The ClassicalMap HyperNettedChain (CHNC) method and associated novel densityfunctional methods for Warm Dense Matter^{1}^{1}1Sanibel Symposium 2011 novel DFT/WDM
Abstract
he advent of shortpulse lasers, nanotechnology, as well as shockwave techniques have created new states of matter (e.g., warm dense matter) that call for new theoretical tools. Ion correlations, electron correlations as well as bound states, continuum states, partial degeneracies and quasiequilibrium systems need to be addressed. Bogoliubov’s ideas of timescales can be used to discuss the quasithermodynamics of nonequilibrium systems. A rigorous approach to the associated manybody problem turns out to be the computation of the underlying pairdistribution functions , and , that directly yield nonlocal exchangecorrelation potentials, free energies etc., valid within the timescales of each evolving system. An accurate classical map of the stronglyquantum uniform electrongas problem given by Dharmawardana and Perrot is reviewed. This replaces the quantum electrons at by an equivalent classical fluid at a finite temperature , and having the same correlation energy. It has been shown, but not proven, that the classical fluid are excellent approximations to the quantum . The classical map is used with classical molecular dynamics (CMMD) or hypernettedchain integral equations (CHNC) to determine the pairdistribution functions (PDFs), and hence their thermodynamic and linear transport properties. The CHNC is very efficient for calculating the PDFs of uniform systems, while CMMD is more adapted to nonuniform systems. Applications to 2D and 3D quantum fluids, Si metaloxidefieldeffect transistors, Al plasmas, shockcompressed deuterium, twotemperature plasmas, pseudopotentials, as well as calculations for parabolic quantum dots are reviewed.
pacs:
PACS Numbers: 71.10.Lp,75.70.Ak,73.22fI Motivation
The advent of powerful shortpulse lasers as well as other new tools for manipulating matter presents new challenges to existing theory. Warm dense matter (WDM) is such a regime where we have highly correlated ions, electrons, finite temperature as well as partial degeneracy effects that have to be taken into account. Sufficiently thin nanoslabs of WDM can be studies with a variety of probes Ping06 (); Ng11 (). Bound states as well as continuum states have to be treated without sinking in a morass of computations. The BornOppenheimer approximation cannot be used if coupledmode effects are important. In this paper we examine new theoretical approaches that extend beyond the familiar territory of densityfunctional theory (DFT) to treat these and other intractable problems in manybody physics.
The HohenbergKohn and Mermin (HKM) theorems hohen () of DFT assert that the onebody density of an inhomogeneous system completely determines its physics. However, implementations of DFT use the more laborious KohnSham (KS) approach kohnsham () in lieu of an accurate kineticenergy functional PerrotH (); Karasiev09 (). The KohnSham of an electron system is:
(1) 
The KS eigenstates, with “energies” , occupations factors at the temperature for all the quantum numbers have to be determined, selfconsistently, using a onebody KohnSham potential in the KohnSham equation. The inclusion of continuum states in this summation consistently, to satisfy sum rules etc., is a challenge discussed in dwp82 (). The KohnSham potential contains an ‘exchangecorrelation potential’ that maps the manybody effects to a functional of the onebody density. Model potentials have been constructed using microscopic theories of systems like the uniform electron liquid (UEF). Such UEFcalculations are equivalent to a couplingconstant integration over the electronelectron pair distribution function (PDF), viz., . Calculating these PDFS, even for uniform systems, is a challenge that is treated in this paper.
