Robert Balawender 1,2,a and Andrzej Holas 1

Transcription

1 herodynaic extension of density-functional theory. I. Basic Massieu function its Legendre and Massieu-Planck transfors for uilibriu state in ters of density atrix Robert Balawender 12a and Andrzej Holas 1 1) Institute of Physical Cheistry of Polish Acadey of Sciences Kasprzaka 44/52 PL Warsaw Poland 2) Eenheid Algeene Cheie (ALGC) Vrije Universiteit Brussel Pleinlaan 2 B-1050 Brussels Belgiu A general forulation of the uilibriu state of a any-electron syste in ters of a (ixedstate enseble) density atrix operator in the ock space based on the axiu entropy principle is introduced. Various state functions/functionals are defined and investigated: the basic Massieu function for fully open therodynaic syste the effective action function for the fully closed (isolated) syste and a series of Legendre transfors for partially open/closed ones the Massieu functions. Convexity and concavity properties of these functions are deterined their first and second derivatives with respect to all arguents are obtained. Other state functions the Gibbs-Helholtz functions are obtained fro previous ones as their Massieu-Planck transfors i.e. by ultiplying the by inus teperature and by specific transforation of arguents (which involves the teperature). Such functions are closer to traditional (Gibbs Helholtz) therodynaic potentials. However the first and second derivatives of these functions represent ore coplicated expressions than derivatives of the Massieu functions. All introduced state functions are suitable for application to various extensions of the density functional theory (D) both at finite teperature and at zeroteperature liit. Derivatives of state functions are essential for deterining the cheical reactivity and other descriptors of the conceptual D extensions. Key words: quantu statistical echanics Massieu function Legendre transfor Massieu- Planck transfor Gibbs-Helholtz potentials Boltzann-Gibbs distributions therodynaic uilibriu state density atrix density-functional theory. a Author to who correspondence should be addressed rbalawender@ichf.edu.pl 1

2 I. INRODUCION Density-functional theory (D) is a rigorous approach for describing the ground state of any electronic syste. 1-5 he success and popularity of D are based on the fact that the fundaental variable the electron density is an observable and also that it is a uch sipler object than the any-body wave or Green s function. Siple approxiations of the exchange-correlation energy functional used by ipleentations of D perfor rearkably well for a wide range of probles in cheistry and physics 6-8 particularly for predictions of the structure and therodynaic properties of olecules and solids. Despite successes of D its applications still suffer fro errors that cause qualitative failures in soe predicted properties. A systeatic approach to constructing universally applicable functionals is a hard proble and has reained elusive. A possible step forward is to recover a balance between the qualitative (conceptual) and the quantitative (coputational) branches of D and to have a deeper look at violations by currently used functionals of the exact conditions that should be satisfied by D or exaple the delocalization error originates fro the violation of linearity of the energy as a function of fractional charges the static correlation error eerges fro the violation of constancy of the energy as a function of fractional spins Both are iportant to assess errors of functionals. 25 Distressingly the status of nonintegral electron nuber fractional spins and fractional occupations in the D is probleatic he answer to the question how syste properties depend on the electron nuber is crucial. Conceptual D offers a perspective on the interpretation and prediction of experiental and theoretical reactivity data based on a series of response functions to perturbations in the nuber of electrons and/or external potential. his approach enables a sharp definition and then coputation fro first principles of a series of well-known but soeties vaguely defined cheical concepts such as electronegativity hardness and ukui function. Identification of cheical concepts and principles in ters of derivatives with respect to the electron density (local cheical reactivity descriptors) or with 2

3 respect to the total electron nuber (global properties) ruires an extension of the density doain to enseble density which integrates to a real positive electron nuber. here are two ain approaches to the introduction of a nonintegral (fractional) particle nuber into D irst approach applies directly a sooth extension of the expression for the electron density and the kinetic energy by introducing fractional occupations of the involved orbitals and siultaneously applies such density as an arguent of the approxiate functionals (like LDA or GGA ones) that were originally derived as approxiations for the integral-electron-nuber systes but reain atheatically eaningful for arbitrary density. In the second therodynaic approach the extension fro an integral to fractional electron nuber is realized through a grand canonical enseble of Merin s finite-teperature D at zero-teperature liit. Besides these two approaches one can also work out the connection with the correct integral-n functionals by applying suitable interpolation ethods e.g. a quadratic approxiation of E(N). While various ways for an extension fro integral- to fractional-nuber approaches ay be atheatically acceptable (correct) nevertheless iportant physical arguents exist that reove arbitrariness. Based on the axioatic forulations (foral atheatical properties) of the Hohenberg-Kohn functional only such functional that satisfies the variational principle for the electronic energy as a functional of the density and is continuous convex and size consistent can be accepted as the exact functional he functional defined with the therodynaic approach satisfies these conditions. Moreover this functional is identical to the functional obtained using the Legendre transforation and density-atrix constrained search techniques. No functional consistent with the variational principle can reach a lower iniu than this functional. 52 Another arguent is that only the therodynaic approach is the proper basis for investigation of open systes and is capable of tackling all issues in the cheical reactivity. 53 3

