [26]: interaction between the Schiff moment and do not interact with the Schiff moment directly (up and electrons mixes close opposite parity levels in the to many-body corrections). Therefore, da/S for Th is molecule, polarises the molecule along its angular mo- approximately equal to da/S for Ra. Using calculations mentum and creates T,P-violating EDM proportional to of Ra atom EDM from Refs. [21, 22] we have the large intrinsic electric dipole moment D - see the dis- cussion below Eq. (4). This enhanced EDM interacts 17 S 16 ¯ with the external electric field ǫ. The experiment has da(Th) 9 10− 3 e cm = 2 10− θ e cm. ≈− · e fm | | − · | | been performed with the TlF molecule [42]. In the pa- | | 225 (16) per Ref. [43] it was proposed to study molecule RaO da(Th) as a function of other T,P and CP–violating in- where the effect may be 500 times larger than in TlF teraction constants η, g,¯ d˜ can be found by the substitu- due to the enhanced Schiff moment and larger nuclear tion of the Th Schiff moment from the equations in the charge Z. The best sensitivity to the electron EDM has nuclear Schiff moment section above. This value of Th been obtained using molecules ThO [46] and HfF+ [47] 3 EDM is 3 orders of magnitude larger than Hg EDM and in the excited metastable electronic state ∆1 which con- 4 orders of magnitude larger than Xe EDM. However, Th tains doublets of very close opposite parity levels. Fi- atom has non-zero electron angular momentum, J = 2, nally, in the recent paper [45] it was suggested that lin- and this reduces the signal coherence time and increases ear molecules MOH, molecular ions MOH+ (M is a heavy systematic errors. In principle, one may use Th4+ ion atom, e.g. Ra in the molecule RaOH+ ) and symmetric which has closed shells or look for zero electron angular top molecules (such as MCH3 or MOCH3) may be bet- momentum Th ions in solid state materials. ter systems than molecules MO since such polyatomic Note that the measurements of the effects produced molecules have a doublet of the close opposite parity en- by the 229Th Schiff moment may be used to search for ergy levels in the bending mode and may be polarised axions. Indeed, the axion dark matter produces oscil- by a weak electric field. The reduction of the strength of lating neutron EDM [38] and oscillating Schiff moment the necessary electric field simplifies the experiment and 4 dramatically reduces systematic effects. Comaparison with existing and proposed exper- The T,P-violating effect in 229ThO is much larger than iments. We should compare suggested experiments with in TlF due to the enhanced Schiff moment and larger 229ThO, ThF+ and 229ThOH+ molecules with other ex- nuclear charge. An additional advantage may appear in isting and proposed experiments. The best limit on the ThOH+ which is expected to have very close opposite nuclear Schiff moment has been obtained in the measure- parity states (similar to RaOH+). Another possibility ment of Hg EDM [17]. However, there is a theoretical 3 may be to use the doublet in ∆1 metastable state of problem here: the most recent sophisticated calculation 229ThO (used to improve the limit on electron EDM ) [35] was not able to find out even the sign of the Hg 3 + and the ground state doublet ∆1 in ThF . Schiff moment, different interaction models give very dif- The interaction constant WS for the effective T,P- ferent results. There are two reasons for this: firstly, the violating interaction in molecules Schiff moment is determined by the charge distribution of the protons. However, it is directed along the nuclear S 199 W = W I n (17) spin which in Hg is carried by the valence neutron, i.e. T,P S I · the Schiff moment in 199Hg is determined by the many- body effects which are harder to calculate. The second (here I is the nuclear spin, n is the unit vector along + + reason is in the formula for the Schiff moment defined the molecular axis) in ThO, ThOH and RaOH may by Eq. (1). There are two terms of opposite sign in be estimated by the comparison with the RaO molecule. this formula which tend to cancel each other, the main Calculation of WS for RaO has been done in Ref. [44]: 4 term and the screening term (remind the reader that the WS (RaO) = 45192 atomic units (here a.u.=e/aB). In + screening term kills the nuclear EDM contribution to the RaOH ion the electron density on the Ra nucleus is atomic EDM). If we do not know each term sufficiently slightly smaller than in RaO (since a part of the elec- accurately, the final sign and the magnitude of the Schiff tron charge density moves to hydrogen), therefore, we as- moment are unknown. sume W (RaOH+) 30000 a.u. In ThO and in ThOH+ S Recently the interest in EDM experiments has moved the electron density≈ on the Th nucleus is expected to towards molecules where the effects are very strongly en- be slightly larger than that for Ra due to the higher hanced by the close rotational levels and very strong in- Th charge and two extra electrons. Therefore, we as- + ternal ”effective electric field”. For example, the limit on sume W (ThO) 50000 a.u. and W (ThOH ) 30000 S S electron EDM in ThO and HgF+ experiments have been a.u. Note that the≈ electron wave function in the≈ bend- improved by more than an order of magnitude in compar- ing molecular mode of RaOH+ and ThOH+ is the same ison with the atomic EDM experiments. The Tl nuclear as in their gound states, therefore, the parameters W S Schiff moment has been measured in the TlF experiment are