Quantum systems at high temperatures behave classically. Then the KohnSham procedure simplifies. The density is given by the Boltzmann form:
(2) 
where is a reference density, and is a classical KohnSham potential that has to be obtained from a microscopic classical manybody theory. The ‘potential of meanforce’ used in classical liquidstate theory is just this classical . If the center of coordinates is selected to be one of the classical particles, and if we consider a uniform fluid, then becomes the density profile of field particles around the central particle which acts like an external potential. The density profile is directly related to the pairdistribution function, i.e.,
(3) 
Hence one may attempt to go beyond traditional DFT and proceed directly to the underlying calculation of the pairdensities themselves. The extension of the HohenbergKohn theorem given by Gilbert, using the onebody reduced density matrix is actually entirely in this spirit Gilbert75 (). However, the PDF is conceptually easier to use than the density matrix. Such considerations suggest that the kineticenergy functional may be sidestepped by: (i) the use of an equivalent “classicalfluid” at a temperature for the (uniform) quantum fluid whose actual physical temperature may even be zero; (ii) the use of effective classical pairpotentials inclusive of quantum effects to calculate classical pair distribution functions which can then be used to compute most of the usual physical properties prl1 ().
The advantage of such a classicalmap approach is that the ions, being essentially classical particles, can be treated together with the electrons in the same classical computational scheme. Unlike quantum electron schemes which, in principle, grow in complexity nonpolynomially in , classical methods are essentially independent of . Here we should note that traditional quantum chemistry and condensedmatter physics treat only the electrons by DFT. On the other hand, Gross and collaborators have attempted to present a completely quantum mechanical nonadiabatic DFT theory of electronnuclear systems, and given an application to the H system GrossKre01 (). In standard calculations, the ion positions are explicitly included and form the external potential for the motion of the KohnSham electron. In warm dense matter (WDM), e.g., highly compressed hot hydrogen, there are as many protons as there are electrons in a given volume of the sample. Ion motion couples with electronplasma oscillations to generate ionacoustic coupled modes. Their effects may be missed out in standard DFT formulations as well as in MD simulations.
In any case, the quantumchemistry approach (e.g., as in the Gaussian package) rapidly becomes intractable, esp. when continuum states have to be included  as in a plasma. The solidstate approach of using a periodic cell is more flexible here, as in the Viennasimulation package known as the VASP. However, WDM applications demand large unit cells and calculations of energy bands for many ionic configurations. The classicalmap approach, where both ions and electrons are treated as classical fluids inclusive of particle motions, provides a new paradigm for warm dense matter and other novel systems which are computationally very demanding by standard methods. Such standard methods could be regarded as microscopic bench marks for more global methods like the classicalmap approach discussed here.
The philosophy of the classicalmap technique is to treat the zerothorder Hamiltonian exactly, i.e., using the known quantum solution, and then use the classical map for dealing with the manybody effects generated from the Coulomb interaction. For uniform systems, the eigensolutions of the problem are plane waves. Fermi statistics imposes a determinantal form to the wavefunctions, and hence the noninteracting PDFs are different from unity if the spin indices are identical. Thus exhibits a Fermi hole, which can be exactly represented by a classical repulsive potential known as the Pauli exclusion potential (PEP) lado (). This should perhaps be called the ‘Fermihole potential’ as it should not be confused with the ‘Pauli Potential’ defined in DFT March86 (); Trickey09 () via the densityfunctional derivative of the difference between the noninteracting kinetic energy and the full von Weizsäcker kinetic energy. In the interest of historical accuracy, it should however be noted that the name ‘Pauli potential’ was already in use for the Fermihole potential since the work of Lado. We use the names ‘Fermihole potential’ and Pauliexclusion potential’ as synonymous, and different from the DFT correction to the von Weizsäcker term known as the Pauli potential.
The key ingredients of the method are the following.

Replacement of the electron system at by a classical Coulomb fluid at an effective classicalfluid temperature given by
(4) where is a ‘quantum temperature’ which depends only on the electron density. is such that the classical fluid has the same correlation energy as the initial quantum fluid at . The motivation for defining by Eq. 4 is given in prb2000 ().

Inclusion of a “Pauli exclusion potential”, i.e., a Fermihole potential (FHP) to reproduce the Fermi hole of spinparallel electrons exactly.