4 he finite-teperature foralis of D was forally set up long ago by Merin. 45 Only a few generalizations have been reported so far We wish to extend the density doain of D to the spin-polarized density which can integrate to a real positive nuber of particles and a real nuber of spins. Before developing this foralis (which will be done in the next papers in the series) we will concentrate in this paper the first in the series on properties of the uilibriu state of the any-electron syste expressed in ters of the density atrix operator in the ock space. Results of the present paper ay also provide foundations for other extensions of D like the current-density functional theory the (one-particle) density-atrix functional theory the pair-density functional theory etc. In Sec. II we provide the entropic forulation of the uilibriu state using the axiu entropy principle. Based on the effective action foralis 6061 and Jensen inuality convexity and concavity properties of the basic Massieu function 62 and its functional Legendre transfor the effective action function are established. Although in the general therostatics the exponential function appearing in the Boltzann-Gibbs distributions can be replaced by soe other increasing function 6364 only the well-known Boltzann-Gibbs-Shannon entropy is considered here. he results derived in Sec. II which characterize a fully open and fully closed syste are translated in Sec. III to partially open/closed systes by applying the Legendre transforation echanis. Properties of all these functions naed the Massieu functions are discussed in particular their convexity and concavity. he relation between the secondderivative atrix and the covariance atrix is derived. In Sec. IV the relations between uilibriu state functions defined by the above foral atheatical procedure (Massieu functions) and by those defined in analogy with the traditional (e.g. Gibbs and Helholtz) potentials (here Gibbs-Helholtz functions) are investigated using the Massieu-Planck transforation. 65 able I displays relations between all entioned state functions. he expression for the second derivative atrix of the Gibbs-Helholtz functions in ters of the 4

5 second-derivative atrix of the corresponding Massieu functions akes a link between a real therodynaic transforation (easy controlled in the laboratory) and that due to iposing constraints that appear ore natural in the axiu entropy principle. he preeinence of Massieu functions over the Gibbs-Helholtz functions in a copact forulation of a generalized statistical echanics and therodynaics 66 and in undergraduate instruction is evident. However the standard reference books on therodynaics usually do not give enough inforation about these functions; the axioatic therodynaics introduced by Callen 71 is to our best knowledge an exception. Advantages of the Massieu functions over the Gibbs- Helholtz functions in analyzing the properties of the uilibriu state becoe apparent when we copare coplexity of the expressions for their second derivatives. Sec. V is devoted to conclusions. In Appendix A the derivatives of the Massieu functions and the Gibbs-Helholtz functions are obtained using the atrix notation. his Appendix ay be useful also for a pedagogical reason. In Appendix B the notion of positive (negative) definiteness of a atrix is recalled. In Appendix C the definition and properties of Legendre transfors are recalled. Appendix D provides a collection of identities satisfied by the vector functions of therodynaic variable transforations. Note: Atoic units are used throughout the paper. he abbreviation w.r.t. eans with respect to fn. eans function. r denotes the trace operation in the ock space. eans a suation over all indices however if the suation index (variable) is continuous eans an integration. See Appendix A for details concerning the vector and atrix notations. 5

6 II. ENROPIC ORMULAION O HE EQUILIBRIUM SAE USING HE MAXIMUM ENROPY PRINCIPLE. CHARACERISIC SAE UNCIONS. he atheatical description of a olecule as a any-electron syste of interest is based on a state-vector linear space; generally it will be the ock space in the second quantization forulation and operators in this space written in ters of the field operators r and ˆ r ˆ. Microscopically (quantu echanically) such a syste is defined by its tie-independent non-relativistic Hailtonian operator Ĥ whereas in the acroscopic doain of the therodynaics it is specified by a set of observables O ˆ j the generating operators. he corresponding syste will be naed the O ˆ j therodynaic syste for this olecule. All generating operators are assued to be tie-independent heritian local (r-dependent) or global (r-independent) and utually couting Oˆ ˆ i O j 0. he therodynaic syste is characterized by a density atrix (DM) operator ˆ (heritian positive seidefinite and the unit-trace one). or a general ixed state ˆ can be expanded in ters of its eigenvalues and eigenstates ˆ ˆ g g 0 g rˆ g 1 (1) K K K K K K L KL K K K K where the set K of ock-space vectors is coplete K K ˆ1. he expectation (average) value of any operator O ˆ j is defined in ters of ˆ as Oˆ rˆ Oˆ g Oˆ. (2) j j K K j K K he expansion (1) was applied for the second for. Along with averages the first-order covariance atrix of expectation values ay also be of interest. Its eleents are defined as K 6

7 O ˆ O ˆ OO ˆ ˆ O ˆ O ˆ O ˆ O ˆ O ˆ O ˆ. (3) i j i j i j i i j j Diagonal eleents correspond to the variance of the expectation values where Oˆ ˆ i i is the fluctuation of the expectation value. All sets of possible average values of the operators of the O ˆ j syste define a space of so-called therodynaic states. o each average-value variable corresponds a conjugate variable of the environent the external source. Starting fro an all-average state (its independent variables are only average values) one can get different states (with different sets of independent variables) by replacing soe average values by their conjugates the corresponding external sources. he therodynaic syste can be classified according to the state variables used to describe it: the fully closed (isolated) one eans that only average values are used as the state variables; the fully open one eans that only external sources are used as the state variables; and various interediate partially open/closed systes can be considered too. As a therodynaic process we consider a suence of state changes defined in the state space of a set of average values and/or external sources. We will consider only reversible processes which ust consist of uilibriu states only; the existence of uilibriu state is taken as a fundaental fact of experience. So there are various types of the uilibriu state: fully closed fully open and interediate types. hese types will be naed also therodynaic ensebles see Sec. III he uilibriu state is copletely decrypted by the uilibriu density atrix (-DM) operator ˆ. or its deterination we use the principle of the axiu for the inforational entropy based on the phenoenological description In order to find the uilibriu state for fully closed systes the axiization is constrained by the deand that the expectation values of the observables ˆ j O have chosen values o j O. hese values represent the independent 7

