Proc. Indian Acad. Sci. (Chem. Sci.), Vol. 106, No. 2, April 1994, pp. 217-227.
9 Printed in India.
Density functional theory as thermodynamics
A NAGY i and ROBERT G PARR b*
~Institute of Theoretical Physics, Kossuth Lajos University, H-4010 Debrecen, Hungary bDepartment of Chemistry, University of North Carolina. Chapel Hill, North Carolina 27599, USA
Abstract. Thermodynamical interpretation of the density functional theory is developed for an electronic ground state, with particular emphasis on the fact that when the system is not homogeneous the electronic kinetic energy varies from point to point. Local thermo- dynamical quantities, temperature, free energy, and entropy, are defined. The formalism yields the same Sackur-Tetrode expression for the local entropy earlier obtained by other methods, plus corrections of that equation for "nonideality'. Various aspects of the thermo- dynamical description are discussed.
Keywords. Density functional theory; thermodynamical description; local entropy; modified Sackur-Tetrode equation; electron gas.
1. Introduction
The present paper takes up from a paper by Ghosh et al (1984) entitled "Transcription of ground-state density functional theory into a local thermodynamics". The
"transcription" is an approximate rather than an exact density functional theory. We explore here the question of whether there exists a more general "thermodynamical"
theory that is exact, or at least more accurate than the GBP theory. We also remark on several subtle aspects of the theory.
We are concerned with the ground state of an electronic system, exactly described by the density functional theory of Hohenberg and Kohn (1964). The central idea is that even for this ground state, the local kinetic energy varies from point to point.
If we denote two thirds of the kinetic energy density per particle at r by T(r), we enlarge the language of the density functional theory to include T(r), which we call the local temperature. We then find various quantities to be functionals or functions of the electron density p(r) and the local temperature T(r). The resulting formulation takes the shape of the classical thermodynamics of a nonhomogeneous system under local equilibrium assumptions:
The GBP theory developed this basic idea. Two derivations have been given of this theory. In the first (Ghosh et a11984), statistical methods were used. A Wigner-like phase-space distribution function f(r, p) was found by maximizing a phase-space information entropy subject to the conditions that f yields density p(r) and the local
* For correspondence
217
218 ft Nagy and Robert G Parr
kinetic energy density. The results are a total kinetic energy of the form
E~[p] = f (3/2)p(r)kT(r)dr,
(1)
and an exchange energy K = (Tt/2)Sp2(r)/kT(r)dr. It is found that local entropy, free energy and so on can be defined, and the analogy with the classical thermodynamics of fluids proves close (Ghosh and Berkowitz 1985). Further, if T is assumed to be a function (rather than a functional) of p, formulas of Thomas-Fermi-Dirac form result. Also, and this provides compelling validation of the local temperature concept, numeral values of the coefficients c~ and Cx of the Thomas-Fermi-Dirac theory are found which are improvements of the original values (as measured by fits to atomic Hartree-Fock data). Identical results were later found by Berkowitz (1986) to follow from a Gaussian-resummation approximation to the Hartree-Fock first-order density matrix. Predicted exchange energies (Lee and Parr 1987) and predicted Compton profiles (Parr et ai 1986) are very good. The whole theory comfortably takes the form of thermodynamics with local temperature. See also Ghosh and Parr (1986).
The GBP theory is, however, not exact. There are assumptions at the beginning, either regarding the existence of an everywhere positivef(r, p) (not possibly precisely true for a quantum-mechanical many-body system) (Ghosh et al 1984) or as to the general appropriateness of a resummation of a density-matrix expansion (a mathe- matical simplification at the beginning) (Berkowitz 1986). There has been an attempt to add accuracy tof(r, p) by adding constraints in the entropy maximization (Morrison
1991), but this approach has not yet proved helpful.