The use of a diffractioncorrected Coulomb interaction to account for the finitesize of the de Broglie thermal wavelength of the electrons at the finite temperature .

Calculation of the pairdistribution functions of the classical fluid using an integralequation method (CHNC), or molecular dynamics. When MD is used in this manner we call it classicalmap molecular dynamics (CMMD).

use of the PDFs in couplingconstant integrations to calculate the Helmholtz free energy and all other thermodynamic properties of the quantum fluid. The linear transport properties (e.g., conductivity) are available from Kubo or Zimantype formulations which use the PDFs and potentials as inputs.
Formulations which use this method have been successfully applied
to a number of quantum systems:
(i) The 3D electron fluid at and at finite prl1 ().
(ii) The 2D electron fluid both at prl2 (); bulutay (); prl3 (); totsuji1 (),
and at finite prl3 ()
(iii)
The calculation of
Fermiliquid properties like the electron effective mass , the enhancement of
the Landé factor quasi (),
and localfield corrections to the response functions lfc ().
(iv) The multicomponent electron fluid in SiSiO metaloxidefieldeffect
transistors 2valley (); preliminary applications to
multivalley massless Dirac fermions in graphene grap07 ().
(v) Electrons confined in parabolic potentials (quantum dots) miyake (); qdot09 ().
(vi) Twomass twotemperature plasmas cdwmur ().
(vii) Equation of state and Hugoniot of Shockcompress hydrogen hyd ().
(viiI) Liquid Al under WDM conditions; linear transport properties of some WDM systems,
where some of the PDFs were calculated using CHNC res2006 ().
i.0.1 The QHNC method of Chihara
For the sake of completeness we also mention Chihara’s ‘quantalHNC’ (QHNC) method QHNC (). Here an HNCtype equation is solved for the electron subsystem. The electronelectron pairdistribution function is calculated by solving the ‘quantal HNC equation’ with “a fixed electron” at the origin. However, the electron pairdistribution functions obtained by this method for jellium are in poor agreement with those from quantum Monte Carlo methods. In fact, if noninteracting electrons are considered, the zeroth order PDF, which is known analytically at and in terms of a Fermi integral at finite (as discussed below) is not recovered correctly by Chihara’s method. The small limit of the ionion structure factors calculated by QHNC fail to reproduce the correct compressibility. Nevertheless, Chihara’s QHNC recovers some of the shortranged order in the ionion pairdistribution functions, where the oscillations and peak heights are in rough agreement with microscopic simulations. The shortcomings in Chira’s formulation are overcome in the CHNC method.
In the following we discuss details of some of the implementations of CHNC using integralequation methods since they are conceptually more transparent and far cheaper than molecular dynamics (CMMD), let alone QMC.
Ii A Classical representation for the uniform electron liquid
A system of electrons held in place by an external potential (as in a solid, a quantum well, or in a molecule) at is necessarily a quantum system. The uniform electron fluid (UEF) at a density , WignerSeitz radius , is the key paradigm for treating exchange and correlation in DFT. The pairdistribution functions (PDFs) of the UEF at are known from quantumMonte Carlo (QMC) studies. They are the basis of exchangeand correlation energies of the UEF. Hence, if the classicalmap scheme could successfully calculate the PDFs of the electron fluid at arbitrary coupling and spin polarization, in 2D and 3D, then the idea that the quantum fluid can be represented by a classical Coulomb fluid stands justified.
Consider a fluid of mean density containing two spin species with concentrations = . We deal with the physical temperature of the UEF, while the temperature of the classical fluid is . Since the leading dependence of the energy on temperature is quadratic, we construct as in Eq. 4. This is clearly valid for and for high . This assumption has been examined in greater detail by various applications where it has been found successful.
