Calculation of ground state energy of harmonically confined two dipolar fermions
CCalculation of ground state energy of harmonically confined two dipolar fermions
Amit K. Das a and Arup Banerjee b ∗ a Laser Material Processing Division, Raja Ramanna Centre for Advanced Technology, Indore 452013, India b BARC Training School at RRCAT, Raja Ramanna Centre for Advanced Technology, Indore 452013, India andHomi Bhabha National Institute, Indore 452013, India (Dated: October 8, 2018)We calculate the ground state energies of a system of two dipolar fermions trapped in a harmonicoscillator potential. The dipoles are assumed to be aligned parallel to each other. We perform thecalculations of ground state energy as a function of strength of interaction between two fermions byemploying variational method with Hylleraas-like explicitly correlated wave function. Furthermore,we perform calculations of ground state energy within Hartree-Fock approximation and the mag-nitude of correlation energy is estimated by subtracting these results from the corresponding wavefunction based results. We also carry out calculations of ground state energies within the realm ofdensity functional theory by using recently reported expressions for exchange and correlation ener-gies under local density approximation. By comparing correlated wave function based results withthose obtained using density functional theory approach we examine the role of fermion-fermion cor-relation and assess the accuracy of local density approximation based expression for the correlationenergy functional.
I. INTRODUCTION
Recently, several studies on the static and dynamicproperties of dipolar Fermi gas have been reported inthe literature [1–8]. For general review on both dipo-lar bosons and fermions we refer the readers to Ref.[9]. The characteristic properties of dipolar Fermi gasare distinctly different from that of an electron gas orusual Fermi gas made of non-dipolar atoms due to theanisotropic and long-range nature of the interaction be-tween the dipoles. The dipolar Fermi gases have beenrealized experimentally by using various atomic specieswith large magnetic moment. For example, isotopes of Cr,
Yb, and
Dy atoms with large magnetic mo-ment of 6 µ B ( µ B denotes Bohr magnetron), 3 µ B , andaround 10 µ B , respectively have been cooled down toreach quantum degeneracy [11–13]. On the other hand,heteronuclear polar diatomic molecules with large electricdipole moment are promising candidates for realizationof dipolar Fermi gases. Recently, quantum degeneracy ina sample of around 4 × fermionic K Rb moleculesin their rovibrational ground state with permanent elec-tric dipole moment of 0.566 Debye has been achieved [14].Furthermore, the anisotropic characteristic of the dipole-dipole interaction has also been experimentally investi-gated in these KRb samples [15].The successful creation of degenerate dipolar Fermigas of KRb molecules has given rise to a new class ofmany-body system and this has rekindled the interestsof theorists to look into these systems. The quantummechanical description of degenerate Fermi gas is morecomplicated than its bosonic counterpart due to the an-tisymmetric nature of the many-fermion wave function.Therefore, for theoretical description of many-fermion ∗ [email protected] systems it becomes necessary to invoke some approxi-mations. We note here that most of theoretical stud-ies on dipolar Fermi gases have been carried out eitherwithin the semi-classical Thomas-Fermi (TF) approxima-tions [1], which neglects effects of both exchange and cor-relation between the fermions or within the Hartee-Fock(HF) theory [3–6], which takes into account the quan-tum mechanical effect of exchange of identical particles.It is only recently Liu and Yin [7] reported calculationof correlation energy of homogeneous dipolar Fermi gasby going beyond HF approximation. They estimated thecorrelation energy, which is defined as the difference be-tween the true ground state energy and the correspond-ing energy in the HF approximation using perturbativeBrueckner-Goldstone formalism along with Monte Carlointegration to fix the coefficient. This study showed thatthe ground state energy of a homogeneous dipolar Fermigas is reduced when correlation effect is included result-ing in significant lowering of critical density required formechanical collapse as compared to the HF case. Theseauthors also employed the expression for the correla-tion energy to calculate the properties of inhomogeneoustrapped dipolar Fermi gas by using the expressions forexchange and correlation energy functionals within localdensity approximation (LDA).We wish to note here that for a system consisting offew (two to three) interacting