What we do in the present paper is to proceed beyond the GBP procedure by postulating thermodynamic description without resorting to either statistical methods or density-matriX analysis. We place special emphasis on the local temperature and local entropy concepts, finding new properties of the former and suggesting new uses for the latter. A general "local density" version of the theory is described, and the GBP theory is recovered as a special case. While the GBP theory gave an ideal-gas-lik e formula for the local entropy, the new theory yields a corrected formula incorporating effects of what we could call "nonideality".
That the electrons in an atom or molecule in its ground state could be treated statistically is really a fundamental idea of the Thomas-Fermi theory, and "statistically"
is general means, of course, "thermodynamically". The original Thomas-Fermi theory is just the zero-temperature limit of a corresponding finite-temperature theory (Feynman et al 1949), and as such embodies ideas of equilibrium and ensemble average. (Note, for example, the use of just these terms in the paper by Cowan and Ashkin 1957).
Not so well known, perhaps, is that the idea of varying local temperature also is old. The equilibrium in question in general leaves gradients in local electronic kinetic energy. Identifying kinetic energy with temperature, one then has for an atom a temperature that is zero at the boundary but much larger at an atomic nucleus (about which more below). Even Fermi seemed to have been aware of this in his first paper on statistical theory (Fermi 1928).
Consider the following from the Kirkwood-Mazo paper on liquid helium (Maze and Kirkwood 1958):
Density functional theory as thermodynamics 219 Thus we may say that, subject to the approximations we have made, a quantum fluid at temperature T behaves like a classical fluid at temperature and at the same density. ~ is, of course, a function of T and v . . . . In this situation the kinetic temperature ~ depends implicitly upon position through its relation to the electronic density, which is not constant because of the nuclear potential.
What was done in the Ghosh-Berkowitz-Parr paper (Ghosh et al 1984) was but to put this idea forward once again. Among subsequent discussions of the idea we call special attention to those of Deb and Ghosh (1987), Parr and Yang (1989), and Dreisler and Gross (1990).
2. The exact thermodynamics
Consider an electron system of interest with the ground-state electron energy functional E:
[p] = j el-p, r] dr, (2)
E
where e is the energy density, a functional of p. From the Hohenberg-Kohn theorem the energy functional takes its minimal value E o at the exact ground-state electron density Po:
Erpo] = Eo, (3)
when the Euler-Lagrange equation,
~E
p=po : " (4)
holds. The Lagrange multiplier/z enters the equation because of the constraint
fpdr
= N. (5)That is, the number of electrons, N, is fixed. The quantity/~, constant through the system, is the chemical potential of the system, the escaping tendency of the electronic cloud.
In order to obtain "thermodynamics" we assume that there exists a new functional
= fp~[p,,p;r]dr,
(6)g[p,p,]
where p, is the entropy density and the energy density PE is a functional of both p and Pr With entropy now in the problem, in the minimal condition we need another constraint for the entropy with a Lagrange multiplier T, in addition to the constraint for the density p having a Lagrange multiplier/Z The extremum principle for the ground-state will be
~{f[pe[p,,p;r]-T(r)p,-~(r)p]dr}=O,
(7)220 ,4 N a g y and Robert G Parr
where the Lagrange multipliers T and/~ are presumed to be functions of r and will be referred to as the local temperature and local chemical potential, respectively. The Euler-Lagrange equations of the resulting "thermodynamics" then are
and
O/~ = T(r),
(8)
,9,
P~
As we are seeking a full thermodynamical formulation, we also require that the local form of the fundamental equation,
g = T S - p V + fiN, (10)
holds, that is
p e = T p s - p + ft p. (ll)
Here p is the local pressure. This must, of course, be defined, for which see the paper of Bartolotti and Parr (1980). At this point we decide that our thermodynamics will be the thermodynamics of the Kohn-Sham noninteracting reference system. One does not need to do this but it appears natural because of the expectation that the properties of the interacting system will be less simple to model. The noninteracting system is more ideal-gas-like!