The properties of classical fluids interacting via pair potentials can be calculated using classical molecular dynamics (MD) or using an integral equation like the modified hypernettedchain equation. The pairdistribution functions for a classical fluid at an inverse temperature can be written as
(5) 
Here is the pair potential between the species . For two electrons this is just the Coulomb potential . If the spins are parallel, the Pauli exclusion principle prevents them from occupying the same spatial orbital. Following the earlier work, notably by Lado lado (), we also introduce a “Pauli exclusion potential” or Fermihole potential (FHP), . Thus becomes . The FHP, , is constructed to recover the PDFs of the noninteracting UEF, i.e., is exactly recovered. The function =  1; it is related to the structure factor by a Fourier transform. The is the “direct correlation function (DCF)” of the OrnsteinZernike (OZ) equations.
(6) 
The term in Eq. 5 is the “bridge” term arising from certain cluster interactions. If this is neglected Eqs. 56 form a closed set providing the HNC approximation to the PDF of a classical fluid. Since the cluster terms beyond the HNC approximation are difficult to calculate, they have been modeled approximately using the theory of hardsphere liquids rosen (). We have provided explicit functions for the 2D electron fluid where it is important even at low coupling br2d (). is important in 3D when the coupling constant for electronelectron interactions exceeds, say, 20. The range of relevant to most WDM work (e.g., even for = 10 ) is such that the HNCapproximation holds well.
Consider the noninteracting system at temperature , with = 0.5 for the paramagnetic case. The parallelspin PDF, i.e, , will be denoted by for simplicity, since , i j is unity. Denoting by , it is easy to show, as in sec. 5.1 of Mahan Mahan (), that:
(7) 
Here is the Fermi occupation number at the temperature . Eq. 7 reduces to:
(8)  
(9) 
Here is the Fermi momentum. Thus is obtained from the Fourier transform of the Fermi function. The zerothorder PDF is a universal function of . It is shown in the inset to Fig. 1.
Assuming that can be modeled by an HNC fluid with the pair interaction , the “Fermihole potential”, viz., , is easily seen to be given by
(10) 
The can be evaluated from using the OZ relations. The case can be evaluated analytically lado ().
We can determine only the product . The classical fluid “temperature” is still undefined and clearly cannot be the thermodynamic temperature as . The Pauliexclusion potential, i.e., the FHP, is a universal function of at each . It is long ranged and mimics the exclusion effects of Fermi statistics that produces quantum entanglement. At finite the range of the Pauliexclusion potential is comparable to the de Broglie thermal wavelength and is increasingly hardsphere like. Plots of and are given in Fig. 1.
The next step in the CHNC method is to use the full pairpotential , and solve the coupled HNC and OZ equations for the binary (up, and down spins) interacting fluid. For the paramagnetic case, = , we have:
(11)  
(12)  
The Coulomb potential needs some discussion. For two pointcharge electrons this is . However, depending on the temperature , an electron is localized to within a thermal wavelength. Thus, following earlier work, e.g., Morita, and Minoo et al. minoo (), we use a “diffractioncorrected” form:
(14) 
Here is the reduced mass of the electron pair, i.e., a.u., where is the electron effective mass. It is weakly dependent, e.g, 0.96 for = 1. In this work we take =1. The “diffraction correction” ensures the correct behaviour of for all .
In solving the above equations for a given and at =0, we have =. A trial is adjusted to obtain an equal to the known paramagnetic at each , via a coupling constant integration.
(15) 
( alone is obtained if is fixed at 0). The resulting “quantum” temperatures could be fitted to the form:
(16) 
We have also presented a fit to the of the 2D electron system, and discussed how the 2D and 3D fits could be related by a dimensional argument. Bulutay and Tanatar have also examined the CHNC method, and provided fits to the of the 2D electron fluid bulutay ().
For any given , given the from the paramagnetic case, we can obtain and prl1 (), at arbitrary unexplored values of spinpolarization by solving the coupled HNC equations, or doing an MD calculation using the Fermihole potential and the diffractioncorrected Coulomb potential. Many analytic theories of electron fluids, e.g., those of Singwi, Tosi, Land and Sjölander Mahan (), Tanaka and Ichimaru, predict which become negative for some values of even for moderate . The PDFs obtained from the HNCprocedure are positive definite at all . In Fig. 2 we show typical results for and comparisons with QMCsimulations. Our results are in excellent agreement with the DMC results prl1 ().