fermions it is possible toconstruct a fully correlated wave function satisfying an-tisymmetric property for the calculation of its groundstate energy. For example, in atomic and molecularphysics Hylleraas-like correlated wave functions [16, 17]with explicit dependence on the inter-electronic coordi-nates have yielded very accurate results for two-electronhelium-like ions and H molecule [18]. It should be men-tioned here that beside exchange effect a correlated wavefunction also takes into account the effect of electron-electron correlation arising from the electron-electron in-teraction through the dependence of wave function on the a r X i v : . [ phy s i c s . a t m - c l u s ] O c t inter-electronic coordinate. However, both HF approach( which takes into account the effect of exchange) andsemi-classical TF approximation neglect effect of two-particle correlations. Our main objective of this paper isto first calculate ground state energy of a system of twodipolar fermions confined in a harmonic trap by usingHylleraas-like correlated wave function. For the purposeof comparison we also carry out the calculation of groundstate energy of a confined two-dipolar fermion system byusing HF approach.Albeit, the correlated wave function approach takesinto account the effect of both exchange and correlationhowever, the form of the wave function becomes verycumbersome with increasing number of particles makingthis approach impractical for systems beyond three par-ticles. A formalism which takes into account the effectsof both exchange and correlation in many-fermion sys-tems efficiently is the Density functional theory (DFT)[19–21]. This formalism circumvents the use of compli-cated many-body wave function of 3N spatial variables( where N is number of particles), rather works withparticle density - a function of just three spatial vari-ables. As a result of this DFT based method turns outto be quite suitable for studying fermionic systems withlarge number of particles. Although the theory is ex-act in principle but the exact forms for exchange andcorrelation functional in term of density are not known.Therefore, to carry out any DFT based calculations it be-comes necessary to use approximate expressions for theexchange and correlation energies. For electronic systemsa tremendous amount of effort has gone into the devel-opment of exchange and correlation functionals with in-creasing degree of accuracy [20, 21]. The availability ofaccurate exchange-correlation functional along with sig-nificantly lower computational cost of DFT-based cal- culations has made DFT an indispensable tool for han-dling many-electron systems like atoms, molecules, andsolids. For electronic systems the simplest and one ofthe most extensively used forms for exchange and corre-lation energy functionals are constructed within LDA byreplacing the constant electron density in the expressionsfor exchange and correlation energies of a homogeneouselectronic gas by the local density of the inhomogeneoussystem. As mentioned above it is only recently expres-sions for the exchange and correlation energies of a ho-mogeneous dipolar Fermi gas have been derived [4, 7].The availability of these expressions has motivated us tocarry out DFT based calculations of the ground state en-ergy of two harmonically trapped dipolar Fermi particlewithin LDA and compare these results with the numbersobtained via correlated wave function approach to assesstheir accuracies.The remaining paper is organized in the following man-ner. In section II we present the model and describe vari-ous theoretical methods employed to calculate the groundstate energy. The results are presented and discussed insection III. The paper is concluded in section IV. II. MODEL AND VARIOUS METHODS TOOBTAIN GROUND STATE ENERGY OF TWODIPOLAR FERMIONS CONFINED IN AHARMONIC TRAP
In this paper we consider a system of harmonicallytrapped two identical fermionic particles of mass m pos-sessing electric dipole moment d , which are polarizedalong the z-axis. The non-relativistic Schr¨ o dinger equa-tion for such a pair of identical dipolar fermions confinedin a harmonic oscillator potential with frequency ω isgiven by, (cid:20) − (cid:126) ∇ − (cid:126) ∇ + 12 (cid:0) r + r (cid:1) + v dd ( r , r ) (cid:21) Ψ( r , r ) = E Ψ( r , r ) . (1)The above equation has been written in the units oflength l ( l = (cid:112) ¯ h/mω ) and energy (cid:15) ( (cid:15) = ¯ hω ) ofcofining harmonic oscillator potential. The potential v dd in Eq. (1) represents the dipole-dipole interaction be-tween the fermions, v dd ( r , r ) = 3 a dd l (1 − θ ) r , (2)where 3 a dd = md / π(cid:15) ¯ h is the effective interactionstrength and θ is the angle between