Using (1) the Kohn-Sham kinetic energy density can be written as
t, = (3/2)pkT, (12)
while the total energy density is
e = ( 3 / 2 ) p k T + PC + Pexc, (13)
where PC and pexc are the classical Coulomb and the exchange-correlation energy density. Now we certainly have
so that
T = T [ p ] , (14)
PE[ T, p] = e[p]. (15)
Note that (14) does not say that T is a function of p but more generally that it is a functional of p.
Equations (8) and (9), together with (13) and (15), constitute an outline of a thermo- dynamical description. However, these equations do not yet uniquely define the entropy density p,. There may still be many pj satisfying (8) and (9).
The independent "thermodynamical" variables of choice will be T and p, and so it will be convenient to use the Helmholtz free energy density PA defined in the usual way,
PA = Pr, -- Tps. (16)
D e n s i t y .functional theory as t h e r m o d y n a m i c s
The functional derivatives of the free energy
A [ T , p ] = f p A [ T , p ; r ] d r
are given by 6A
~ a = -- Ps,
and
6A = f t .
We similarly define the enthalpy density,
Pn = PE + P = Tps + fzp,
the density of the Gibbs potential,
Pc = P~ - T p s + P = ftp,
and the density of the grand potential,
to = Pe - TPs - ~P = - P.
Corresponding Legendre transforms of the entropy density are p~ - (P/T) =
(p~/T)
- ~ / T ) pp s - ( p E / T ) - ( p / T ) = - ( f ~ p / T ) ,
and
Ps - ( P e / T ) + (fliT) = (p/T).
221
(17)
(18)
(19)
(20)
(21)
(22)
(23) (24)
(25) At this stage we have the whole set of customary thermodynamic relations. But the results are purely formal unless we can provide specific implementations, which in turn require determination of a formula or formulas for the entropy. This we now consider.
3. Thermodynamics in the local density approximation
In the local density approximation, all quantities are functions of p. For simplicity, we may take the exchange-correlation energy density exc to be a function of p only:
exc(p ). Simple partial derivatives then replace functional derivatives in (8), (9), and (18).
From the definition of the Helmholtz free energy density PA (16) and the local form of (18)
0T p = - p~' (18a)
222 ,4 N a g y and R o b e r t G P a r r
we. arrive at the well-known Gibbs-Helmholtz differential equation
o r \ T f ~ " (26)
It can be readily integrated with the aid of (13) for the energy density PE:
PA = -- ( 3 / 2 ) p k T l n T + p ~ , + T f ( p ) , (27)
where
P ~ , = PC + Pexc (28)
is the total potential energy density and f(p) is an arbitrary function ofp. Equations (16) and (27) lead to an expression for the entropy,
Ps = ( 3 / 2 ) p k + ( 3 / 2 ) p k In T - f ( p ) , (29)
while from (18) the local chemical potential reads,
[: = - ( 3 / 2 ) k T l n T t vk, + T(6f/Op), (30)
where
is the functional derivation of the total potential energy Eno t with respect to the density, i.e. the Kohn-Sham potential. The fundamental equation (11) provides an expression for pressure,
P = PP -- PA = p v ~ -- Ppot + T [ p ( 6 f / f p ) - f ] . (32) The difference P V k ~ - p ~ , can be given by
PVk~ - P ~ t = p v - p e ~ + (1/2)pvr (33)
where
v out(r) = | p(r) dr', (34)
J l r - r ' l
is the classical Coulomb repulsion potential. So the pressure takes the form
p = p ( v - e~c) + (1/2)PVr + T [ p ( 6 f / 6 p ) - f ] . (35) In the local density approximation for the exchange we obtain
p = p 2 ( d e x c / d p ) + (1/2)pVcou~ + T [ p ( 6 f / 6 p ) - - f ] . (36) Expression (35) or (36) constitutes the local virial theorem of this new theory.