The determined from the unpolarized is used to calculate at any . The QMC results for at agree with ours, since our agree with those from MC. For example, at = 10, the spinpolarized is: CeperleyAlder, 0.0209 Ry; OrtizBallone, 0.0206 Ry ortiz94 (); our method (CHNC), 0.0201 Ry; Kallio and Piilo, 0.0171 Ry KP ().
Most of the recent work using CHNC has been on the 2Delectron fluid owing to its accrued interest in nanostructures and technological applications. The electronelectrons interactions are stronger in reduced dimensions, and the use of a bridge function to supplement the CHNC equation is essential for accurate work br2d (). However, even the appropriately chosen harddisc bridge works quite well, as seen in Fig. 3.
ii.1 Fermiliquid parameters of electron fluids
It is in fact possible that in some circumstances, WDM may fall into the category of a Fermi liquid. Highly compressed electron systems have correspondingly high Fermi energies and hence may have a physical temperature . In any case, we review the calculation of Fermiliquid parameters as it is an important aspect of the capability of a classical map to extract results in the strong quantum domain.
Microscopic manybody physics allows one to calculate various quantities like the effective mass or the Landé factor that enter into Landau’s theory of Fermi liquids. One would perhaps assume that a classical representation of a Fermi liquid would hardly be successful in attacking such problems. For instance, is usually calculated from the solutions of the Dyson equation for the oneparticle interacting Green’s function. If the real part of the retarded selfenergy is , the Landau quasiparticle excitation energy , measured with respect to the chemical potential is used in calculating the effective mass .
(17)  
(18)  
(19) 
This is a very arduous calculation, and there are technical questions about the difficulties of satisfying sum rules, Ward identities etc., when the Dyson equation is truncated in some approximation. The values of calculated by different authors using different perturbation expansions differ significantly, and from QMC results vigmstr05 (); quasi ().
By contrast, the calculation of , and also using CHNC is very simple because it can evaluate the free energy of the electron fluid as a function of the physical temperature as well as the spin polarization . The ratio of the interacting and noninteracting specific heats provides a simple evaluation of the , while the ratio of the interacting and noninteracting susceptibilities, determined from the second derivative (with respect to ) of the exchangecorrelation correction to the free energy provides the product quasi ().
(20)  
(21) 
Detailed calculations for ideal 2D electron layers (see Fig. 4), thick layers as well as for multivalley systems using the CHNC method have been presented in our publications quasi (). Calculations of and for the 3D electron liquid using CHNC have not yet been undertaken, while RPA results have been given by Rice Mahan ().
Iii Dense Hydrogen and other plasmas
Dense hydrogen, or any other fully ionized plasma is a direct generalization of the uniform electronfluid problem to include an additional component (e.g., protons), while removing the positive neutralizing background. Let us consider a fully ionized plasma with ions of charge , and density . Then the electron density , and we assume that both subsystems are at the same physical temperature . The electron subsystem will have to be calculated at a classicalfluid temperature and the electronelectron interactions have to be diffraction corrected. On the other hand, the ions are classical particles and the simulations (or integral equations) for the ions will use the physical temperature . The quantum correction can be neglected for ion, as discussed in hyd (). The total Hamiltonian now contains the three terms, , and the electronion interaction . The electron system contains two spin components, while the ion system adds another component. Thus, a threecomponent problem involving six pairdistribution functions have to be calculated. If spin effects could be neglected, then the two spincomponents of the electrons could be replaced by an effective onecomponent electron fluid where the Pauliexclusion potential (i.e, FHP) is included after averaging over the two components.