the vector r = r − r and the direction of the dipoles, which is chosento be along the z-axis.To calculate ground state energy of trapped dipolartwo-fermion system we need to solve the eigenvalue equa-tion given by Eq. (1). For this purpose we applyRayleigh-Ritz variational method with an appropriatetrial wave function for Ψ( r , r ) . To describe the cor-relation effect accurately we choose to use a trial wavefunction for the two-fermion system with explicit depen-dence on the fermion-fermion separation r . To this endwe employ wave function of the form (S-symmetry)Ψ( r , r ) = N (cid:2) exp( − αr − βr ) + exp( − αr − βr ) (cid:3) [1 − exp( − λr )] , (3)where N is normalization constant. The total wave func-tion is product of Ψ( r , r ) and the the spinor wave func- tion χ (1 , r , r )) is symmetric withrespect to the interchange of coordinates the spinor partmust be antisymmetric to ensure the antisymmetric na-ture of the total wave function. In the above equation(Eq. (3)) the three nonlinear coefficients α , β , and λ arevariational parameters, which are determined by mini-mizing the ground state energy as described below. Wenote here that keeping in mind the harmonic nature ofthe confining potential we choose gaussian functions in-volving the coordinate variables r and r . The form of the correlation function is motivated by the fact thatunless the wave function goes to zero as r →
0, the in-tegral for the interaction energy diverges. Also becauseof the finite size of the dipoles it is physically not possiblefor the dipoles to come within distance 1 /λ .The Rayleigh-Ritz variational method for solving Eq.(1) involves finding the minimum value of the followingenergy functional of trial wave function Ψ E [Ψ] = (cid:82) (cid:82) Ψ ∗ ( r , r ) (cid:104) − (cid:126) ∇ − (cid:126) ∇ + (cid:0) r + r (cid:1) + v dd (cid:105) Ψ( r , r ) d r d r (cid:82) (cid:82) Ψ ∗ ( r , r )Ψ( r , r ) d r d r . (4)By substituting the wave function given by Eq. (3) in theabove equation and using the coordinate system involv-ing r , r and r , it can be reduced to sum of integralsof the form (cid:90) ∞ dr (cid:90) ∞ dr (cid:90) r + r | r − r | dr f ( r , r , r ) (5)by using the differential volume element as [23] dV = 8 π r r r dr dr dr , (6)with f ( r , r , r ) being a function of r , r and r . Theenergy functional is minimized by varying three nonlinearparameters α , β , λ to obtain the ground state energy.Having described the correlated wave function basedapproach for the calculation of ground state energy weproceed with a brief description of the method employedto calculate the ground state energies of dipolar Fermisystem within the realm of DFT. In accordance with theHohenberg-Kohn theorem [22] of DFT the ground stateenergy of a dipolar two-fermion system can be uniquelyexpressed as a functional of the fermion density ρ : E [ ρ ] = T [ ρ ] + V conf [ ρ ] + E dd [ ρ ] + E xc [ ρ ] , (7)where T [ ρ ] and V conf [ ρ ] are the non-interacting kineticand confinement energies respectively. For a two-fermionsystem considered in this paper T [ ρ ] and V conf [ ρ ] aregiven by, T [ ρ ] = − (cid:90) (cid:112) ρ ( r ) ∇ (cid:112) ρ ( r ) d r , (8)and V conf [ ρ ] = 12 (cid:90) r ρ ( r ) d r . (9)The third term E dd [ ρ ] is the dipolar counterpart of theHartree energy representing classical electrostatic energybetween two dipoles and it is given by E dd [ ρ ] = 12 (cid:90) v dd ( r , r ) ρ ( r ) ρ ( r ) d r d r , (10) where v dd ( r , r ) is the dipole interaction potential asdefined in Eqs. (2). The last term E xc [ ρ ] represents thethe exchange-correlation (XC) energy arising purely dueto quantum mechanical nature of two trapped fermions.Furthermore, the second part of Hohenberg-Kohn theo-rem provides the energy variational principle accordingto which the ground state density can be obtained bymaking the energy functional (Eq. (7)) stationary. Asmentioned above, in general the exact forms for the ex-change and correlation energy functionals are not knownand thus for implementing DFT based calculations oneneeds to use approximate forms for these energy function-als. In the present paper we use the expressions for ex-change and correlation energies of homogeneous dipolarFermi gas derived recently in Ref. [3, 7] within LDA byreplacing the uniform fermion density by the local den-sity ρ ( r ) of the inhomogeneous system. In accordancewith Ref. [7] we employ following expressions for theexchange ( E x ) and correlation ( E c ) energy functionalswithin LDA: E x [ ρ ] = −
445 (6 π ) / (3 a dd ) (cid:90) ρ / ( r ) d r , (11) E c [ ρ ] = −