practically not affected by these bending vibrations [42]. Similar to 199Hg, calculations of the 203,205Tl Schiff (where we have the doublet of the opposite parity lev- 3 229 moments suffer from the problem of the cancellation be- els). The parameter W in the ∆1 state of the ThO S tween two approximately equal terms in Eq. (1) and molecule should have comparable value to their ground 3 1 the problem of the nuclear core polarization contribu- state values since ∆1 and the ground Σ0 state differ by 3 + tion (since there is a strong cancellation between the two one electron orbital only. The ∆1 state in ThF molec- 3 229 terms in the valence proton contribution in Tl [9, 11, 36]). ular ion is similar to the ∆1 state in ThO molecule. The estimates presented above are based on compari- Actually, the interpretation of the TlF experiment [42] son with the numerical calculations of the Schiff moment was done in terms of the proton EDM. However, here we contribution in RaO. Estimates based on the Z depen- probably have even a more serious problem (below we dence extrapolation Eq. (15) from TlF give 2 times larger will follow the discussion in Ref. [36]). Firstly, calcula- results. tions with different choices of the strong interaction give Substitution of the Schiff moment (12) to the energy different signs and magnitudes of the Schiff moment Sp S induced by the proton EDM ( since we also have here the shift WT,P = WS I n gives for the fully polarised I · cancellation between the main and the screening contri- molecule the energy difference between the Iz = I and 229 butions). The authors of the molecular calculation [48] Iz = I states in ThO: − selected the maximal value out of 4 numbers calculated 7 by A. Brown (this maximal number leads to the strongest 2WSS =1. 10 θ¯Hz (18) · limit on the proton EDM), and this value of Sp was used A similar estimate is valid for molecular ions ThF+ and in all other molecular calculations for TlF [49–51] (see ThOH+. The measured shift in the 1991 TlF experiment also [21]). There was no such accuracy investigation for [42] was 0.13 0.22 mHz. The same sensitivity in the the proton EDM contribution to the Hg Schiff moment 229ThO,− ThF+ ±or ThOH+ experiments would allow one but naively we may expect that the accuracy is actually 10 199 to improve the current limit θ¯ < 10− and also the lim- lower than in Tl since the valence nucleon in Hg is its on other fundamental parameters| | of the CP-violation neutron. theories such as the strength of T,P-violating potential η, The second problem is that in practically any model the πNN interaction constantsg ¯ and the quark chromo- the contribution of the T,P-violating nuclear forces to EDMs d˜. the Schiff moment is 1-2 orders magnitude larger than 5 the proton EDM contribution (the ratio is ”model inde- nuclei where there is no single dominating contribution. pendent” since the πNN interaction constant appears in Thus, calculations of the collective Schiff moments look both contributions and cancels out in the ratio). There- more ”clean” theoretically. More importantly, the collec- fore, to obtain the limit on the proton EDM we neglect tive Schiff moment is enhanced by 2-3 orders of magni- much larger contribution of the P,T-odd nuclear forces. tude. Thus, in the Particle Data tables the limit on the pro- Conclusion. We propose to search for the T,P and ton EDM is presented assuming that there are no other CP violating effects in the molecule 229ThO where the contributions to atomic and molecular EDM. However, if effects are 2-3 orders of magnitude larger than in TlF we wish to test CP-violation theories such limits on the due to the enahnced Schiff moment of the 229Th nu- proton EDM from Hg and Tl EDM can hardly be used. cleus and large nuclear charge. An additional advantage 229 + 3 These theoretical problems do not exist for the collec- may be in ThOH molecular ion, in ∆1 state of the tive Schiff moments in the nuclei with the octupole defor- 229ThO molecule and in 229ThF+ molecular ion which mation. The second screening term is very small in this have very close opposite parity energy levels and may be 3 case since it is proportional to a very small intrinsic dipole polarised by a weak electric field. The ∆1 state of the moment D of the ”frozen” nucleus. If the distributions 229ThO molecule has already been used to measure elec- of the neutrons and protons are the same, D = 0. Thus, tron EDM. The enhanced effects in these molecules may there is no cancellation and the intrinsic Schiff moment also be used to search for axions. 229Th lives 7917 years, is proportional to the known electric octupole moment may be produced in macroscopic quantities (as it is done (which may actually be measured using probabilities of for the medical applications) and is very well studied in the octupole transitions between the rotational levels). numerous experiments. Then the calculation is reduced to the expectation value This work is supported by the Australian Research of the T,P-odd interaction < Ω W Ω >, here Ω > is the Council and Gutenberg Fellowship. I am grateful to M. | | | ground state of the ”frozen” deformed nucleus [15, 16]. Kozlov, N. Hutzler, A. Palffy, Jun Ye, D. DeMiIle, H. Calculation of one expectation value looks more reliable Feldmeier, N. Minkov, A. Afanasiev, P. Ring and TAC- n> | | | than the calculation of the infinite sum Pn E0 En in TICA collaboration for useful discussions. −

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