4. The modified Sackur-Tetrode equation and the local virial theorem
Recently, the local virial theorem has been derived in the density functional theory (Nagy and Parr 1990)
p = p k T + Pxc" (37)
Density functional theory as thermodynamics 223 The most general expression for Pxc is given by the exchange correlation stress tensor (Deb and Ghosh 1987)
Pxc = - (l/a) TrOxc, (38)
while in the local density approximation (Nagy and Parr 1990) we have
Pxc = p2 (dexffdp)" (39)
We expect the two expressions for the pressure [(35) and (37)] to be the same if we return to the original density functional theory, i.e., if p and T are not independent any more,
V(r) = ~(p(r)). (40)
This provides an expression for the function f ,
p(vx~ -- ex~) + (1/2)pvo,z + T[p(6f/t~p) -- f ] = p k T + Px~. (41) It can be proved by a simple substitution that
f --e _1 _
f = k p l n p - k p ( 1 - b ) - p p - i v . . . . + 2vc~ P ~ / P dp, (42)
where b is an arbitrary constant. Using this expression for the function f we get for the entropy
= p~d + p p - ~ G~ - e ~ + -i ~o~l -- P ~ / P dp'
ps ~(p)
where
p~d = p [k { (5/2) - In p + (3/2) In T} - b],
(43)
(44)
is the entropy of the ideal gas, i.e. the Sackur-Tetrode equation if the constant b is chosen to be
b = - (3/2) k ln(2nk). (45)
So the entropy contains a correction to the Sackur-Tetrode equation arising from the nonideality.
However, assumption (40) causes the present thermodynamic description to become ambiguous. Assume that a relation between the quantities T and p exists,
#( T, p) = 0. (46)
Then we may write
0o,
and
\ c 3 T J p " \ d p J r
a0
(47)
(48)
224 ,4 Nagy and Robert G Parr
Multiply (47) by an arbitrary function h and add (47). There results
= 9 9
We see the ambiguity in the expressions of ps and/~:
+ = _ p~,
\ dTJp \dT]p
+ =•.
\
dp
,I T \dP,Ir(49)
(50) (51) Using the relation (40) between T and p,
we obtain
and
g = T - 2:(p) = 0, + h = --Ps
\ dT/p
dO ]r do
(40a) (52)
(53) Since h is arbitrary, ps and/~ are not completely determined.
We can take advantage of this ambiguity. With one choice of the function h,
t l )
h = p p - 1 l)xc -- exc + -~ co=t - P,,~/P
dp, (54)
~(p)
we obtain the ideal gas expression (44) for the entropy
ps
=p:.
( 5 5 )So in the limiting case of the original
density
functional theory the G B P theory is recovered.It is possible to make use of the ambiguity in another way. We may choose h so as to give
/~ = # = const. (56)
That is, the local chemical potential is constant and equal to the chemical potential of the original density functional theory. This requires
(dr)-'[ d/ 3krln r_3kr_3 kdr].
h = ~pp L d p - 2 2 2 d p / (57)
In this case the Euler-Lagrange equation of the original density functional theory, 3 d T
~
k r + ~pk~p + Ors=#, (58)Density functional theory as thermodynamics 225 is obtained. If we use the scaling argument of Ghosh et al 1984, T(r) has the form
2C 02/3 r
T(r) = ~-k,. , ,, (59)
and for the kinetic density we obtain the well-known Thomas-Fermi kinetic energy expression
t~ = cp s/3 (r). (60)
Then the Euler-Lagrange equation (58) takes the form of the Thomas-Fermi equation if exc is neglected and the Thomas-Fermi-Dirac equation if exc ~
pl/3.
The thermodynamical theory introduced here enlarges the original density functional theory. A generalized expression for the local entropy density is obtained:
in addition to the Sackur-Tetrode expression it contains a term arising from the nonideality of the system. Furthermore, a more general expression is gained for the pressure, i.e. for the local virial theorem. When this new thermodynamical picture breaks down, i.e. the temperature is not independent of the density, all the earlier results of the GBP theory are nicely recovered.