23 (3 a dd ) (cid:90) ρ / ( r ) d r . (12)In Ref [7] the above forms for the exchange and cor-relation energy functionals have been applied to studythe mechanical collapse of trapped fermionic KRb polarmolecules.Once again we make use of variational principle asguaranteed by Hohenberg-Kohn theorem to carry out cal-culation of the ground state energy and the correspond-ing density. To this end we need to make a judiciouschoice for the form of trial ground state density ρ ( r ).Keeping in mind the harmonic trapping potential andthe anisotropic nature of the dipole-dipole interactionterm we choose following form for the trial density forthe ground state: ρ ( r ) = 2 | φ ( r ) | , (13)with φ ( r ) = (cid:88) p C p exp (cid:2) − ( w p x + w p y + γ p z ) (cid:3) . (14) In Eqs. (14) w p and γ p are the variational parameters andthe above form of density is also normalized to number offermions. The constants C p ’s are such that φ ( r ) remainsalways normalized. With the above choice for the form ofthe trial density, the kinetic, confinement, and interactionenergy terms in the energy functional given by Eq. (7)can be simplified to, T [ ρ ] = 2 π / (cid:88) pq C p C q ( w p + w q ) √ γ p + γ q (cid:34) w q + γ q − w q ( w p + w q ) − γ q ( γ p + γ q ) (cid:35) , (15) V [ ρ ] = 2 π / (cid:88) pq C p C q w p + w q ) √ γ p + γ q (cid:20) w p + w q ) + 1( γ p + γ q ) (cid:21) , (16) E dd [ ρ ] = − (cid:88) pqrs π / C p C q C r C s w p + w q + w r + w s ) (cid:112) ( γ p + γ q + γ r + γ s ) 3 a dd l f ( κ ) , (17)where κ = ( γ p + γ q )( γ r + γ s )( w p + w q + w r + w s )( w p + w q )( w r + w s )( γ p + γ q + γ r + γ s ) , (18)and f ( κ ) = κ − κ − κ tanh − √ − κ (1 − κ ) / , if | κ | < , if | κ | = 1 − κ κ − + κ tan − √ κ − κ − / , if | κ | > . (19)Using above expressions (Eqs. (11) - (19)) we minimizethe total energy by varying the parameter w p ’s, γ p ’s and C p ’s to obtain ground state density and the correspond-ing energy. The the number of basis functions ( p ) in Eq.(14) is fixed by checking the convergence of the groundstate energy with increasing number of basis functions.In order to assess the accuracy of LDA correlation en-ergy functional employed by us (Eq. 12) we first esti-mate the correlation energy component from the totalenergy obtained via wave function based approach. Forthis purpose we use HF based definition of the correlationenergy. As mentioned before the correlation energy isdefined as the difference between the exact ground stateenergy and the corresponding energy in the HF approx-imation. Therefore, in order to estimate correlation en-ergy we also carry out calculations of the ground stateenergy E HF within HF approximation. The correlationenergy is then obtained by subtracting E HF from thecorresponding result obtained with correlated wave func-tion approach. The calculations within HF approxima-tion can be performed employing the procedure describedabove for DFT based calculations by substituting E c = 0 and an exact expression for the exchange energy for twofermion system given by E x = − E dd . (20)In the next section we present the results of our calcula-tions obtained by applying various approaches mentionedabove and compare them to test their accuracies. Be-fore proceeding to the next section it should be men-tioned here that in general the correlation energy ob-tained within DFT differs from the above definition dueto the presence of kinetic energy component in the DFTbased definition of the correlation energy. However, fortwo-particle system considered in this paper the contri-bution of kinetic energy is zero and two definitions ofcorrelation energy coincide [21]. III. RESULTS AND DISCUSSIONS
The correlated wave function based variational calcu-lation for determining the ground state energy has beenperformed using GNU Scientific Library (GSL) routines[24]. The variational integrations were carried out us-ing VEGAS algorithm of Monte-Carlo multidimensionalintegration method and the energy minimization calcula-tions have been performed using Nelder-Mead algorithmas implemented in GSL. In Table I we present the resultsfor the ground state energies obtained by employing cor-related wave function for different values of interactionstrength determined by the dimensionless η = 3 a dd /l ranging from value 0.0 to 1.0. From this Table we ob-serve that for η = 1 . E GS ) of aharmonically confined two dipolar fermions is 2 . TABLE I. Ground state energy as a function of strength ( η )of the dipolar interaction as calculated from the correlatedvariational function. η = 3 a dd /l Ground state energy1 . . . . . . . . . . . . . . . . . . . . . . thermore, on decreasing the strength of the dipolar inter-action, the ground state energy increases gradually andit correctly approaches the energy of two non-interactingharmonically confined fermions ( E non − int = 3 .