8 variables characterizing the state of the closed syste. Irrespectively of the specific for of the entropy S ent ˆ the realization of the principle of the conditional axiu reduces to the variational procedure for finding the unconditional axiu of the Lagrange function (fn.) ent ent j j j j ˆ Oˆ Oˆ S ˆ Oˆ S ˆ rˆ Oˆ (4) ( j is a set of Lagrange ultipliers) where the general conditions operator is j Oˆ Oˆ ˆ ˆ i O i joj. (5) j When ˆ ˆ O is deterined fro axiization of the Lagrange fn. Eq.(4) the ruired values of the paraeters o j ultipliers j that the ruireents of the closed syste can be attained by choosing such values of O ˆ ˆ ˆ ˆ ˆ j r ˆ O i O i Oj o j. (6) are satisfied. But before iposing the ruireents (6) we can consider the uilibriu state described by j as independent paraeters. By definition it is tered the uilibriu state of a fully open syste while the paraeters (Lagrange ultipliers) j represent the external sources entioned earlier. o ake our paper an easier read we are going to indicate in various places the exeplary case which covers statistical echanics probles (ensebles therodynaic functions etc.) considered in the Parr-Yang book. 1 Here the therodynaic syste of this case is specified by O ˆ 1 O ˆ 2 H ˆ N ˆ where ˆN is the particle-nuber operator. Using for the entropy the Boltzann-Gibbs-Shannon (BGS) expression the expectation value of the natural logarith of ˆ (with a inus sign; the Boltzann constant is oitted because we will use teperature expressed in energy unit) 8

9 S ent ˆ BGS ˆ ˆ ˆ S r ln (7) fro the usual Euler approach the ruireent 0 the well-known canonical distribution aduately describing the physical syste is obtained ˆ ˆ ˆ ˆ O exp O O (8) where Oˆ rexp Oˆ (9) is the partition fn. he result of axiization of the Lagrange fn. Eq.(4) will be naed the basic Massieu fn.: BGS ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ O Max O O S jr Oj ˆ:r ˆ 1 (10) j a diensionless quantity while the axiizer ˆ ˆ O ˆ the -DM operator Eq.(8). When the label j of an independent arguent variable j of is a continuous paraeter e.g. the position vector r check that j r the fn. becoes a functional of this variable. It is easy to can be expressed in ters of as (see Eqs.(4) and (8)) ˆ ˆ ˆ ˆ ln ˆ O O O O. (11) his for (11) of allows for investigation of its properties (like convexity) using the effective action foralis. he above expressions for ˆ ˆ O and Ô are eaningful provided the partition fn. Eq.(9) with Eq.(5) attains a finite value: ˆ cev K K exp ˆ cev O K exp O O. (12) K K 9

10 Here the state vectors cev K are due to assued Oˆ ˆ i O j 0 the coon eigenvectors (cev) of all operators O ˆ j chosen for the definition of the therodynaic syste: Oˆ o. (13) cev K cev j K j K cev he set K is coplete its eleents are noralized and utually orthogonal. It will be convenient now to apply the convention described in Appendix A and instead of using M- M eleent sets like j j 1 j M o o o o o ˆ j Oˆ O ˆ ˆ ˆ 1O2... OM 1 2 O to introduce vectors. or exeplary case we have M M 12 Oˆ H ˆ Nˆ. he conditions operator Eq.(5) represents a scalar product ˆ ˆ ˆ O O O. herefore its eigenvalue at the eigenvector cev K occurring in Eq.(12) is O K K o. (14) he restriction written in Eq.(12) with (14) ay be violated in two cases: (i) the eigenvalue spectra K K o o Oˆ happen to be such that the suation over K is divergent for every point fro the external-sources space (ii) while the suation is convergent for soe it ay be divergent for another. hus not all sets of observables are suitable for the specification of the therodynaic syste. he valid set is such that the restriction Eq.(12) with (14) is satisfied for at least one point. or exaple considering the set Ô of observables M K in K having positive extending to infinity spectra Inf o j o j 0 Supo j K the point in j M 1 o sees to be the entioned point suitable for checking the validity. All further j j1 considerations in the present paper will be confined to valid only sets of observables. K j1 10

11 or the valid set Ô we define the doain Ô of the acceptable externalsources points by K o Oˆ exp. (15) K or exaple the point Oˆ because 1 for infinite set of state K vectors. inally the codoain of acceptable expectation-values points apping (see Eq.(6)) Ô is defined by the ˆ ˆ ˆ ˆ ˆ o o O r O O (16) naely ˆ ˆ ˆ O o O O. (17) It is iportant that ˆ given in Eq.(8) not only satisfies 0 but also it is the true axiizer i.e. that the variational principle holds: ˆ Oˆ ˆ ˆ ˆ O O. he proof is based on the Jensen s inuality for convex functions of an operator. or the proof it ay be convenient to have ˆ in the for (1). hen let us perfor the axiization Eq.(10) first w.r.t. the weights g of the expanded ˆ ˆ g K K Eq.(1) at fixed. he optiu weights p K are found in the for of the Boltzann-Gibbs distribution the canonical distribution p p O ˆ exp O ˆ exp O ˆ w.r.t.. he next axiization K K L K K L L L cev leads to K which are the eigenfunctions of ˆ and due to Eq.(8) siultaneously the eigenfunctions of Ô Eqs.(13)(14). he obtained -DM is cev cev ˆ cev O pk K K with p ˆ K p K L O K ˆ. he following inualities (based on the Gibbs inuality and the Jensen s inuality) hold 11

