hydrogen-helium plasma via path integral Monte Carlo. D. Wendland,1, 2 V. Ballenegger,2 and A. Alastuey1. 1)Laboratoire de Physique, Ecole Normale ...
Quantum partition functions of composite particles in a hydrogen-helium plasma via path integral Monte Carlo D. Wendland,1, 2 V. Ballenegger,2 and A. Alastuey1 1) Laboratoire
de Physique, Ecole Normale Sup´erieure de Lyon, UMR CNRS 5672, 46 all´ee d’Italie, 69364 Lyon Cedex 07, France 2) Institut UTINAM, Universit´ e de Franche-Comt´e, UMR CNRS 6213, 16, route de Gray, 25030 Besan¸con Cedex, France (Dated: 6 October 2014)
We compute two- and three-body cluster functions that describe contributions of composite entities, like hydrogen atoms, ions H− , H+ 2 and helium atoms, and also charge-charge and atom-charge interactions, to the equation of state of a hydrogen-helium mixture at low density. A cluster function has the structure of a truncated virial coefficient and behaves, at low temperatures, like a usual partition function for the composite entity. Our path integral Monte Carlo calculations use importance sampling to sample efficiently the cluster partition functions even at low temperatures where bound state contributions dominate. We employ also a new and efficient adaptive discretization scheme that allows one not only to eliminate Coulomb divergencies in discretized path integrals, but also to direct the computational effort where particles are close and thus strongly interacting. The numerical results for the two-body function agree with the analytically known quantum second virial coefficient. The three-body cluster functions are compared at low temperatures with familiar partition functions for composite entities.
I.
INTRODUCTION
Internal partition functions for composite particles in a plasma, like atoms, molecules or ions with inner degrees of freedom, determine reaction constants at equilibrium and are important ingredients in equation of state (EOS) calculations. Within the chemical picture, such functions are introduced on phenomenological grounds where the longrange Coulomb divergencies are removed via ad-hoc regularizations, as for example in the Brillouin-Planck-Larkin partition function for the hydrogen atom.1–3 In this paper, we consider a hydrogen-helium mixture described in the physical picture as a Coulomb gas of point electrons, protons and α particles, and we focus on the contributions to the EOS of composite entities, made with three elementary particles, namely ions H− , H+ 2 and helium atoms. The contributions of three particle effects are described at low densities by the third virial coefficient. Virial coefficients would naively be divergent in a system with longrange Coulomb forces, but are in fact convergent thanks to the screening provided by ionized charges.4–7 This screening erases in particular the contributions of infinitely extended Rydberg states. A proper estimation of internal partition functions for composite entities requires thus to deal simultaneously with recombination and screening effects. The Screened Cluster Representation (SCR)8 , presented synthetically in refs9,10 , achieves precisely such a description within the physical picture. It establishes that the usual virial coefficients are replaced, in a system with long-range forces, by truncated virial coefficients, the so-called cluster functions, and further contributions associated with screened
Coulomb interactions between these clusters of particles. The cluster functions converge even in the vacuum, where particles interact via the bare Coulomb potential.8 The large-distance integrability is ensured by regularizing truncation terms whose origin can be traced back to screening. The cluster functions are entirely determined by the spectral properties of the associated few-body Coulomb Hamiltonians and depend, in the vacuum, solely on the inverse temperature β and on fundamental constants. They contain contributions from both bound and diffusive states. At low temperatures, a cluster function tends to the internal partition function of the entity described by the cluster, i.e to the sum of Boltzmann weights e−βEj of all energy levels Ej of the composite entity identified by those bound states. The ground state, with energy E0 = min(Ej ) < 0, provides, in the low temperature limit β → ∞, the most important contribution as it grows exponentially fast (exp(−βE0 )). The Screened Cluster Representation has been used to study van der Waals interactions in the presence of free charges11 and to compute the first corrections to the Saha EOS for a partially ionized hydrogen plasma, in a low-density and low-temperature limit appropriate to describe the Saha regime.12 Those corrections involve the cluster functions in the vacuum for two-, three- and four-body clusters, as well as further functions describing in particular atom-charge interactions. They have been computed numerically, in some approximations, in ref.13 . Beyond the two-body function, the cluster functions are currently known only at very low temperatures. An accurate calculation of those functions at higher temperatures, where many states in the cluster are excited or it is even dissociated, is highly desirable to improve our knowledge of the thermodynamical properties of the hydrogen gas and of the hydrogen-helium mixture.13 The goal of the present paper is to calculate, from first principles, the three-body cluster functions, i.e. truncated quantum virial coefficients, appearing in the EOS of the hydrogen-helium mixture. We compute also the contributions to the EOS from atomcharge interactions. We employ Path Integral Monte Carlo (PIMC) methods14,15 without introducing any other approximation than the use of discretized paths. The calculations are thus exact, apart from the statistical uncertainties/errors inherent to the Monte Carlo approach and from the finite discretization. PIMC methods have been applied previously to compute internal partition functions of model systems16 and of small to medium-size molecules17,18 . Our calculations differ from those works (see also19–22 ) for several reasons. First, a cluster function is not a plain partition function, to which it reduces only at low temperatures. This prevents us from applying numerical methods specifically designed to compute free-energy differences. Secondly, those works aimed to calculate molecular partition functions and introduced therefore ad-hoc cutoffs to restrict the path integral calculation to molecular states only, excluding diffusive states. No such cutoff appears in our calculations, which need to capture not only molecular contributions but also large-distance ionized contributions. Thirdly, previous works used the Born-Oppenheimer approximation and an effective potential between the nuclei determined from densityfunctional theory, similarly to works23,24 on calculations of the third virial coefficient in an atomic helium gas. Here, the electronic problem is neither solved separately, namely no Born-Oppenheimer approximation is made, nor do we assume that the temperature is sufficiently low for the electrons to be in the ground state. Rather, we compute the partition function, or more generally the truncated virial coefficient, of the full threebody Coulomb system by sampling directly, at finite temperature, over both nuclei and electron paths interacting via the Coulomb potential. This increases significantly the computational cost, but the results are then approximation-free and also valid at high 2
temperatures where electrons can be highly excited or even ionized, i.e. in states not described by a ground state potential energy surface. An algorithm to compute few-body cluster functions must overcome several difficulties. The quantity of interest is not an expectation value over some probability distribution, since the cluster function has the structure of a virial coefficient, which is not necessarily positive. Thus, the standard Metropolis-Hastings Monte Carlo algorithm cannot be applied directly as in usual PIMC simulations. We employ a partially-adaptive importance sampling approach, and well-chosen coordinates, to sample efficiently the high-dimensional configuration space of three mutually interacting paths. At the considered temperatures, electrons behave quantum mechanically and interact strongly with the surrounding particles, so that bound states can be formed. The aforementioned works used free-particle sampling since only the almost classical nuclei paths intervene, once the Born-Oppenheimer approximation has been used and the electronic problem is solved separately. For temperatures below the dissociation energy, free-particle sampling cannot capture properly bound state electronic paths. In our proposed algorithm, the sampling of electron paths is governed by a simple importance sampling function that mimics the exact two-body density matrix. This approach allows us to sample efficiently the truncated virial coefficient, and other functions if wished, even at quite low temperatures where contributions from bound states dominate. The divergence of the Coulomb potential at the origin requires special care in discretized path integrals. Notice that this difficulty is not present if the Born-Oppenheimer approximation is used since the effective potential between nuclei is regular at the origin. The common solution to this problem consists in using an effective two-body potential.15,25,26 This potential can be derived from the analytically known two-body Coulomb density matrix or one can use high-temperature approximations25,26 . Paths need to be highly discretized in order to capture the correlations, beyond the two-body ones, between the various interacting particles. In PIMC simulations, paths are usually discretized uniformly up to a maximum level.15 The chosen discretization might be too low for strongly interacting close-by particles, while being too high for far-away particles. We avoid this by introducing an adaptive discretization scheme in which portions of paths that are strongly interacting are more discretized than portions where the interaction is weak, thereby concentrating the numerical effort in the region where it is most needed. It has to be emphasized that for finely discretized paths, the various effective potentials all reduce, at leading order, to the straight-line Coulomb potential.15 Thus, we employ in our calculations the straight-line potential in combination with the adaptive discretization of paths. When discretizing finely portions of paths that interact strongly, all short-distance Coulomb divergences are effectively eradicated. Our importance sampling approach, together with the adaptive discretization scheme, allows us to calculate the three-body cluster functions precisely, from high temperatures, where the path integral approach is particularly adequate, down to quite low temperatures where the cluster function tends to known partition functions of composite particles. This paper is organized as follows. In Sct. 2, we recall the origin and the definition of the cluster functions in the Screened Cluster Representation of the hydrogen-helium mixture. For pedagogical reasons, those functions are first expressed in the operatorial language in terms of truncated traces over Mayer virial operators. Their equivalent path integral expressions, used in the numerical calculations, are then written down explicitly. Sct. 3 presents our Monte Carlo algorithm for computing such few-body cluster functions. The numerical results for the functions describing contributions to the EOS of H− , H+ 2 3
and helium atoms, as well as atom-charge interactions, are shown in Sct. 4. Comparisons are made, at low temperatures, with semi-phenomenological partition functions available in the literature. Some concluding remarks are made in Sct. V.