An example of a classicalmap calculation of the EOS of lasershock compressed hydrogen has been given by Dharmawardana and Perrot hyd (), where a Hugoniot has been calculated and compared with those from other methods (see Fig. 5). The article by Michael Desjarlais in this issue also refers to the problem of the equation of state (EOS) of highly compressed hydrogen Dejarlais11 (). A proper experimental probe of such lasercompression experiments needs to address some method of independent measurement of the electron temperature and the ion temperature . If the electrons and ions are in equilibrium, . Then the usual DFT methods using the BornOppenheimer decoupling would be expected to give a good prediction of the EOS, and also the Hugoniot. The EOS calculation is essentially a calculation of the partition function. This requires the evaluation of
(22) 
Here the total Hamiltonian is rewritten in terms of , , and the electronion interaction which is again a Coulomb potential. We have included a crosssubsystem temperature which is simply for equilibrium systems. If a BornOppenheimer approximation is used, the electrons ‘do not know’ the temperature of the ions, and vice versa. For equilibrium systems, a BornOppenheimer correction can be introduced, e.g., as in Morales et al. Morales10 (). However, the add on correction introduced by Morales et al. will change the virial compressibility, leaving the small behaviour of the protonproton structure factor unaffected, and hence the effect on the compressibility sum rules has to be examined. In any case the DFT implementations in codes like VASP, or SIESTA cannot deal correctly with the case , and it is not clear if they treat the term in the partition function correctly even in the equilibrium case, due to the use of the BornOppenheimer approximation which prevents the possibility of coupled electronion plasma modes in the system.
The CHNC technique is a nondynamical method that does not need the BornOppenheimer approximation. It correctly treats the crossinteraction even for twotemperature systems, as established by direct MD simulations cdwmur (). Fig. 5 shows that the SESEME and other standard EOS agree with the CHNCBO calculation where is set to , while the Lasershock experiments, where may hold, should agree with chosen as the temperature of the scattering pair. Ion masses are much larger than , and hence approaches the electron temperature, as demonstrated in Dharmawardana and Murillo via MD simulations cdwmur (). In effect, the calculation of the Lasershock hydrogen Hugoniot has to address nonequilibrium effects, as well as nonadiabatic effects associated with the use of the BornOppenheimer approximation in standard simulations. The conclusions of Galli et al. Galli02 () also point to nonequilibrium effects associated with the electronion interaction, i.e., precisely the term in the Hamiltonian indicated in Eq. 22. Our own views have evolved beyond what we stated in Ref. hyd (), and the subject probably needs to be revisited, within a twotemperature quasiequilibrium setting, without making the BornOppenheimer approximation, especially at very high compressions.
iii.1 Pseudopotentials
We may also consider the case when the ions are not fully ionized into bare nuclei, but carry a group of core electrons. For instance, Alplasmas at 0.5 eV and normal compression have a charge and a core of 10 electrons. Although it is sufficient for many problems to treat the Al as point charges, a more accurate theory may wish to include the effect of the the core radius and welldepth of the electronion interaction via a pseudopotential. Such pseudopotentials are well known at zero temperature. A very simple model is that of Ashcroft, while modern implementations are very sophisticated.
Alpseudopotentials suitable for WDM have been given in parametrized from by Perrot and Dharmawardana elr98 (). The basic idea is to generate the charge density around a given nucleus of charge , immersed in a UEF of density parameter , at a temperature . The ion is place in a spherical cavity in the positive background (for details see Ref. elr98 ()) and is determined by a KohnSham calculation which satisfies the Friedel sum rule and other properties. Then we define a weak nonlocal pseudopotential by the following relations in space.