12 cev gk K p K K pk K ˆ ˆ ˆ ˆ (18) (dependence on Oˆ Oˆ is suppressed here for brevity). he detailed proof of these inualities can be perfored analogously as in Ref.1 pp where it is done on the exaple of O ˆ H ˆ N ˆ. It corresponds to the exeplary case with 1 2. he fn. ˆ Hˆ Nˆ ˆ ˆ where 1 2 here Eq.(4) is the sae as is the grand potential of Ref.1 ˆ ˆ r 1 ln ˆ H ˆ N ˆ. At uilibriu analogous relations hold between and 0. he basic Massieu fn. has the generating property for expectation values: ˆ O r ˆ ˆ ˆ ˆ O O O (19) where the new notation ˆ eans previous ˆ ˆ O O (siilar convention will O be applied to the arguents of ˆ ). his generating property can be easily shown by differentiation w.r.t. i of Eq.(11) with inserted Eq.(5). he generating property of an open syste Eq.(19) can be used now to obtain the properties of the closed syste. he vector fn. ˆ the uilibriu ultipliers guaranteeing satisfaction of the constraints (6) can be o O deterined fro Eq.(19) rewritten in a for of a vector uation ˆ O o (20) 12

13 as its solution w.r.t. at given o Ô (the gradient of a fn. is a row vector see Appendix A). It eans that the identity ˆ ˆ ˆ ˆ r o O O O o should be satisfied at each point o. or the exeplary case o E N. he proble of existence and uniqueness of the solution of this vector uation needs special attention. Eq.(20) can be viewed as the direct ap o in ters of ˆ O or uivalently of ˆ O ˆ naely ˆ O o o Oˆ r ˆ ˆ ˆ ˆ O O O (21) (obtained earlier in Eq.(16) in ters of ˆ only). In the last uality the notation of Eq.(A2) for the gradient was applied together with suppression of the subscript. Based on the property (proven in Appendix A of Ref. 61 ) of Heritian operators  and ˆB that the relation r exp Aˆ 1 Bˆ rexp Aˆ rexp Bˆ 1 is satisfied for 0 1 (uality holds if and only if operators  and ˆB differ by an ordinary nuber) the strict convexity of ˆ as a function of can be proven.61 It eans that inuality O ˆ ˆ ˆ 1 O O 1 O (22) holds for 0 1 and for 0 1. his property is of fundaental iportance: based on it the one-to-one ap o can be proven. 61 herefore the solution o O of ˆ Eq.(20) is unique and it represents the ap o reciprocal to that in Eq.(21). 13

14 Due to the generating property (20) of the change of variables fro to o can be accoplished via the Legendre transfor ˆ o O of (see Appendix C) known as the effective action fn.: ˆ ˆ ˆ ˆ o O o O O o o O (23) which is strictly concave fn. of the expectation values o. 61 ro differentiation of Eq.(23) w.r.t. o and after applying Eq.(20) one obtains the explicit ap ˆ o O ˆ ˆ o o o O o O (24) showing the generating property of. he apping (24) defines also the relation between Ô and Ô as ˆ ˆ ˆ (25) O O O which is reciprocal to the relation in Eq.(17). herefore we observe the one-to-one correspondence between two doains Ô and Ô. Since the vector of observable operators Ô is fixed dependence on it will be often suppressed e.g. ˆ Oˆ ˆ. he proven aps o yield identities (cop. Eqs.(C9) and (C8)) o o o i.e. o o (26) o i.e. By expressing fn. is in fact just the entropy. (27) in Eq.(23) in ters of the entropy Eq.(10) we find that the effective action BGS av o S ˆ o (28) 14

15 av where ˆ o ˆ o is the -DM operator expressed as a function of average values (av) o (as independent variables). Siilarly as the Massieu fn. fro appropriate partition fn. av o ln Oˆ o in Eq.(11) the effective action fn. can be also obtained (29) using the specific conditions operator ˆ ˆ av O o o O o (30) see Eqs.(8) and (9) with Ô replaced by av Ô. As discussed in Appendix C when is the Legendre transfor of Eq.(23) induced by the generating property of induced by the generating property of Eq.(20) then Eq.(24) naely is the Legendre transfor of o o S ˆ o BGS (31) (the relations (27) and (28) were used to obtain the entropy representation). hus the fns. and are Legendre-transforation uivalent (see able I). Equations (21) and (24) together with the definitions (10) and (23) for the content of the usual Gibbs-Helholtz relations and provide grounds to call and the characteristic state fns. in the corresponding variables (for closed and open syste respectively). Note that fro the strict convexity of Eq.(22) follows positive seidefiniteness (see Appendix B) of the 2 second-derivatives atrix i j. Since it is identical with the first-order 2 covariance atrix Oˆ ˆ ˆ i O j O i j [differentiate both sides of Eq.(19) w.r.t. j and use the definition (3)] our result is in agreeent with the well known positive 15