II.
FEW-BODY CLUSTER FUNCTIONS
A.
The hydrogen-helium mixture within the SCR
In the physical picture, the hydrogen-helium mixture is described as a quantum Coulomb system of point protons (p), electrons (e) and α particles governed by the Coulomb Hamiltonian HNp ,Ne ,Nα = −
N X h ¯2 1 X eγi eγj . ∆i + 2m 2 |r − r | γ i j i i=1 i6=j
(1)
Here N = Np + Ne + Nα denotes the total number of particles and γi the species (e, p or α) of the i-th particle with mass mγi , charge eγi and spin σγi . The particles are enclosed in a box with volume Λ, in contact with a thermostat at temperature T and a reservoir of particles that fixes the chemical potentials {µγ }, or equivalently the fugacities zγi = exp(βµγi ). For a short-ranged interaction, the second virial coefficient in the expansion of the pressure in terms of the fugacities would involve traces of the form12 1 Tr[e−βHNp ,Ne ,Nα ]Mayer , Λ
(2)
over properly anti-symmetrized states, in for instance position and spin space, of Mayer virial operators [e−βHNp ,Ne ,Nα ]Mayer . Examples of two-body Mayer operators are [e−βHp,e ]Mayer = e−βHp,e − e−βHp e−βHe ,
(3)
[e−βH2p ]Mayer = e−βHp1 ,p2 − 21 e−βHp1 e−βHp2 ,
(4)
with an obvious short-hand notation for Hamiltonians HNp ,Ne ,Nα . Analogous expressions hold for Mayer operators associated to other two-body clusters like (α, e) or (2e)=(e1 ,e2 ). Similarly, the Mayer operator for the three-body cluster (2p,e) reads (0)
[e−βH2p,e ]Mayer = e−βH2p,e −e−βH2p e−βHe − 21 e−βHp1 ,e e−βHp2 − 21 e−βHp2 ,e e−βHp1 +e−βH2p,e , (5) (0)
where H2p,e = Hp1 +Hp2 +He is the free Hamiltonian. In a system with long-range forces, the trace (2) is divergent. These divergences are suppressed by screening effects. Three different techniques can be used to account for screening effects in a quantum plasma: the effective-potential method introduced by Morita27,28 , RPA chain resummations in many-body perturbation theory within the thermodynamic Green function formalism7 , and Abe-Meeron chain resummations in the path integral representation of the system, known as the loop or polymer formalism29 . These formalisms, though quite different, lead to the same formula for the virial EOS, as it should.4–7,30 The Screened Cluster Representation (SCR), derived in the polymer formalism, is a suitable technique 4
able to deal simultaneously, and exactly, with recombination and screening effects at arbitrary orders in the fugacity series. This approach is efficient for handling recombination effects since it is not perturbative with respect to the interaction, contrary to the Green function formalism. The SCR provides explicit formulas for all terms contributing at given order in the fugacity series for any thermodynamic quantity. The expressions of the SCR are represented by Mayer-like diagrams built with clusters of particles interacting via an effective screened interaction φ resulting from chain resummations. All long-range Coulomb divergences are removed via the introduction of φ, which is the quantum analog of the Debye potential.31 A cluster C is a group of strongly interacting particles that may form bound states. In the zero-density limit, the contribution of a cluster in the SCR is given by a truncated virial coefficient, known as a cluster function ZC (β). These functions can be interpreted, at temperatures below the dissociation temperature of the cluster, as an internal partition function of the cluster. They are finite thanks to truncation terms, inherited from screening effects, that ensure large-distance integrability. The truncation terms do not introduce any arbitrariness because the associated counter-terms are also included in the SCR: only the full contribution of ZC (β) plus that of the corresponding counter-terms has an unambiguous sense. The counter-terms appear as diagrams describing screened interactions between all possible dissociation products of C. The SCR has been used to derive the first corrections to the ideal Saha EOS in a partially ionized hydrogen gas in a low-density and low-temperature limit appropriate to describe the Saha regime.12,13 In this limit, known as Scaled Low Temperature (SLT limit), those corrections to the Saha equation can be ordered, providing the SLT equation of state.12,13 The terms in the SLT EOS involve the cluster functions in the vacuum for two-, three- and four-body clusters, as well as further functions describing charge-charge, charge-atom and atom-atom interactions (see Sct. II-D in ref13 ). We recall that only the sum of the three-body cluster functions and of the associated screened interactions between dissociated products has an intrinsic meaning, free of any arbitrariness. It is precisely this sum that appears in the SLT EOS. The contribution of a cluster C(Np , Ne , Nα ) can be written as a trace over a truncated Mayer operator ZC (β) =
(2πλ2Np ,Ne ,Nα )3/2 Λ
Tr[e−βHNp ,Ne ,Nα ]TMayer .
(6)
The operators [exp(−βHNp ,Ne ,Nα )]TMayer are similar to plain Mayer operators, for instance eqs. (3)-(5), but involve further truncation terms (see formulas in the next subsections) that ensure large-distance integrability. The factor (2πλ2Np ,Ne ,Nα )3/2 , where λNp ,Ne ,Nα = (¯ h2 β/(Np mp + Ne me + Nα mα ))1/2 is the de Broglie thermal wavelength associated with the center-of-mass motion of the cluster, is included in definition (6) so that ZC (β) is dimensionless and linked to a sum over internal states only. In the following we study clusters build by two particles, namely (p,e) and (α,e), describing the formation of atoms H and ions He+ , as well as charge-charge interactions. Furthermore we consider the three-body clusters (p,2e), (2p,e) and (α,2e), which account for the formation of ions H− , H+ 2 and atoms He and for interactions between their dissociation products. 5
B.
Two-particle clusters
The proton-electron cluster function Zp,e (β), which describes free hydrogen atoms and ionized p-e interactions, is defined by (2πλ2H )3/2 Tr[e−βHp,e ]TMayer = 4(2πλ2H )3/2 Zp,e (β) = Λ
Z
dre DT (0, r e ).
(7)
The factor 4 arises from the proton and electron spins, r p = 0 from translational invariance, and (8) DT (r p , r e ) = h0, r e |[e−βHp,e ]TMayer |0, r e i, where [e−βHp,e ]TMayer
−βHp,e
=e
−βHp|e
−e
β
Z i h 1 β I I + 1− dτ1 dτ2 T Vp−e (τ1 )Vp−e (τ2 ) 2! 0 0 Z i h 1 β I I I − dτ1 dτ2 dτ3 T Vp−e (τ1 )Vp−e (τ2 )Vp−e (τ3 ) . (9) 3! 0
Z
I (τ1 ) dτ1 Vp−e
The Hamiltonian Hp,e = Hp|e + Vp−e contains the purely kinetic part Hp|e and the interI (τ ) denotes the Coulomb operator in particle potential Vp−e = −e2 /|r e − r p |, while Vp−e the interaction picture and T the time ordering operator. The cluster function Zp,e (β) can be expressed in terms of Ebeling’s virial Q(xpe ) function for which an exact series representation is available (see eq. (47)). The cluster function Zα,e (β), describing ion He+ , is defined as (7) but where the proton is replaced by an α particle. A simple scaling shows that Zα,e (β) is given by the same series (47) in which x = xαe . C.
Three-particle clusters
• Cluster (2p,e)
The two proton and one electron cluster function Z2p,e (β) is defined, similarly to (7), as (2πλ2H+ )3/2
2 Tr[e−βH2p,e ]TMayer , (10) Λ where the truncation terms are generated according to the rules of the Screened Cluster Representation.8 The trace (10) can be split into a direct and an exchange term Z Z h i 3/2 T T 2 dr p dr e 4D (0, r p , r e ) − 2E (0, r p ; r e ) , (11) Z2p,e (β) = (2πλH+ )
Z2p,e (β) =
2
where we identify DT (r 1 , r 2 , r 3 ) = hr 1 , r 2 , r 3 | DT |r 1 , r 2 , r 3 i, T
T
E (r 1 , r 2 ; r 3 ) = hr 2 , r 1 , r 3 | E |r 1 , r 2 , r 3 i. 6
(12) (13)
The trace of the pure Mayer operator (5) (without long-range truncation terms), taken over globally anti-symmetrized states in position-spin space, leads to the identifications (0)
D = e−βH2p,e − e−β(H2p +He ) − e−β(Hp1 ,e +Hp2 ) − e−β(Hp2 ,e +Hp1 ) + 2e−βH2p,e ,
(14)
E = e−βH2p,e − e−β(H2p +He ) .