(23)  
Here is the Lindhard response function at finite and electron WignerSeitz radius , and is a local field correction consistent with the density and temperature of the UEF. Further more, is the Fourier transform of the realspace freeelectrondensity pileup calculated at the jellium density and temperature , for the nucleus . That is
(24)  
(25) 
The bound electron density is obtained from the orbitals of the finite KohnSham equation as in Perrot Perrot93 (). Here it should be noted that the bound electrons have to be assigned to a nucleus keeping in mind that some bound states are those of ‘hopping electrons’ which form a band of localized states near the continuum hopping92 (). Equation 23 defines the pseudopotential to be capable of recovering the chargepile up via linear response. Hence it has to render a weak potential. It is not very satisfactory if the resulting pseudopotential proves to be strong. However, the method seems to work in most cases. The pseudopotential can usually be parametrized (as in an Ashcroft emptycore potential), with a core depth and a core radius such that
(26)  
(27) 
This is evidently a very simple form, compared to modern, hard, nonlocal pseudopotentials used in solidstate calculations at . Such modern potentials remove the core, but a KohnSham equation has to be solved as they are not weak, and cannot be treated using linear response. However, we have found that simple potentials as in Eqs. 2326 are adequate for even the liquidmetal regime close to the melting point, even for metals which require nonlocal pseudopotentials at . Excellent accuracy is obtained if the response functions are calculated for electrons with an effective mass specified for each case. It is particularly important to note that the ‘mean ionization’, i.e., is a parameter which appears in the pseudopotential. The is also the Lagrange parameter defining the total charge neutrality of the plasma, as discussed in Refs. dwp82 (), Perrot93 (). A few examples of this type of simple pseudopotentials are given in Table 1.
element  

Al  3.141  3.0  0.3701  0.3054  0.998 
C  2.718  4.0  0.0  0.3955  1.658 
Si  3.073  4.0  0.0  0.9475  0.98 
The C and Si pseudopotentials were used to generate PDFs of these ionic liquids and compared with CarParinello simulations in Ref. csige90 (). Thus these pseudopotentials can be used in the CHNC equations, or in the CMMD simulations, to take account of the existence of a finitesized core. Such methods can be used to discuss properties of warm dense matter, thus providing a complementary approach to the simulations based on statistical potentials discussed by Graziani et al. in the context of the Cimarron project for simulations of warm dense matter Graziani11 ().
Iv Twotemperature quasiequilibria and nonequilibrium systems.
When energy is deposited rapidly in matter using laser radiation, the electrons absorb the energy directly and equilibriate among themselves, achieving a very high electron temperature . The ion subsystem, at temperature , takes much longer to heat up due to the slow temperature relaxation via the electronion interaction. Hence, in laserheated systems, it is common to find . The inverse situation prevails in shockheated materials since the energy of the shock wave couples to the heavy ions and not to the electrons Ng11 ().
The possibility of using a static approach like the CHNC for nonequilibrium systems resides on Bogoliubov’s idea of timescales and conserved quantities. We have exploited these ideas in our work on hotelectron relaxation, both within Green’sfunction methods, and within CHNC methods elr98 (). The parameters , in a twotemperature system are merely Lagrange parameters which assert that, for certain time scales , , the subsystem Hamiltonians , are conserved quantities. Similarly, a number of other parameters, e.g., quasiequilibrium chemical potentials, thermodynamic potentials, pseudopotentials, , etc., attached to the subsystems may be conserved for the selected time scales. In fact, the original discussions of quasiequilibria by Bogoluibov were used in Zubarev’s theory of nonequilibrium Green’s functions, and RPAlike results for the quasithermodynamics as well as energy relaxation were addressed therein. However, RPAlike theories are of limited value. In strongly coupled regimes, the PDFs associated with the given subsystems can be constructed using CHNC, where the use of the correct intersubsystem temperatures (e.g., ) for evaluating intersystem PDFS (e.g., ) is essential. The nature of this intersystem temperature is revealed by its appearance in the intersubsystem energyrelaxation formula elr08 (). A calculation of the distribution functions of two temperature plasmas using HNC methods as well as MD methods was given recently cdwmur ().