16 sei-definiteness property of the covariance atrix. It should be noted that the positive 2 definiteness of the atrix i j iplies the existence and positive definiteness of the atrix reciprocal to it see Appendix B. Siilarly fro the strict concavity of the action function o follows negative seidefiniteness of its second-derivatives atrix o o o 2 i j. We obtain here explicitly this property of fro the property of. Let us differentiate Eq.(24) w.r.t. o to obtain But leads to. (32) o is the solution of Eq.(20). When differentiated w.r.t. o the indicated uation o o 1. (33) (here 1 denotes the unit atrix ij ). his uation allows the interpretation of the atrix o as a reciprocal to the atrix (provided it is a nonsingular atrix). herefore Eq.(32) can be rewritten as 1 (34) III. MASSIEU UNCIONS PARIAL LEGENDRE RANSORMS O SAE UNCIONS. Let us consider now the uilibriu state of a generic M syste with the (fixed nuber) M degrees of freedo (i.e. M generating operators j j 1 M Oˆ ) for which only M expectation values have been iposed as constraints (with 0 M ). he coponents of the 16

17 ˆ ˆ ˆ 1 2 ˆM are reordered to have first operators conjugate to the independent vector M external sources j followed by j 1 M operators with their expectation values o j j1 iposed as the independent variables. he considered uilibriu state of the M syste (for a given order of coponents of the observable vector Ô ) will be naed also the M Oˆ therodynaic enseble of the syste j j 1 M. his nae is a generalization of such traditional naes as the grand canonical enseble canonical enseble etc. In the exeplary case of M 2 we take 2. he independent variables are two external sources corresponding to observables Ĥ and ˆN : 1 (the reciprocal teperature) and 2. It will be convenient to introduce the sets of indices: U M L L U of the lower (L ) upper (U ) and full ( ) range respectively. Next notations for the colun vectors of variables (functions operators) x L x U x are defined by:... xl x1x 2 x x x... 1x 2 xm U x x x x x x... L U 1 2 M. However for the typographic reasons the colun vector x coposed of subvectors x L and x U will be written alternatively as x x x besides L U x x x L U. A notation for the atrix calculus used in this chapter is shortly recalled in Appendix A. In the exeplary case: L 12 U. So x L is a two-coponent vector x U is epty. he independent variables L o U of the M therodynaic syste can be considered as obtained via trivial apping of the vector o o L U naely M (35) L U L U L U M where as it follows fro the generating property of Eqs.(20)(21) 17

18 U L U ou ou L (36) U (see Appendix D for the description of x ). y he doain M of acceptable points L o U is defined by the apping (35) as M M M M (37) M M M M M (see able II). he convenient therodynaic fn. for the M syste is the Legendre transfor o of L U L U connected with Eq.(36) as the generating uation (see Appendix C) o M L ou L U L U U L U. (38) U his function generates the reverse (to ) apping M M o o o M M (39) L U L U L U L U M L U because it is connected with the first-derivatives vector Eq.(C7) M M o o U L U U L U. (40) herefore can be obtained as the Legendre transfor of M o M o M o o (41) L U L U L U U L U U his eans that and 22 are Legendre-transfor uivalent. Interpretation of for the exeplary case in postponed to the next Section. he relations (38) and (41) expressed in ters of the transfored variables are 1 2 M o M o o M o L U L U L U U U L U (42) M o o L U L U L U U U L U. (43) 18

19 he first derivatives of these two functions w.r.t. L are ual Eq.(C4) and they can be written in two versions Eqs.(C10) (C11) M o M o o M o o M (44) L L U L L U L U L L U L U L U L L U L U L L U L L U. (45) M o o he independent variables of the M syste can be considered alternatively as o o M 0 obtained via trivial apping of the vector o L U o o o o o o o o M (46) L U L U L L U U M 0 L U where the generating property of Eq.(24) gives o o o o L L U L L L U. (47) he Legendre transfor of o o L U connected with the generating Eq.(47) is M o o o o o o o o L L U U L U L L L U. (48) his fn. generates the reverse (to M 0 ) apping M o o o o o o o M 0 (49) L U L U L L U U M L U because of its connection with the first derivatives vector Eq.(C7) M M o o o L L U L L U. (50) herefore can be obtained as the Legendre transfor of o M o o M o o M o L L U U L U L L L U. (51) his eans that and are Legendre-transfor uivalent. he relations (48) and (51) in ters of the transfored variables are M o M o o o o M o L U L L U U L L L U (52) 19

20 M o o o o o o o o L U L L U U L L L U. (53) he first derivatives of these two functions w.r.t. o U are ual Eq.(C.4) and they can be written in two versions M o o M o o o M o o o M (54) U L U U L L U U U L L U U L U M o o o o o o o o. (55) U L L U U U L U U L U Since the M syste is unique its independent variables obtained directly fro the M L ou ML U fully open syste variables Eq.(35) should be the sae as obtained o o M M 0 M via the fully closed syste variables Eqs.(21) (46) able II so uality L U M 0 M M L U M 0 L U o o o L U L U L L U U L U L U (56) should hold. Really it is true due to Eq.(27). After applying this identity (27) to Eq.(48) in which o is substituted by o we find Next with M o o o L U L U L L. (57) obtained fro Eq.(31) as o o o (58) L L U U and substituted to Eq.(57) we see that given in Eq.(57) is identical to given in Eq.(38). hus all relations obtained for Eqs.(50) (55) hold for. wo Eqs. (50) and (40) can be cobined into one uation o o o o M M M L U L L U U L U. (59) 20