(15)
The truncation manifests itself by the subtraction of further terms (see eq. (4.10) in the SCR.8 ) The full expression of the truncated direct term DT is given in appendix A. For the exchange contribution, we have T
−βH2p,e
E =e
−βH2p|e
−e
β
Z h i 1 β I I + 1− (τ2 ) (τ1 )V2p−e dτ1 dτ2 T V2p−e 2! 0 0 Z β i h 1 I I I − dτ1 dτ2 dτ3 T V2p−e (τ1 )V2p−e (τ2 )V2p−e (τ3 ) , (16) 3! 0 Z
I dτ1 V2p−e (τ1 )
where H2p|e = H2p + He (the electron does not interact with the 2 protons), V2p−e = I (τ ) = exp(τ H2p|e )V2p−e exp(−τ H2p|e ). Vp1 −e + Vp2 −e and V2p−e • Clusters (p,2e) and (α,2e)
The cluster functions Zp,2e (β) and Zα,2e (β) are defined in the same way as Z2p,e (β) [eq. (10)] with obvious substitutions for particle species.
D.
Path integral formula for cluster functions
In the genuine Screened Cluster Representation, the statistical weights and bonds are expressed in terms of functional integrals over the paths associated with the particles forming the clusters. The expressions for ZC (β) as traces (6) over truncated Mayer operators are obtained by applying backwards the Feynman-Kac formula.32 The original expressions in terms of path integrals are written explicitly in the following for two-and three-particle clusters.
• Proton-electron cluster function
The matrix element (8) that appears in (7) is obtained from the path integral expression Z 1 (βV )2 (βV )3 T −βV (Lp ,Le ) D(ξ )D(ξ ) e − 1 + βV − D (r p , r e ) = + , p e (2πλ2p )3/2 (2πλ2e )3/2 2 3! | {z } B2T (Lp ,Le )
(17) R where D(ξ) denotes path integration over a Brownian bridge ξ(s), s ∈ [0, 1], (ξ(0) = ξ(1) = 0) with normalized Gaussian measure D(ξ). Here Lp (Le ) is a loop made up of a single proton (electron), characterized by its position r p (r e ) and by its shape ξ p (s) 7
(ξ e (s)). The attractive Coulomb interaction between these two one-particle loops is given by Z 1 1 2 ds . (18) V (Lp , Le ) = −e |r e + λe ξ e (s) − r p − λp ξ p (s)| 0 • Three-particle cluster function
The genuine path integral expression of the direct function DT (r 1 , r 2 , r 3 ) [eq. (12)] is Z 1 1 1 T D (r 1 , r 2 , r 3 ) = D(ξ 1 )D(ξ 2 )D(ξ 3 ) B3T (L1 , L2 , L3 ), 2 3/2 2 3/2 2 3/2 (2πλ1 ) (2πλ2 ) (2πλ3 ) (19) where B3T (L1 , L2 , L3 ) is identical to expression (A1) of DT in Appendix A in which one sets a = −βV (L1 , L2 ), b = −βV (L1 , L3 ) and c = −βV (L2 , L3 ). (20) The function DT (r 1 , r 2 , r 3 ) takes a quite simple form in the path integral language (it involves simple combination of loop-loop interactions as expressed by eqs. (A1) and (20)), in contrast to the operatorial language in which each truncation term involves more cumbersome time ordered Coulomb interactions as for example in eq. (A3). The exchange term [eqs. (13) and (16)] of the three-particle cluster function is given by Z 1 2 T E (r 1 , r 2 ; r 3 ) = D(ξ 1 )D(ξ 2 )D(ξ 3 )e−(r2 /λ1 ) e−βVself (L12 ) B2T (L12 , L3 ). 2 3 2 3/2 (2πλ1 ) (2πλ3 ) (21) Here L12 is a 2-particle closed loop made with the two open paths [r 12 (s) = (1 − s)r 1 + sr 2 + λ1 ξ 1 (s) and r 21 (s) = (1 − s)r 2 + sr 1 + λ1 ξ 2 (s)] of the exchanged particles 1 and 2. The interactions between the particles along their paths which appear in (21) are R1 2 Vself (L12 ) = 0 ds e |r 12 (s) − r 21 (s)|−1 and Z 1 eα1 eα3 eα2 eα3 V (L12 , L3 ) = ds + , (22) |r 12 (s) − r 3 (s)| |r 21 (s) − r 3 (s)| 0 with r 3 (s) = r 3 + λ3 ξ 3 (s). III. A.
ALGORITHM Position of the problem
A numerical method for computing few-particle cluster functions must overcome several difficulties. We underline that both short and long distance effects of the Coulomb potential must be treated simultaneously when computing ZC (β). A first difficulty arises because the cluster functions do not have the standard form of an expectation value, as required for using the Metropolis-Hastings algorithm. Instead, ZC (β) takes the form of an integral over the whole phase space, like a partition function, whose calculation is significantly more challenging. The cluster functions are moreover virial partition functions that differ from plain partition functions, the integrand being not a pure exponential, but a truncated exponential. The subtracted terms, arising either from the Mayer truncation 8
or from the long-range truncation inherited from screening effects, can cause ZC (β) to become negative. Standard methods for computing partition functions, such as free energy perturbation, Bennett’s Monte Carlo or thermodynamic integration are not applicable to such virial partition functions. The free-energy perturbation method can be used to sample Mayer integrals in classical systems33 , but its generalization to a quantum system poses the problem of the choice of the reference system (for which the exact cluster function needs to be known and to be similar to that of the system of interest). A thermodynamic integration approach is possible in the two-particle case, where one can compute Zp,e (β) by sampling efficiently the bond B2T (Lp , Le ) at any temperature via a Markov chain random walk, but such an approach does not generalize conveniently to three-particle clusters because the truncated Mayer function B3T (L1 , L2 , L3 ) is not always positive (and cannot therefore be used as equilibrium probability distribution of a Markov chain). Without the benefit of importance sampling brought by Markov chain Monte Carlo, one must develop a numerical method that is able to sample sufficiently well the integrand, in particular at low temperatures where small regions in configuration space can lead to exponentially large contributions arising from the exponentials in eqs. (17), (19) and (21). Those large contributions at low temperatures occur due to the formation of bound states, and capturing them correctly in a numerical path integral calculation is not trivial.
B.
Adaptive discretization and effective action of paths
Paths integrals are usually discretized uniformly in the imaginary time interval [0, β] by using the convolution property of the density matrix at times βk = kτ with k = 1, ..., M and τ = β/M . For a single particle of mass m in a potential V (r), the density matrix ρ(r, r; β) = hr|exp(−β(H0 + V ))|ri is rewritten for instance as an integration over products of density matrices at a higher temperature τ : Z ρ(r, r; β) = dr 1 ...dr M −1 ρ(r, r 1 ; τ )...ρ(r k , r k+1 ; τ )...ρ(r M −1 , r; τ ). (23) The Feynman-Ka¸c formula provides the path integral (PI) representation Z R1 ρ(r 1 , r 2 ; τ ) = ρfree (r 1 , r 2 ; τ ) D(ξ) e−τ 0 ds V (r1,2 (s)+λτ ξ(s))
(24)
where ρfree (r 1 , r 2 ; τ ) = (2πλ2τ )−3/2 exp(−(r 1 −r 2 )2 /2λ2τ ) is the free particle density matrix, r 1,2 (s) = (1−s)r 1 +sr 2 is the straight-line path and λ2τ = h ¯ 2 τ /m. Expressing the average of the exponential in (24) with the cumulant expansion, the high-temperature behavior of the density matrix is seen to be ρ(r 1 , r 2 ; τ ) ≈ ρfree (r 1 , r 2 ; τ ) exp −τ Veff (r 1 , r 2 ; τ ) + O(τ 3 ) . (25) where the effective potential Z Veff (r 1 , r 2 ; τ ) =
Z D(ξ)
1
ds V (r 1,2 (s) + λτ ξ(s)) 0
9
(26)
corresponds to the average of the genuine potential over free paths connecting r 1 to r 2 . For the Coulomb interaction, the cumulant effective potential reads15,25 Z 1 |r (s)| eγi eγj 1,2 cumul. ds (r 1 , r 2 ; τ ) = erf p Veff . (27) 2 |r 1,2 (s)| 2λτ s(1 − s) 0
Combining (23) and (25) yields the well known discretized path integral " M −1 # Z M −1 X (r k − r k+1 )2 X 3M ρ(r, r 0 ; β) ≈ (2πλ2τ )− 2 dr 1 ...dr M −1 exp − −τ Veff (r k , r k+1 ; τ ) 2λ2τ k=0 k=0 (28) β β 0 (r 0 = r, r M = r ). If the Trotter decomposition exp(−β(H0 +V )) = limM →∞ (exp(− M H0 ) exp(− M V ))M had been used to discretize the PI, eq. (28) would have been obtained but with the cumul. smoothed potential Veff replaced by the genuine Coulomb potential. It would then be divergent since the exponentiated Coulomb singularity is not integrable. By contrast, cumul. remains bounded everywhere and eq. (28) converges towards the continuous result Veff as 1/M 2 .15 Other bounded effective potentials have been proposed for performing path integral simulations of coulombic systems, such as partially averaged potentials26,34 and pair also the exact effective pair potential Veff . The latter potential is inferred from the exact two-body density matrix via pair
ρ(r 1 , r 2 ; τ ) = ρfree (r 1 , r 2 ; τ ) e−τ Veff
(r 1 ,r 2 ;τ )
.