V Inhomogeneous systems
The classicalmap technique uses a classical fluid at a finite temperature to represent a uniformdensity quantum fluid at . The parameter is density dependent, and hence the extension to a system with an inhomogeneous density is not straightforward. Furthermore, integralequation techniques like the HNC become very complicated when applied to inhomogeneous systems. Molecular dynamics can be applied if a viable mapping can be constructed. However in this connection we should note that studies of confined classical electrons in parabolic traps have also yielded useful insightsWrighton09 ().
The classicalmap technique treats the zerothorder Hamiltonian exactly, i.e., the map is constructed to reproduce the known quantum solution classically, requiring the confining potential to be mapped as well. Even when there is no confining potential (other than a uniform background), the zerothorder problem of has to be correctly treated. This was done in the UEF problem by constructing a Pauli exclusion potential (i.e., the Fermihole potential) to recover the Fermi hole in the exactly.
When noninteracting electrons are placed in an external potential, e.g., a parabolic trap, the uniform density modifies to a new distribution . Classically, this distribution is of the Boltzmann form, Eq. 2 where contains all the terms found in the exponent of the HNC equation. Thus, given the calculated from a quantum mechanical treatment of which contains the parabolic external potential, it is necessary to invert the HNC equation to get the effective classical potential which corresponds to . A simplified approach to this was used by us in ref. qdot09 (). At this stage the calculation is somewhat similar to the determination of the Pauli exclusion potential, and hence the specification of an effective fluid temperature does not become necessary. The classical Coulomb fluid at a finite temperature is still necessary for dealing with the manybody effects generated from the Coulomb interaction. However, given a nonuniform distribution, there is no evident method of defining a unique and the simplicity of the original CHNC method is lost. Further more, the electronelectron pairdistribution functions now depend explicitly on two coordinates, viz., . The use of moleculardynamics simulations is more convenient in dealing with systems where the simplicity of homogeneous systems is lost. Another advantage an MD simulation is that the the bridgefunction approximations are avoided.
In mapping an inhomogeneous system of density to a homogeneous slab of density we have used the form quasi (); Jost05 (); qdot09 (),
(28) 
in dealing with 2D systems. The same method has been used by GoriGiorgi and Savin for 3D systems ggsavin (). Using such a uniform density to define a unique temperature of an equivalent classical fluid, we were able to reproduce the charge distribution of interacting electrons in 2D quantum dots obtained from QuantumMonte Carlo methods. However, as we used CHNC, it was necessary to introduce bridgefunctions and boundary corrections which impaired the transparency of the classical map. Hence this work qdot09 () may be regarded as a preliminary attempt.
Vi Conclusion
We have outlined the classicalmap technique of treating the quantum manybody problem in Fermi systems via a mapping to an equivalent classical system at a densitydependent effective temperature different from the physical temperature, and where the particles interact by a pair potential consisting of a Pauliexclusion potential and a diffractioncorrected Coulomb potential. Large numbers of particles, and their thermodynamics or quasithermodynamics can be easily calculated. Since pairdistribution functions can be calculated accurately, and at any value of the coupling constant, the adiabatic connection formula provides results for the nonlocal exchangecorrelation functionals in an entirely unambiguous, rigorous manner. No gradient corrections, metafunctionals etc., are needed. The BornOppenheimer approximation is not necessary as the CHNC technique is not dynamical. Hence the method would be of great interest from the point of view of equationsofstate studies, both for equilibrium, and for quasiequilibrium systems.
Since suitable derivatives of the free energy with respect to density, temperature, and spin polarization lead to Landau Fermiliquid parameters, the method is capable of easily furnishing alternative results for the effective mass , Landé factor, localfield factors of response functions etc., which are difficult to determine by standard Greensfunction perturbation techniques of quantum manybody theory.
The application of the method to inhomogeneous systems is still poorly developed. Similarly, the method, being a technique for the total energy as a functional of the pair density, is similar to DFT in not yielding spectral information within its own formal structure.
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