21 ro the appings (39) and (49) follow two uivalent fors of the -DM of the M syste ˆ M ˆ M ˆ av M o o o o o L U L U L U L L U U. (60) he functions ol ou (for 0 M ) will be naed the state functions in the entropy representation. In suary the copletely open syste is characterized by MM (U is the epty set) the independent variables are all external sources. he fully isolated syste is M 0 characterized by o o (L is the epty set) the independent variables are all expectation values. he partially open/closed syste 0 M is characterized by the fn. M L ou it will be naed the ( ) Massieu fn. (see able I). It is easy to verify that the conditions operator (a generalization of Eq.(5)) M ˆ M ˆ ˆ L U L U L U L U U (61) O o o O O o generates directly the ( ) Massieu fn. ˆ M M ln r exp O (62) [copare Eqs.(11) and (9)] and when inserted to Eq.(8) it also leads to the sae -DM as in Eq.(60) ˆ ˆ Oˆ exp Oˆ r exp Oˆ M M M M (63) due to cancellations of the contributions o U U in the nuerator and denoinator in Eq.(8). According to Eq.(28) the effective action function M ol ou in Eq.(52) can be replaced by the entropy BGS S ˆ M. 21

22 he atrix of second derivatives of L ou with respect to its arguents vector is obtained in Appendix A in two uivalent fors in ters of subatrices of either 2 2 or 2 o o 2 naely L ou LL LU UU UL LU UU o UUUL L U UU 1 1 LL LL LU 1 1 UL LL UU UL LL LU. (64) Note that is syetric as it should be e.g. UL LL LL UL LL LU the last step due to the syetry of. 2 Reebering that j Oˆ Oˆ i i j i.e. Oˆ Oˆ for the AB A B blocks Eq.(64) can be rewritten in ters of the first-order covariance atrix as 1 1 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ OL OL OL OU OU OU OU OL OL OU OU OU M 1 1 ˆ ˆ ˆ ˆ ˆ ˆ OU OU OU OL OU OU. (65) he diagonal blocks of have particular definiteness: LL is positive definite UU is negative definite. o find these properties we recall fro Appendix B that fro negative definiteness of follows the sae property of LL and therefore of 1 LL finally is positive definite. Siilarly fro positive definiteness of follows negative M 1 LL LL definiteness of M 1 UU UU. (While in general the atrices and have only particular seidefinitness the last two conclusions concern their arguent points at which atrices LL and UU are not singular i.e. have particular definiteness. Next the diagonal blocks of have particular definiteness rather than seidefiniteness because infinite eigenvalues of LL 22

23 and UU ust be excluded as unphysical). he established properties of LL and UU can be M sued up as strict convexity of o L U w.r.t L and strict concavity w.r.t. o U. M As a concave function of o U (at fixed L ) o L U can reach its axiu at soe 0 M 0 unique o o. Obviously at this point the first derivatives vanish o U U L M 0 and therefore o 0 U L U L U 0 follows fro Eq.(59) (all coponents of the vector 0 are U L U L U U zero). After inserting this result M U 0 operator Ô U of the total generating vector operator O ˆ ˆ ˆ OL OU U into Eq.(61) we see that the upper-range vector is in fact not involved in the general conditions operators ˆ M O therefore the nuber of degrees of freedo in the considered state of the therodynaic systes is reduced fro M to (as specified by the coponents of Ô L ). Of course one can consider also a state of this syste having another nuber k of its first derivatives w.r.t. o ual zero 1 k M j. hen at uilibriu this syste is uivalent syste with the nuber of degrees of freedo reduced to M k. IV. GIBBS-HELMHOLZ UNCIONS HE MASSIEU-PLANCK RANSORMS O MASSIEU UNCIONS. o be closer to traditional therodynaic functions used by cheical and physical counities which prefer to use the criteria based on Gibbs and Helholtz potential we define now the M Gibbs-Helholtz fn. of the M Massieu fn. given by M as the Massieu-Planck transfor 65 (for 1 M ) M 1 M o o L U L U (66) i.e. by introducing a prefactor 1 at and inserting to it the specific external sources 23

24 1 L L L L (67) (here L 2... siilarly 2...M ). Using Eq.(42) and (52) this M can be also rewritten as o o M 1 M M L U L U U U o o o o 1 M M M L U 1 L L 1 BGS M M S o L U o1 LoL (68) where we defined ˆ ˆ L U L U see Eq.(61) and BGS BGS M S o S O o M M 1 M o o. (69) U U L U U L U In therodynaic applications eans the reciprocal teperature (in an energy unit) 0 being the external source conjugate to the expectation value o 1 of soe energy operator Ô 1 (like the total energy the internal energy etc.). herefore M is a quantity with the diension of the energy. or the exeplary case we take 2 and then find 22 0 the (22) Gibbs-Helholtz fn. coincides with the grand potential of the grand-canonical enseble. he independent paraeter 2 is known as the cheical potential of the syste. he explicit for of the grand potential is 0 E N 1 S BGS 22 the (22) Massieu fn. 0 transfor of 1 2. inally of previous section is the reverese Massieu-Planck. In general for the distinctive external source one can choose any coponent i of the vector L and then reorder the coponents appropriately to obtain again M in the for (66) having the sae diension as that of the chosen O ˆi ; acceptable sign of depends on the nature of O ˆi. Note that to each ( ) Massieu fn. there corresponds a set of Gibbs-Helholtz fns. with 1 M M (any i fro L can be chosen to be 24