(29)
pair When discretizing the PI for the density matrix of a many-particle system using Veff , 2 errors still scale as 1/M , due to the correlations between the paths beyond the two-body ones.15 To capture those correlations, a large number M of time slices needs to be used, the exact potential reducing then to the cumulant potential.
In this paper, we introduce a new way of discretizing Coulombic path integrals that does not require the effective potential to remain bounded thanks to an adaptive nonuniform discretization of paths. Note that the M time intervals in eq. (23) need not all have the same length τ . In the limit M → ∞, where eq. (28) becomes exact, the thermal wavelength λτ tends to 0 and the cumulant potential (26) reduces to the semi-classical straight-line potential −r 2 | Z 1 2arctanh |r|r11|+|r 2| Vline (r 1 , r 2 ) = ds V (r 1,2 (s)) = eγi eγj . (30) |r 1 − r 2 | 0 which is integrable but unbounded. The unphysical exponentiated 1/r singularity can be avoided by discretizing the paths, via recursive bisections, until the exponentiated action e−τk Vline (rk ,rk+1 ) < 1
(31)
of all path segments are smaller than one (one fixes in practice a threshold value < 1). Here τk = β/(2Lk ) is the length of the time interval associated to the path segment k (Lk is the discretization level). Notice that the stopping criterion (31) relies on Vline rather than on the cumulant potential to ensure that the discretization of the paths is not stopped prematurely. The absolute value |Vline | is indeed an upper bound for 10
path segment to be further discretized
FIG. 1. A path segment (r0 , r1 ) of an electron (blue circle) near a proton (red circle). The use of the cumulant potential instead of Vline in the stopping criterion would lead to such a path segment not being further refined, because of the smoothing embedded in the cumulant potential, even though the potential is large in-between the path segment’s end points. cumul. |. Had we used the primitive action −τk (V (r k ) + V (r k+1 ))/2 or the cumulant |Veff potential in the criterion, the algorithm could stop refining paths early even though the particle passes right next to the Coulomb singularity (see Fig. 1). Portions of paths that approach the singularity are thus discretized to a degree that is inversely proportional to the closest distance of approach to the singularity. The Coulomb divergence, which does not exist in the continuous path integral, is effectively eradicated via this adaptive discretization, even in the case where the genuine Coulomb potential (i.e. the primitive action) is used in eq. (28). Most importantly, the computational effort is directed, as it should, in regions where the potential is strong. The adaptive discretization avoids furthermore the systematic error associated to a uniform discretization. This adaptive scheme is simple to implement, provides accurate results (see Sct. IV A) and agrees with the physical discussion of how a collapse of paths on the Coulombic singularity is prevented by the entropy associated with highly discretized paths.32
When the paths are discretized adaptively, a good accuracy is obtained by computing the path actions with the simple effective potential Vline , i.e. the high-temperature limit of the cumulant potential. The choice of this action allows not only to reuse the value of Vline already computed when checking the stopping criterion (31), but, more importantly, it provides results with the same accuracy as with the full cumulant potential. To understand this surprising result, which is a consequence of the harmonicity of the Coulomb potential, let us compare the straight-line potential with the cumulant potential. The latter can be written in Fourier space as Z 1 Z 1 2 2 1 cumul. ds Veff (r 1 , r 2 ; τ ) = dk eik·r1,2 (s) e− 2 k λτ s(1−s) Vˆ (k) (32) 3 (2π) 0 cumul. In the case of a short-ranged potential, Veff = Vline + O(λ2τ ) at high temperatures, or equivalently at a high discretization level (λτ → 0). When evaluating the action cumul. −τ Vline instead of −τ Veff , one neglects terms of order O(1/M 2 ) since λ2τ = O(1/M ) assuming a uniform discretization. Since this error is made M times, once in each slice, the resulting global error is O(1/M ) when paths actions are computed with Vline . However, this argument no longer holds if the potential is long-ranged because the coefficient in the O(λ2τ ) term becomes ill-defined in general. This coefficient vanishes in fact in the case of the Coulomb potential, because of its harmonicity, and the errors decay then faster than 1/M . When the paths are discretized adaptively, the argument of the error function
11
in the Coulombic cumulant potential (27) is almost always large. Indeed Nβi e2 /r(s) < p (adaptive discretization) implies λτ /r(s) < r(s)aB /r(s) and hence λτ /r(s) < if r(s) ≥ aB (the latter condition is satisfied most of the time in molecular systems).35 The Coulombic cumulant potential can then be expanded as ! p 2 Z 1 2 e e 2λ s(1 − s) |r (s)| γ γ 1,2 τ cumul. 1 − exp − 2 Veff ' ds i j ·P (33) |r (s)| 2λ s(1 − s) |r 1,2 1,2 (s)| 0 τ | {z } Vline
3
where the polynomial P (x) = √xπ − 2x√π + ... arises from the large-x expansion of erf(x). From eq. (33), Vline deviates from the cumulant potential only by exponentially small terms which have no impact on the order of the errors (recall that they scale as 1/M 2 when using the cumulant potential). When using Vline together with the adaptive discretization, the errors are therefore expected to scale as 1/hM i2 thanks to the harmonicity of the Coulomb potential. The effective potential associated to the exact pair-action could be used in place of Vline , but it is more expensive to evaluate and provides errors that scale similarly as 1/M 2 (though presumably with a smaller prefactor). C.
Importance sampling of paths at low temperatures
After discretization of the particle paths r i (s) (i = 1, ..., Np + Ne + Nα ) into their root point r i,0 = r i (0) = Ri and Mi − 1 intermediate points r i,k (k = 1, ..., Mi − 1), a cluster function ZC (β) becomes a multidimensional integral of a function f ({Ri }, {r i,k }; {τi,k }). The number of points Mi ≥ 1 in the discretized path for particle i is determined by the stopping criterion (31) and τi,k = β/(2Li,k ) is the length of the time interval between points r i,k and r i,k+1 . For instance, the discretization of eq. (17) expressed in relative coordinates takes the form Z Z f (R; β) = D(ξ)f (R, ξ; β) ≈ dr 1 . . . r M −1 f (R, {r k }; {τk }), (34) with f (R, {r k }; {τk }) =
M −1 Y k=0
1 3/2 − e 2πλ2τk
(r k −r k+1 )2 2λ2 τk
e−τk Vline (rk ,rk+1 ) − . . . ,
(35)
where r 0 = r M = R, λ2τk = h ¯ 2 τk /mpe , and the dots stand for the truncation terms P3 n 2 3/2 n=0 (−τk Veff ) /n!. In (34), function f (R; β) is equivalent to the matrix element (2πλ ) rel hR|e−βH + . . . |Ri of the Hamiltonian H rel describing relative the motion of a protonelectron pair. The integrand (35) is sampled at low temperatures from a well chosen sampling function P{Ri },β ({r i,k }) for fixed root points {Ri } and temperature β, while the remaining integrations over those root points are performed by an adaptive importance sampling scheme. At fixed {Ri }, the integration over the inner path points {r i,k }, Z Y f ({Ri }, {r i,k }; {τi,k }) dr i,k = f ({Ri }; β), (36) i,k
12
is performed by rewriting the integral as an average f ({Ri }; β) = f ({Ri }, {r i,k }; {τi,k })
N{Ri },β P{Ri };β ({r i,k })
,
(37)
P
over a well chosen probability distribution P{Ri };β ({r i,k }), with known normalization constant N{Ri },β , and by evaluating this average by direct importance sampling according to the distribution P . The simplest choice for P is the free-particle distribution15 , i.e. eq. (35) where Veff is set to zero, from which one can sample efficiently, since it is Gaussian, using a L´evy construction14 . Free-particle sampling is adequate at high temperatures, but fails to sample the integrand sufficiently √ well at low temperatures, due to the formation of bound states. The extension λ ∝ β of the free Brownian paths becomes very large at low temperatures whereas the size of bound states is of the order of the Bohr radius aB λ, leading to undersampling of the dominating region. By this undersampling, the variance of the Monte Carlo (MC) estimator grows quickly when the temperature is decreased, so that the MC calculation converges extremely slowly. We recall that this problem is avoided in usual PIMC simulations by the Metropolis-Hastings algorithm in which the most contributing regions are explored preferentially, but such an approach does not seem to be applicable to the computation of virial partition functions as stated previously. Since the truncation terms in (35) are well sampled by free-particle paths, we focus on the sampling of the exponential terms. Several approaches, in particular matrix squaring and direct summation over eigenfunctions, see15 , allow to compute efficiently the two-body Coulomb density matrix (35), but they are limited to the two-particle case. We introduce here a sampling of paths, based on a L´evy construction, that is adequate to compute, even at quite low temperatures, few-body virial partition functions. The L´evy construction consists in generating paths via bisections. One begins with the two fixed end points, for instance r 0 at “time” 0 and r M = r 0 at “time” β, and samples a bisecting point r 1 at time β/2 according to some distribution Tr0 ,r0 ,L=1 (r 1 ) given below. Once that point has been chosen, one bisects the two new intervals [0, β/2] and [β/2, β], generating points at “times” β/4 and 3β/4. In our implementation, one recursively bisects all intervals for which β|Vline (r k , r k+1 )|/(2L ) is not small in comparison to < 1. The optimal sampling distribution for a point r, for given end-points points x1 and x2 of a path segment at recursion level L, is, in the two-particle case, L
L
Tx∗1 ,x2 ,L (r) ∝ hx1 |e−βH/2 |rihr|e−βH/2 |x2 i.