5. Discussion
One important point not yet mentioned has to do with an ambiguity in the definition of local kinetic energy. This arises from the well-known equations between two global kinetic energy formulas, in the simplest case,
Ek,. = f g r ~V2)~bdz = ~ f V r 1 6 2 dT. (61) Following an argument in favor of it by Cohen (1984), for a Hartree-Fock or K o h n - Sham single determinant of orbitals, we take Ek~" = Stdr, where
~, 1VpiVp, 1 V2p (62)
t = i 8 pi 8
For an exact wavefunction expressed in terms of natural orbitals, one has a similar formula in terms of natural orbital densities, with occupation numbers ni weighting the individual orbital components. The arbitrariness is in the coefficient of the V2p term, which has no effect on the global Eki .. In favor of the coefficient - 1/8, among other arguments, is the fact that it not only makes the local temperature zero at the boundary of an atom or molecule, but it makes it infinity at any nucleus (a point not made in the literature heretofore) and ordinarily positive eleswhere. [An exception is the midpoint of H~- for large internuclear distance, as pointed out to RGP by Conyers Herring (1986).] This ambiguity deserves further study, especially in view of the importance Bader (1990) attaches to the quantity V2p in his theory of atoms and molecules.
Gadre and coworkers have studied the "information entropy" of an electronic system, defined as a position,space entropy plus a momentum-space entropy, both
226 A Nagy and Robert G Parr
defined as information measures for the density without regard for the temperature distribution. Our local temperature is a momentum measure, of course. So we believe that the Sackur-Tetrode-like entropy of Eq. (43) is a quantity of considerable intrinsic interest, which may well serve the purpose to which Gadre now puts information entropy.
Here, an analogue to the classical steady-state thermodynamics has been presented.
One of the basic assumptions of the steady-state thermodynamics is the assumption of local equilibrium (Cowan and Ashkin 1957). This states that all thermodynamic functions of state exist for each small volume element of the system and that the thermodynamic quantities for the nonequilibrium system are the same functions of the local variables as the corresponding equilibrium quantities. Our treatment accords with this assumption, as our main assumption is the existence of the local form of the fundamental thermodynamical Eq. (11). Based on this we have constructed the thermodynamic quantities, including the local temperature, the local chemical potential and the local pressure- all r-dependent quantities because the electrons constitute an inhomogeneous system. The description may prove useful for understanding dynamic as well as static aspects of many-electron systems.
It is well known that the local density functional theory in which the exchange- correlation energy that is a component of Eq. (2) is a function of p and not a functional of p has wide applicability but has limited numerical accuracy. Our Eq. (5) plus a local exc also define a "local" theory, but involves two functions, p and Ps (or p and T) and so is more general and potentially possibly more accurate. Demonstrations already are in the literature that quantitative improvements indeed do result from such generalized "local" functionals (Parr et al 1986; Ghosh and Parr 1986).
We conclude by expressing the thought that the formulation here described may aid in solving the longstanding problem of finding for nonhomogeneous systems a simple yet accurate statistical model to replace the Thomas-Fermi-Dirac model. It is known that much of the (considerable) inaccuracy ~f the Thomas-Fermi description is removed once one requires the electron density to behave properly at a nucleus (Parr and Ghosh 1986). Now that we know that for an atom there is a temperature varying from zero at the boundary to infinity at the nucleus, ought one not try to find a thermodynamic model inputing that information at the beginning rather than discovering it at the end? The Thomas-Fermi-Dirac models do not, it seems, appropriately incorporate nucleus cusp conditions. Perhaps the temperature idea will enable this somehow to be done.
Acknowledgements
This publication is based on work sponsored by the Hungarian-US Science and Technology Joint Fund in cooperation with the National Science Foundation and the Hungarian Academy of Sciences, under Project 146/91.
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