25 ). Siilarly to the property exhibited by the ( ) Gibbs-Helholtz fn. is a convenient generating function for the considered syste. As shown in Appendix A its first derivative is M o M 2 BGS M M o L U S o L U o o L U L U L U (70) where L L should be inserted for the arguent L of M o L and of M U in the above result. he atrix of the second derivatives obtained in Appendix A is 2 BGS 2 S 0 2 o L U M L U 1 M 2 0 y y L o 0 L U U (71) where the Jacobian atrix y o L U is given in Eq.(A23). It should be noted that the above second-derivatives atrix of the fn. M is written in ters of the second-derivatives atrix of the fn. being therefore uch ore coplicated expression than itself. or the exeplary case of 12 S N 22 2 BGS L we have the first-derivative vector (72) and the second-derivative atrix BGS M 2 S y y 0 0 (73) where y 1 0 (74) Hˆ Hˆ Hˆ Nˆ. (75) Nˆ Hˆ Nˆ Nˆ

26 It should be noted that 22 is the second-derivatives atrix of the (22) Massieu fn. of the 22 previous Section. Its firs-derivative vector is 1 2 E1 2 N1 2. o analyze the stability criteria of the Gibbs-Helholtz fn. M we evaluate two diagonal blocks of its second-derivatives atrix Eq.(71): 1 0 M 2 BGS L M LL 2 S yl L LL yl L 0 0 L LL (76) M M UU (77) UU and for arbitrary nonzero vector v L the scalar M vl LL vl 2 BGS 2 2 S v 1 M y v y v L L L LL L L L based on Eq.(76). Its first ter is nonnegative due to BGS S 0. Since y z1 1 1 det LL 0 the vector y v L L L is nonzero too. Due to this fact and the positive definiteness of M LL the second ter of the scalar is positive [see Eq.(B1)]. As the considered scalar is positive LL in Eq.(76) is negative definite when 0 according to Appendix B. Due to the negative definiteness of M UU the block UU of Eq.(77) is positive definite when 0. herefore M o is a strictly concave fn. of L U L and a strictly convex fn. of o U. When 0 (possible when the external source is not the reciprocal teperature) the properties of the definiteness and curvature are to be reversed. Since the first variable is treated differently than L in the definition of M L ou Eq.(66) we evaluate also the corresponding blocks of LL to find M 3 BGS M 11 2S L LL L (78) M 1 M. (79) LL LL 26

27 Using Eq.(65) the blocks LL and UU can be easily rewritten in ters of the first-order covariance atrix. uniquely as he Massieu-Planck transforation of into M Eq.(66) can be reversed M M 1 o o L U L U. (80) herefore the state fns. M and are uivalent characteristics of the M therodynaic syste for 1 M (see able I). Although the curvature properties of the M are the sae as in the case of (for 0 and reversed for 0 ) the functional dependence of M on is quite different than that of. Due to this two descriptions of the M syste ay reveal different aspects of its properties. In particular the Massieu fn. M M o v o L U L U where vl (1 L ) is fixed attains its extreu at sae point 0 where its first derivative w.r.t. v o L U v L o U M M vl L vl ou vl ol vl o (81) U vanishes. hus 0 L ou is a solution of the uation M 0 0 M 0 0 o 1 o o o L U L L L U 0. (82) 2 M 2 M his extreu is in fact the iniu because v o L U vl LL vl 0 vl ou due to positive definiteness of LL. On the other hand the M Gibbs-Helholtz fn. M L ou is a onotonically increasing fn. of because its derivative w.r.t (see Eq.(70)) M BGS M o S L U 2 ˆ (83) 27

28 is positive for If the zero value of the liit of this derivative for exists then M L ou attains a finite-value axiu for or iniu for. he functions M o (for 1 M L U ) will be naed the state functions in the energy representation. V. CONCLUSIONS he uilibriu state which stes fro the axiu entropy principle and is characterized by the basic Massieu fn. was defined and investigated. Concavity and convexity properties of the basic Massieu fn. its Legendre transfors and corresponding Massieu-Planck transfors for various ensebles were deterined using the effective action foralis. able I displays relations between various state functions defined and discussed in the paper. Since all the displayed functions are utually connected with soe direct and reverse transforation they represent uivalent descriptors of the uilibriu state of the O ˆ ˆ ˆ ˆ O1 O2 OM therodynaic syste for various possible sets of M independent variables chosen aong eleents of the external sources and expectation values o. Since the arguents used here are general (the axiu entropy principle the effective action foralis) the results obtained in the fraework of the enseble approach can provide a rigorous atheatical foundation for generalizations (to finite teperatures and to fractional nuber of electrons spins pairs) of various extension of D. In particular the spin-density functional theory extension and the analysis of the zero-teperature liit of this extension will be the subject of the next two papers in the series

29 ACKNOWLEDGMEN he authors acknowledge support of Ministry of Science and Higher Education under Grant No. 1 09A Special thanks are due to Professor Paul W. Ayers for his valuable coents and beneficial rearks on the anuscript. 29