(38)
Sampling from this optimal distribution is not feasible since the normalization constant cannot be computed quickly.15 The idea is to sample from a simple, easily normalizable, approximation for T ∗ (r) that mimics, at all temperatures, the optimal distribution (38). The low-temperature behavior of T follows from inserting the spectral decomposition of exp(−τ Hpe ) in (38) and from discarding all excited state contributions βE0
Tx∗1 ,x2 ,L (r) ∝ e− 2L−1 |Ψ0 (r)|2 , β → ∞, (39) √ 3/2 where Ψ0 (r) = exp(−r/aB )/( πaB ) is the ground-state wave function and E0 = EH is the ground-state energy of the hydrogen atom. The probability distribution T ∗ (r) behaves at high temperatures (or L → ∞) as −
Tx∗1 ,x2 ,L (r) ∝ e
(xm −r)2 λ2 /2L
13
,
β → 0,
(40)
where xm = (x1 + x2 )/2. Notice that (40) is the free-particle sampling distribution. A sampling function that imitates T ∗ can hence be constructed by interpolating between the two previous asymptotic limits T ∗ ≈ Tx1 ,x2 ,L (r) =
−r)2 βE0 1 − (xλm2 /2 L e 1 + e− 2L−1 |Ψ0 (r)|2 . N
(41)
Eq. (41) is a sum of two probability distributions Pfree + PΨ whose normalization constants, Nfree and NΨ , can be computed analytically (N = N (xm , L) = Nfree + NΨ ), see Appendix B. Since (40) is a weighted average of two distributions, (Nfree Pfree +NΨ PΨ )/N , one samples with probability Nd /N from distribution Pd (d = free or Ψ). Generating the bisection points from the Gaussian distribution Pfree is straightforward. Sampling from PΨ is first attempted by rejection sampling. If more than 50 attempts are rejected in a row, PΨ is sampled instead via a Markov chain, started and equilibrated for the given interval parameters xm and L. Since the Markov-chain sampling is rarely called in our method (only in the first steps of the discretization when L is small and β is large), the computational efficiency is not crucial and one can let the Markov-chain sampling run sufficiently long compared to the correlation time. The inclusion of distribution PΨ , with the particular weight NΨ /N , brings a large reduction in the variance of the MC calculation of f ({Ri }; β). We stress that the exact ground-state energy E0 and wave function Ψ0 could actually be replaced by approximations to those quantities without loosing the enormous reduction of variance brought by the importance sampling function (41) at low temperatures. In the first steps of the L´evy construction, the temperature associated to the slice is low and therefore the point is chosen according to the product of the free-particle distribution and |Ψ0 (r)|2 (paths of size ≈ aB are thus generated with a high probability), whereas at a high discretization level L (high temperature) the kinetic factor dominates and (41) reduces to free-particle sampling. The expression (37) unfolded up to level L = 2 (hence made up of M = 3 points) yields for example NR,R,L=1 NR,r1 ,L=2 Nr1 ,R,L=2 f (R; β) = f (R, {r 1 , r 2 , r 3 }; β) . (42) TR,R,L=1 (r 1 ) TR,r1 ,L=2 (r 2 ) Tr1 ,R,L=2 (r 3 ) P Expression (42) illustrates our sampling scheme in the case of a single path associated to the relative motion of an electron-proton pair. If no relative coordinates are used, the argument of the wave function in (41) depends on the relative distance Ψ0 (r i,k − r j,k ) of the proton position r j,k , at given time slice k and the one of the electron r i,k to be sampled. The normalization constant of this new probability distribution Tj (which depends on L and on the two points xm and r j,k ) is denoted by Nj = Nj (L, xm , r j,k ) where j is a proton index. D.
Sampling paths in a few-particle cluster
The previously described algorithm can be adapted to sample simultaneously paths of several interacting charged particles. Protons paths are much smaller than electron paths, λp λe , due to the large mass difference. The electrons being more mobile, it is their paths that arrange themselves around the proton paths. Inspired by such a BornOppenheimer-like picture, one generates first proton and α particle paths by free-particle 14
sampling. The electron paths are then generated, using the approximation that each electron interacts only with one heavy particle at a time, by the two-body importance sampling function (41) which involves the hydrogen ground-state. This method for sampling few-particle partition functions ZC (β) turns out to be quite effective. To generate an inner point r i,k of electron path i, one therefore first chooses a particle j, among the Np + Nα heavy particles and samples the bisection point r i,k according to (41).PCloseby particles are favored by setting the probability to pick particle j to be Nj /( j Nj ), where Nj is the normalization constant introduced earlier. The probability distribution for point r i,k is hence the weighted mean of two-body sampling distributions 1 X x1 + x2 P , r j,k , Nj Tj L, 2 j Nj j
(43)
where the normalized probability distribution Tj , introduced in the previous section, includes the factor 1/Nj . The integrand is eventually rewritten, similarly to (42), with factors that are the inverse of (43) for each bisected interval in the L´evy construction. The importance sampling distribution P{Ri },β ({r i,k }) in eq. (37) is hence built with freeparticle Gaussian distributions for the intermediate points in proton and α-particle paths and with products of distributions (43) for electron paths for each bisected path interval. E.
Integration over root points
Sampling over the root points is carried out by an adaptive importance sampling scheme that presents similarities with the Vegas algorithm36 . Regions in root-points coordinate-space where the variance of the MC estimate of the path integral is large, are sampled with higher probability. In a two-particle cluster, the integration over the root point of the second particle reduces, by rotational symmetry, to a one dimensional integral over the relative distance r. In a three-particle cluster, the two integrals over root points R1 = r 1 and R2 = r 2 reduce, thanks to rotational symmetry and to the specific form of the integrand f (r 1 , r 2 ) = F (r1 , r2 , r0 ), to an integration over three distances r1 = |r 1 |, r2 = |r 2 | and r0 = |r 1 − r 2 |: Z
Z dr 1
0
dr 2 F (r1 , r2 , r ) = 8π
2
Z
∞
Z dr1
0
∞
Z
r1 +r2
dr2 0
dr0 r1 r2 r0 F (r1 , r2 , r0 ).
(44)
|r1 −r2 |
Notice that the phase space volume element r1 r2 r0 dr1 dr2 dr0 makes the integrand much smoother in these coordinates than with coordinates r1 , r2 , θ where θ is the angle between vectors r 1 and r 2 . With the latter variables, sharp peaks would indeed be generated at large distances when r1 ≈ r2 and θ ≈ 0. The three distances are not treated on an equal footing in eq. (44), but this drawback can be circumvented. Summing over distances r1 , r2 , r0 corresponds to summing over all possible triangles. Since a triangle can be parametrized by the radii x1 , x2 , x3 of three circles touching each other (see Fig. 2), one can convert (44) into an integration over all radii Z Z ∞ 0 2 dr 1 dr 2 F (r1 , r2 , r ) = (4π) dx1 dx2 dx3 (x1 +x2 )(x1 +x3 )(x2 +x3 )F (x1 +x2 , x1 +x3 , x2 +x3 ), 0
(45) 15
where each radius can vary without constraint. These new radii variables reflect the full symmetry of the problem. The integral (45) is then rescaled to the unit cube via a change of variable yi xi = a , yi ∈ [0, 1], i = 1, 2, 3, (46) (1 − yi )2 where a is a length scale, chosen in practice to be min(aB , λe ). This particular choice is motivated by the observation that the relevant length scale is the Bohr radius at low temperatures, whereas it is the thermal de Broglie wavelength at high temperatures.