0) as theinteraction is completely switched off. We note herethat the ground state energy of two interacting dipolarfermions confined in a harmonic oscillator potential is lessthan its non-interacting counterpart which clearly indi-cates that the dipole-dipole interaction is on the averageattractive in nature.Having obtained the results for the ground state ener-gies by using the correlated wave function given by Eq.(3) we now present the corresponding results obtained viaDFT and HF based approaches. These calculations havebeen performed by employing finite basis sets consist-ing of gaussian functions (Eq. (14)). Therefore, first wecheck the convergence of our results by increasing numberof basis functions. The results for the ground state en-ergy obtained by employing DFT and HF based methodsfor interaction strength η = 1 . η . First of all it can be seenfrom Figure 1 that both HF and DFT methods over-estimate the ground state energy in comparison to thecorresponding wave function based values. Among threeapproaches the wave function based method yields low- est values for the ground state energy which is consis-tent with the fact that this method takes into accountboth effects of exchange and correlation most accurately.Moreover, note that the ground state energy obtainedby employing wave function and DFT based approachesdisplay similar trend with respect to the variation of in-teraction strength. In particular we observe from Fig-ure 1 that ground state energies obtained by employingwave function and DFT base methods decrease from non-interacting ( η = 0) value of E GS = 3 .
00 as η is increased.On the other hand, the ground state energy calculated byHF method remains almost constant at a value of about3 .
00 for the whole range of interaction strength consid-ered in this study. This behaviour clearly indicates thatfor dipolar fermions effect of correlation, which is nottaken into account in HF method, is quite importantand it contributes significantly to the ground state en-ergy of the system. We note here that although DFTbased method overestimates the ground state energy ascompared to the ones obtained through correlated wavefunction approach nonetheless it performs better thanHF method in predicting the variation of energy with theinteraction strength. This is because, correlation energyis taken into account in DFT, albeit in an approximateway. Also it can be noted from Figure 1 that for strongerinteraction ( η ∼ E c ) obtained by these two methodsare displayed in Figure 2 as a function of the strengthof dipolar interaction η . We observe from this Figurethat these two methods yield same sign (negative) forthe correlation energy and display similar trend with re-spect to variation of interaction strength η . However,the LDA based expression for correlation energy under-estimates the magnitude of the correlation energies ascompared to those calculated from Hyllleraas-like wavefunction approach for weaker strength of interaction. Asthe strength of interaction is increased the contribution of TABLE II. Ground state energy as a function of number of basis function in Eqs. (14).Number of basis function Ground state energy from DFT Ground state energy from HF1 −− . .
763 2 . .
755 2 . .
754 2 . .
755 2 . η = 3 a dd /l . All numbersare in harmonic oscillator unit. correlation energy obtained from DFT based method fastapproaches to that calculated from wave function basedmethod and for η ≥ .
00 the magnitude of correlationenergy is overestimated by the LDA based expression.Having discussed the general behaviour of total en-ergy and the contribution of correlation energy to it wenow feel it is worth here to discuss the energetics for thecase of recently achieved degenerate sample of fermionicKRb molecules with very large value of dipole moment[15]. For this system the value of 3 a dd is about 6000 a B ,where a B is the Bohr radius [10]. When converted toharmonic oscillator unit with a typical value of oscillatorlength l = 1 × − m, this corresponds to a strengthof the dipole interaction η = 0 . E GS = 2 . . E GS = 2 . FIG. 2. The variation of correlation energy calculated by cor-related wavefunction (black) and DFT (red) as a function ofinteraction strength η = 3 a dd /l . All numbers are in har-monic oscillator unit. to be -0.082. On the other hand, the value of correla-tion energy obtained by employing LDA based expres-sion Eq. (12) yields E c = − . IV. CONCLUSION
In this paper we have calculated the ground state en-ergy of a harmonically confined two interacting dipo-lar fermions by variational method using a Hylleraas-like correlated wave function. We study the variationof ground state energy as a function of strength of inter-action between the two dipoles. In order to estimate themagnitude of correlation energy we have also performedcalculation of the ground state energy of dipolar fermionswithin HF approximation. By employing recently de-rived expressions for the exchange and correlation energyfunctionals of homogeneous dipolar gas within LDA wehave also calculated DFT based ground state energy oftwo trapped dipolar fermions. The comparison of wavefunction based results with the ones obtained via DFTclearly shows that two methods exhibit similar trend inthe variation of ground state and correlation energies.However, both the energies are underestimated by LDAbased calculations as compared to the results obtainedvia Hylleraas-like correlated wave function. We have alsofound that the difference between the results obtained byLDA and wave function based methods decreases withincreasing interaction strength and they are very close for η ≈ .
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