FIG. 2. A triangle with sides of lengths r1 , r2 , r3 = |r 1 − r 2 | can be characterized by the radii x1 , x2 and x3 of three circles that touch each other. The simple relations r1 = x1 + x3 , r2 = x1 + x2 and r3 = x2 + x3 hold. The contact points between the three circles are the same as the contact points between a circle inscribed in the triangle and the three sides.
The unit cube, or unit interval in the case of Zp,e (β), is then discretized evenly on a grid. To initialize the adaptive importance sampling MC integration over root points, one first evaluates, at a random point in each grid cell, the integrand f (R1 , R2 ; β) = f (y1 , y2 , y3 ; β), and its variance, by averaging over 100 configurations of paths of the three (resp. two) particles. The number of grid cells to consider can actually be halved by using the symmetry of the integrand with respect to a permutation of the two protons in cluster (2p,e) or electrons in clusters (p,2e) and (α,2e). A cell is then chosen at random, with a higher probability to select a cell displaying a large variance. In the next step, 100 additional paths configurations are generated for a random point in the chosen cell and the average value of ZC (β), and of the variance of the integrand in that cell, are updated. The procedure of choosing a random point in a random cell (i.e. positions for root points) and averaging over a number of discretized paths is repeated until the maximum number of iterations is reached.
IV.
NUMERICAL RESULTS
First, we test our sampling method by computing the hydrogen internal partition function Zp,e (β) (Sct. IV A) and apply it then to partition function of three-particle clusters (Sct. IV B). As the three-body truncated virial coefficient (A1) contains large compensating terms at low temperatures, the evaluation of this function is done in our implementation using arithmetics with a precision of 40 digits37,38 rather than with 15 digits as provided by usual double precision numbers. Error bars in all plots are standard deviations evaluated over 8 independent calculations. 16
A.
Proton-electron cluster function
The proton-electron cluster function Zp,e (β) [eq. (7)] can be considered as a benchmark and is known analytically. It can be expressed in terms of Ebeling’s second virial function Q(x) for which an exact series representation is available.7,39 One can write40 ∞ X ζ (n − 2) x n Zp,e (β) = 2 1 2 Γ n + 1 2 n=4 ∞ h x i X x2 nx x3 x2 2 2 4n 2 1 + erf −n − √ − − √ , =2 ne 2n 4 π 6n π n=1
(47) (48)
√ with x P = xpe = 2βe2 /λpe . The second equality is derived by inserting the definition −s ζ(s) = ∞ and carrying out the sum over n in (47). The series in the l.h.s of (48) m=1 m converges rapidly at high temperatures (small x), while the new series representation in the second line converges rapidly at any temperature (including large x). We use atomic units, and measure thus energies in Hartrees (1 Ha = 2 Ry). Let β ∗ = β 2|EH | and notice that x2pe = 2β ∗ . The dimensionless parameter β ∗ is very close to the inverse temperature measured in Ha−1 since |EH | ≈ 1 Ry, where the corresponding difference EH − 1 Ry is due to the finite mass of the proton. At low temperatures, the most important contribution in Zp,e (β ∗ ) arises from the exponentially growing Boltzmann factor exp(−βEH ) associated to the ground-state energy. This behavior is apparent in eq. (48) since Zp,e (β ∗ ) ∼ 4 exp(β ∗ /2) when β ∗ → ∞ with a factor 4 accounting for spin degeneracy. We computed Zp,e (β ∗ ) by sampling the functional integral (17) over 107 particle paths. The adaptive discretization of the paths was stopped if the break-up condition with = 0.1 (see Sct. III B) is fulfilled, provided the discretization level L is at least 4. This cutoff represents the only small source of systematic error in our calculation. Our results for Zp,e are shown in Fig. 3. The numerical calculations are in excellent agreement with the analytical result (48) from high down to quite low temperatures (β ∗ ∼ 40 corresponds to T ' 8000 K). If we had used plain free particle sampling instead of our importance sampling method, the MC calculations would show large deviations for temperatures below β ∗ ' 1 because bound state paths would not be sampled properly. The sum over states embedded in trace (7) is clearly well captured by our PIMC calculation. At high temperatures (β ∗ → 0), the numerical results display the asymptotic behavior 2 Zp,e (β ∗ ) ∼ π96 β ∗2 as expected. The average number hM i of time slices in our calculations is plotted in Fig. 4, for three different temperatures, as a function of the electron-proton distance at time zero (i.e. relative electron-proton distance between the paths’ root points). The number of slices decreases, as expected, when the two particles are far away from each other. The accuracy of the adaptive discretization method can be compared with that of a fixed uniform discretization in M = 2L time slices. Fixing M to be close to hM i (in fact slightly larger), the accuracies of both the adaptive and non-adaptive approaches are reported in Table I. For the same numerical effort, the adaptive discretization provides a better accuracy than a fixed discretization (recall that M > hM i). The accuracy gain is noticeable especially at low temperatures. This gain is brought by the fact that the adaptive algorithm concentrates the numerical effort where it is most needed, i.e. 17
FIG. 3. (color online) Numerical results (crosses) and analytical result (red solid line, eq. (48)) for the cluster function Zp,e in units of 4 exp(βEH ). The phenomenological Brillouin-PlanckP 2 2 2 Larkin partition function ZBPL (β) = ∞ n=1 n (exp(−βEH /n ) − 1 + βEH /n ) is also shown. Β= 1 Ha-1
400
Β= 4 Ha-1 300 Β= 36 Ha-1 200
100
0.2
0.4
0.6
0.8
y 1.0
FIG. 4. Average number of time slices used in the computation of Zp,e (β), as a function of distance, for 3 different temperatures. The dimensionless distance variable y is defined by eq. (46).
in regions where the potential is strong. Portions of paths that explore those regions can be discretized effectively more finely in the adaptive method than they are in the fixed discretization method. Table I also shows that the straight-line effective potential provides results essentially as accurate as those computed with the cumulant effective potential, as expected from the theoretical analysis (see Sct. III B).
18
β (Ha
−1
)
exact
36
6.50 · 107
16
2.98 · 103
1
1.519 · 10−1
Zp,e (β) adaptative discretization uniform discretization value error hM i value error M 7 7 6.39 · 10 -1.7% 440 5.84 · 10 -10% 512 7 6.62 · 10 1.9% 440 (Vline not applicable) 3 2.96 · 10 -0.6% 220 2.86 · 108 -4.0% 256 3.01 · 103 1.0% 220 (Vline not applicable) −1 1.511 · 10 -0.4% 40 1.521 · 10−1 0.1% 64 −1 1.525 · 10 0.5% 40 (Vline not applicable)
effective potential cumulant line cumulant line cumulant line
TABLE I. Average, over 8 independent calculations, of Zp,e (β) calculated with a uniform discretization in M time slices, with M similar to the average hM i in the adaptive discretization. The paths actions are calculated either with the cumulant potential (27), or with the straight-line potential (30), using each time the same 107 paths. The stopping criterion in the adaptive method is | 2βLi Vline |< = 0.1.
B.
Three-body cluster functions
We computed the three-body cluster functions for ions H− , H+ 2 and for the helium atom 7 by sampling over 10 paths. Notice that all paths are independent in our direct sampling approach. The direct (D) and exchange (E) terms appearing in eq. (11) are plotted each time separately and the results are compared with simple models for the partition functions of those entities that hold at low temperatures (kB T dissociation energy). √ The high temperature behavior of the cluster functions is a polynomial in β. We n extracted the exponent n and the prefactorp a of ZC (β) ∼ aβ 2 by fitting the results of eight calculations for two high temperatures β · (1 Ha) = {2.5·10−3 , 5.0·10−3 }, for each considered cluster (see Table II). The fitted exponents agree well with their theoretical values: n = 7 for the direct and n = 4 for the exchange term.
Zp,2e Zα,2e Z2p,e
n a Direct 6.999 ± 0.002 −0.144 ± 0.001 Exchange 4.00 ± 0.02 0.40 ± 0.04 Direct 6.999 ± 0.003 −3.59 ± 0.06 6.57 ± 0.08 Exchange 4.000 ± 0.002 Direct 6.978 ± 0.009 −(1.8 ± 0.1) · 104 n
TABLE II. Coefficients of the high temperature behavior a β 2 for the direct and exchange contributions for the various particle clusters.
1.
H− cluster
The results for the cluster function Zp,2e (β) are shown in Fig. 5. The exchange term is always positive, while the direct term is negative at high temperatures and changes sign at around β = 19 Ha−1 (T ' 17000 K). It is not surprising that D becomes negative at 19
Temperature (K)
Temperature (K) 106 10
15
(a)
105
105
104 10
Cluster (p,2e)
5 ⋅ 103
104
(b)
Cluster (p,2e)
1012
Cluster function
Cluster function
1012
15
109 106
1000
109
106
1000 1 |Direct term (D+WH-e)| Exchange term (E)
10−3 0.1
1
1
10
β
|Zp,2e| |Zp,2e+WH-e| ZGS 10
(Ha-1)
20
30
β
40
50
60
(Ha-1)
FIG. 5. (color online) Cluster function Zp,2e (β) = 4D −2E in absolute values [dotted green line in plot (b)] and its two contributions: the direct term |D| and exchange term E [plot (a)]. Plot (a) is logarithmic while plot (b) is semi-logarithmic. Zp,2e (β) tends to the partition function ZGS (β) = 2 exp(−βEH− ) at low temperatures. The sum of Zp,2e (β) and of the atom - ionized electron interactions described by function WH−e (β) is shown as a solid black line in plot (b).
high temperatures because some subtracted terms in (A1) are negative and provide the leading contribution. At low temperatures, the ground-state of ion H− dominates exponentially the matrix elements D and E, leading to the low-temperature asymptotic behavior ZGS (β) = 2 exp(−βEH− ), with a degeneracy factor of 2 due to proton spin. The numerical results are compared in Figs. 5b and 6 with ZGS (β). Both functions grow approximately with the same rate, but the former is about five times smaller than the latter. This difference is caused by truncation terms in (A1) associated to interactions between an ionized electron and a hydrogen atom, namely terms of the form (eb − 1 + b − b2 /2 + b3 /3! )(a + c)n /n! and (ec − 1 + c − c2 /2 + c3 /3! )(a + b)n /n! (with n = 2 or 3), where a = −βVe1 −e2 , b = −βVe1 −p and c = −βVe2 −p . Notice that these terms are zero by charge symmetry and translational invariance in imaginary time when n = 1. The factors eb and ec , which describe a hydrogen atom formed by the pair e1 − p or e2 − p, can be exponentially large at low temperature (their magnitude is controlled by the binding energy |EH | of the atom which is not much smaller than the energy |EH− |), while factors (a + c)2 and (a+b)3 can also be significant. The term (a+b)3 takes large positive values when the ionized electron is close the proton, since the attractive proton-ionized electron interaction dominates indeed the repulsive electron-ionized electron interaction at short distances, the proton path being more localized. The subtraction of the atom-charge interactions lowers Zp,2e (β) with respect to a partition function defined by summing only over bound states of the three-body Hamiltonian. The sum of the atom-charge interaction terms can be identified with the opposite of function W (1, 1|0, 1) defined in ref.12 if one replaces the screened Coulomb potential in W by the bare Coulomb potential. Those interactions are excluded (subtracted) in the definition of particle clusters8,12 , but the full contribution 20
Temperature (K) 2
105
5 ⋅ 103
104
Cluster function / 2 exp(-β EH-)
Cluster (p,2e)
0
−2
Direct term (D+WH-e) Exchange term (E) Zp,2e+WH-e
−4
10
20
30
40
50
60
β (Ha-1)
FIG. 6. (color online) Cluster function Zp,2e (β) together with atom - ionized electron interactions in units of ZGS (β) = 2 exp(−βEH− ). The direct (D) and exchange (E) contributions are also shown separately.
of three-particle effects to the thermodynamic properties involve not only the contributions of the 3-body particle clusters but also those of the counter-terms that generate the functions W (see eq. (4.10) of ref12 ). By subtracting the atom-charge interactions for all relative distances of the particles, unphysical negative contributions have appeared in Zp,2e at short-distances. Those are compensated by the counter-terms in WH−e which have eventually to be added. We therefore consider the sum Zp,2e + WH−e , which is the combination appearing in the function h4 (β) of the SLT expansion.12 Notice that the function Zp,2e + WH−e accounts simultaneously for the atom-electron interactions at large distances as well as the formation of an H− ion at short distances, without any approximate treatment for configurations at intermediate distances. The numerical results for electron-atom interactions WH−e and for the sum Zp,2e + WH−e are shown in Figs. 5(b) and 6. The atom-charge contribution WH−e is calculated in practice with a Mayer bond ˜ T = B T − 1 (βV (Lp , Le ))4 between the proton and electron paths. At low temperatures B 2 2 4! the sum Zp,2e + WH−e approaches ZGS (β) faster then Zp,2e alone. Fitting Zp,2e + WH−e to 2 exp(−βE) at low temperatures yields the energy −0.525 Ha, in good agreement with the known value EH− ' −0.527 Ha.41 We recall that the present work does not aim for spectroscopic accuracy of the ground state energy, but we study the low temperature behavior of our path integral calculations, well suited for high temperatures, in order to connect to known low temperature asymptotics.
2.
H+ 2 cluster
The results for the cluster function Z2p,e (β) are shown in Fig. 7. Again, Z2p,e (β) is negative at high temperatures. The exchange term (not shown) is negligible at all 21
Temperature (K) 105 1021
104
(a)
5 ⋅ 103
5
Temperature (K)
105
104
(b)
Cluster (2p,e)
5 ⋅ 103
Cluster (2p,e)
Cluster function / Zrot-vib(β)
4
Cluster function
1018
1015
1012
109
|Z2p,e| -WH-p |Z2p,e+WH-p| ZStancil Zrot-vib
106
3 2 1 0
Z2p,e /20 -WH-p /20 Z2p,e+WH-p ZStancil Zrot-vib
−1 −2
1000
0
10
20
50
β (Ha-1)
60
10
70
20
30
40
β (Ha-1)
50
60
70
FIG. 7. (color online) Cluster function Z2p,e (β): (a) in absolute values in a logarithmic scale; (b) in units of Zrot−vib (β) in a linear scale. The partition functions Zrot−vib (β) and ZStancil (β), and the atom-proton interactions WH−p (β) are defined in the main text. In plot (b), Z2p,e (β) and |WH−p (β)| are divided by 20 for visualization purposes.
considered temperatures because of the large mass of the protons. The direct contribution is also not shown since it reduces to Z2p,e (β)/4. At low temperatures, the contributions vib from bound states grow exponentially fast. Since the energy quanta kB THrot + and kB T + H 2
2
rot vib for rotations and vibrations of the molecular ion H+ 2 are small (TH+ ' 41.87 K and TH+ ' 2 2 3150.78 K), many rotational-vibrational states are excited at the considered temperatures. We compare in Fig. 7, for temperatures lower than the dissociation energy (T < (EH+2 − EH )/kB ' 30 000 K), the cluster function Z2p,e (β) with two different models for the partition function of ion H+ 2 . In the first model, the molecular states are described in the simple picture where the electron is in its ground state, while global rotations and vibrations are taken into account with a rigid rotor model and a harmonic oscillator, respectively. When T THrot + , the partition function for this model is given by 2
Zrot−vib (β) ' 2 exp(−βEH+2 )
1 2T . rot TH+ 1 − exp(−THvib + /T ) 2
(49)
2
In the second model, the partition function is calculated via a direct sum over all (423) electronic ground state rotational-vibrational levels whose energies are determined by solving the radial nuclear Schr¨odinger equation within the Born-Oppenheimer approximation (using an effective potential obtained by a density functional theory calculation).42 The latter model42 provides the more refined partition function ZStancil (β) which accounts for rotation-vibration coupling. ZStancil (β) is about 1.5 times larger than Zrot−vib (β) (see Fig. 7). The rotational and vibrational partition functions in (49) depend only weakly on the temperature, in contrast to the exponential growth of the electronic partition function ZGS (β) = 2 exp(−βEH+2 ) as β → ∞. They provide a slowly varying factor whose magnitude is about 1500 when T ∼ 10 000 K. 22
Fig. 7 shows that Z2p,e (β) is about 50 times larger than the Stancil partition function when T < 20 000 K. The difference arises from the subtracted atom-ionized proton interactions W (1, 1|1, 0) = WH−p (β) in the truncation of Z2p,e (β), similarly to the case of Ze,2p (β). When the ionized proton is close to the other proton bound in a hydrogen atom, WH−p takes large negative values, which lead to an increase of Zp,2e (β) since those interactions are subtracted in the cluster function. One considers therefore the sum Z2p,e +WH−p , which is the relevant quantity for thermodynamics. Notice that Z2p,e + WH−p accounts exactly for the effects of ionized proton-hydrogen atom interactions at large distances and for molecular states of H+ 2 at short-distances, without any arbitrariness in how configurations at intermediate distances are treated. The numerical results for WH−p and for the sum Zp,2e + WH−p are shown in Fig. 7. The atom-ionized proton interaction WH−p does compensate the large contributions in Z2p,e in such a way that the sum Zp,2e + WH−p is close, for temperatures below the dissociation energy of H+ 2 , to the partition function ZStancil (β). The fact that Zp,2e + WH−p is slightly larger, at low temperatures, than the partition function of Stancil is probably due to the fact that our calculations are not restricted to bound molecular states, but includes also contributions from dissociated states, such as atom-charge interactions at large distances.
3.
He cluster
The cluster function Zα,2e (β) is shown in Fig. 8. Since the first ionization energy of a helium atom (I1 = 0.9036 Ha) is much larger than the ionization energy of ion H− (IH− = EH− −EH ' 0.027 Ha) and ion H+ = EH+2 −EH = 0.097 Ha), the ground state 2 (IH+ 2 contribution of the helium Hamiltonian quickly dominates all other terms in Zα,2e (β) at low temperatures (T < 60 000 K). Fitting the data at low temperatures to 2 exp(−βE), one finds EHe ' −2.905 Ha, which is close to the ground state energy EHe = −2.9034 Ha of the helium atom. Below 30 000 K, Zα,2e (β) can safely be approximated by 2 exp(−βEHe ).
V.
CONCLUSIONS
This work reports two new developments for the numerical calculations of Coulomb path integrals, as well as their application to cluster functions, appearing in equations of state at low densities. First, we have devised an efficient importance sampling strategy to compute fewbody cluster functions. It relies on the knowledge of the solution of the two-particle Coulomb problem and enables us to overcome to a large extent the exponential variance increase of the MC estimator at low temperatures originating from the formation of bound states. Second, an adaptive discretization procedure is introduced which removes all short-distance Coulomb divergences that would be present at fixed discretization when working with the bare Coulomb potential. This adaptive discretization exempts us from using more sophisticated and computationally demanding effective potentials such as the pair-action potential. With this self-adapting discretization, correlations between the particle paths are not approximated by pair-actions, but are accounted for by discretizing sufficiently. Notice that in the limit of finely discretized paths, as required to capture all correlations, the pair-action effective potential reduces to the plain Coulomb straight-line potential employed in our calculations. The adaptive discretization is efficient because it 23
Temperature (K) 1024
Temperature (K) 2⋅ 104
105
(a)
106
Cluster function
1018
106 |Zα,2e| |Direct term| Exchange term ZGS
1
10−6
2.5
5
7.5 β
10 12.5 (Ha-1)
2⋅104
105
(b)
Cluster (α,2e)
15
17.5
Cluster function / 2 exp(-β EHe)
106
Cluster (α,2e)
1
0.5
0
Direct term Exchange term Zα,2e
−0.5
20
2.5
5
7.5 β
10 12.5 (Ha-1)
15
17.5
20
FIG. 8. (color online) Cluster function Zα,2e (β): (a) in absolute values (logarithmic scale); (b) in units of ZGS (β) = 2 exp(−βEHe ) (linear scale). The direct and exchange contributions and the ground state partition function ZGS (β) = 2 exp(−βEHe ) are also shown.
directs the computational effort where it is most needed, in regions where particles are close and therefore strongly interacting. Numerical tests have shown that our approach combining the adaptive discretization with the straight-line Coulomb action provides results that are more accurate at low temperatures than those computed with the usual uniform discretization and the cumulant action. We have applied our algorithm to calculate two- and three-body cluster functions, which include bound and diffusive states. Our calculation of the two-body function agrees well with its known series representation, even at low temperatures where bound state contributions dominate, underlining the efficacy of our importance sampling approach. The uncertainties on our numerical results for the three-body cluster functions are very small at high and intermediate temperatures. At a low temperature like T ≈ 8000 K, the error on function Zp,2e (β) is about 1%. The three-body cluster functions reduce at low temperatures to known atomic, ionic or molecular partition functions determined by other means, so that they are now known in the whole temperature regime. Ground state energies can be extracted with relatively small errors, at most 1%, highlighting the accuracy and reliability of our computations. For ion H+ 2 , the cluster function agrees at low temperatures with Stancil’s partition function obtained by summing explicitly over many rotational-vibrational energies. We stress that the aim of the present work is not extracting ground state energies with spectroscopic accuracy, but determining precisely the three-body cluster partition functions at temperatures T > 8000 K where the considered clusters are significantly excited and possibly even dissociated. The application of the present results in the extension of the SLT EOS to a partially ionized hydrogenhelium mixture, as well as the computation of four-body cluster functions describing H2 molecules and interactions between its dissociation products is foreseen. 24
ACKNOWLEDGMENTS
The calculations were run on computers from the Institute UTINAM of the University de Franche-Comt´e, supported by the R´egion Franche-Comt´e and the Institut des Sciences de l’Univers (INSU).
Appendix A: Direct term in a 3-body cluster
The direct term in a 3-body cluster [eq. (12)] involves the following operator DT DT = ea+b+c − ea − eb − ec + 2 + 2(a + b + c) + a2 + b2 + c2 + 2(ab + ac + bc) 1 + (a3 + b3 + c3 ) + a2 (b + c) + b2 (a + c) + c2 (a + b) + 2abc 3 1 2 2 1 2 2 + (a b + a c + b2 c2 ) + [a3 (b + c) + b3 (a + c) + c3 (a + b)] + a2 bc + ab2 c + abc2 4 6 1 3 2 1 2 3 2 + [a (b + c ) + b (a + c2 ) + c3 (a2 + b2 )] + (ab2 c2 + a2 bc2 + a2 b2 c) 12 4 1 3 3 1 2 2 2 3 3 3 3 + (a b + a c + b c ) − a b c 36 8 1 a 1 1 a b c 2 − e (b + c) − e (a + c) − e (a + b) − e (b + c) − eb (a + c)2 − ec (a + b)2 2 2 2 1 3 1 2 1 2 1 3 1 2 1 2 1 3 1 2 1 2 a 1 3 b 1 3 c 1 3 −e ( b + c + b c+ bc )−e ( a + c + a c+ ac )−e ( a + b + a b+ ab ), 6 6 2 2 6 6 2 2 6 6 2 2 (A1) where kinetic operators are implicit and where we have set a = −βVp1 −p2 ,
b = −βVp1 −e ,
c = −βVp2 −e .
(A2)
In the first line of (A1), one recognizes the Mayer operator (14), while the truncation terms (besides those of the Mayer truncation) involve one or more time-evolved Coulomb operator(s) possibly associated with a Gibbs operator ea , eb or ec for a sub-set of particles inside the cluster. Truncation term a c eb stands for instance for the expression Z
β
Z dτ1
0
τ1
dτ2 e−βHp1 e|p2 [Vp2 −e (τ1 )Vp1 −p2 (τ2 ) + Vp1 −p2 (τ1 )Vp2 −e (τ2 )] ,
(A3)
0
where Hamiltonian Hp1 e|p2 is the sum of 3 kinetic terms and of the Coulomb operator Vp1 −e and where the time-ordering of the time-evolved operators has been taken into account. By construction, operator DT remains integrable when any of the three particle is sent to infinity. This can be checked by performing a Dyson expansion of (A1). The 1 terms of lowest order in the interaction potential are indeed 24 (a3 b2 c2 + a2 b3 c2 + a2 b2 c3 ) + 1 3 3 3 3 3 3 (a b c + a bc + ab c ), which are all integrable. 36 25
Appendix B: Normalization of the sampling distribution
The norm of the second distribution in eq. (41) is NΨ = exp(−βE0 /M )NΨ0 with NΨ0
(xm −r)2 λ2 M
Z ∞ Z 1 2 2 2 2 dr dz r2 e−(r +xm −2rxm z)/λM −2r/aB |Ψ0 (r)| = 3 = dr e aB 0 −1 Z ∞ 2 λM 2 2 2 2 = 3 dr r e−(xm −r) /λM − e−(xm +r) /λM e−2r/aB aB x m 0 √ λ λ3M π λ2M /aB −2xm /aB λ2M xm M = 3 e e 1− erfc − aB 2 x m aB aB λM 2 λM xm λM 2xm /aB erfc + , (B1) +e 1+ x m aB aB λM Z
−
2
with λ2M = λ2 /M and M = 2L . 1
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W. Ebeling. Coulomb interaction and ionization equilibrium in partially ionized plasmas. Physica A, 43:293–306, 1969. 40 6,43 Denoting Euler’s constant ) ( by γ ' 0.577, function Q(x) is defined, for x > 0, as i h √ i √ R h 2 2 2 (βe ) 2 π βe 3 2R 2π +γ (2πλ2 )3/2 hr|e−βH |ri − 1 − r − 2r2 dr + 3 (βe2 )3 ln Q(x) = (2πλ2 )3/2 lim λ R→∞
r