30 APPENDIX A: VECORS BLOCK MARICES DERIVAIVES O HERMODYNAMIC UNCIONS A colun atrix a vector in M-diensional space is denoted by a variable with a subscript which indicates an ordered set of indices e.g. x. A row atrix is the transpose of a colun atrix x x x x M. Here eans the full range of indices in the set 12...M. It ay be partitioned into subsets e.g. the lower range upper one 1... M L and the U L U. he corresponding block structure of a row atrix is denoted as x xl xu x x x. or typographic reasons the corresponding colun L U ay be alternatively denoted as x xl xu. A square or rectangular atrix is distinguished by two set subscripts e.g. a square atrix can be coposed of its blocks: square ones LL UU and rectangular ones LU and UL naely LL LU UL UU M ij i j 1 (A1) and so on for higher-rank atrices like etc. he gradient of a scalar fn. f of a vector x is a row atrix f x f f f x x1 x2 xm f x f i i f x. (A2) x he atrix of second derivatives of this fn. is a syetric square atrix being the gradient of the first derivative vector M 2 2 f x f x f M 2 ij x x i j 1 x j x ii j1 f x f. (A3) Siilarly the Jacobian atrix which describes locally the transforation x y x vector variables is given by of 30

31 y x y1 y1 x1 x 2 y x y y. (A4) 2 2 x x1 x2 Its typical application is the chain rule e.g. finding the gradient of the fn. f x g y x as f x g y y x f x g y x y x x y x y y x he gradient of a scalar product of the vectors fx and h x is h f f h h f f h f h h f x x x x he second-derivatives atrix of a product of scalar fns. f x and hx is. (A5). (A6) fh f f h f h h h f x x x x x x x (A7) he introduced notation is used now to obtain derivatives of the () Massieu fn. M L o Eq.(42) or Eq.(52) (reebering that M M U ) in ters of derivatives of the basic Massieu fn. Eq.(11) with Eq.(5) or of the effective action fn. o Eq.(29). It will be convenient to introduce the notation for the arguent of M M L U L U and for auxiliary vectors: x x x x x x o M o x o x x L U the vector L U and. his allows to rewrite the appings (39) and (49) as x x and x o x. Considering x M to be the independent variable vector we replace and o in Eqs.(20) and (24) by and o to obtain 31

32 x o x o x x. (A8) In the notation for derivatives of functions and the subscript is suppressed. Using the present notation for Eqs.(42) and (52) we have M x x x x o x x o x U U. (A9) L L he gradient of this fn. evaluated with the help of Eqs.(A5) and (A6) is 0 o x o x ol x 0U xl ol x x x x L U x xu U x (A10) ( 0 L and 0 U denote zero vectors in appropriate subsets of indices siilarly 0 UL 0 UU etc. will denote zero atrices). his gradient is expressed in ters of Jacobian atrices and o. hey can be easily evaluated x x 1 0 LL UL x U L U U o x x ol L o o U L x 0 1 UL UU (A11) (1 LL and 1 UU denote unit atrices ij in appropriate subspaces). After inserting the results (A8) and (A11) into the first or the second for of Eq.(A10) the coon siple result for follows x o x x o o o M M M L U L L U U L U (A12) (i.e. Eq.(59) is confired). Interestingly this result is independent of the Jacobian atrices. he second-derivatives atrix obtained directly fro Eq.(A12) by applying Eq.(A3) o o L L L U x U L U U (A13) is coposed of blocks of the two Jacobian atrices. he uations that allow their deterination are obtained by differentiations in Eq.(A8) o o. (A14) 32

33 his syste of uations is solved now for nontrivial blocks of and o Eq.(A11). Presence of the trivial blocks (zero and unit) is helpful it leads to the following uations for blocks UL UU U L 0UL 1 UU U U UU (A15) LLoL L 1LL o 0 LL L U LU LU (A16) which have the solutions 1 1 U L UU UL U U (A17) UU o 1 1 L L LL ol U LL. (A18) LU After inserting these solutions into reaining uations for blocks one obtains the second for of solutions 1 1 U L UL LL U U UU ULLL (A19) LU o 1 L L LL LUUU 1 UL o L U LU UU. (A20) Since as shown in Sec.II is positive definite and is negative definite hence as discussed in Appendix B their diagonal blocks UU and LL preserve definiteness and therefore their reciprocals 1 UU and 1 LL exist and have the definiteness unchanged. By inserting the results (A17) (A20) into Eq.(A13) two fors of in Eq.(64) are obtained. he derivatives of the Gibbs-Helholtz fn. M Eq.(66) are obtained in a siilar way as the derivatives of. he vector of independent variables of M is denoted z z z z o 1 L U L U and depending on it the vector y z o L L U L o U [see Eq.(67)] denotes the arguent of in Eq.(66). Using the present notation Eq.(66) can be rewritten as M z z y z (A21) M

34 he first-derivative vector according to transposed Eq.(A5) is 1 M M z y z 1 1 z z 1 z z M 2 M 1 z1 y z 10 z1 y z y z (A22) where the Jacobian atrix is y z 1 0L 0 zl z1 1L L 0UL 1 LU UU After inserting it into Eq.(A22) we find. (A23) z z1 z1 1 z1 z z1 2 M 1 M M M 1 M o1 LoL ol U M 2 M 1 M 1 M M 1 M L L L U (A24) where in the last step we took fro Eq.(A12). After taking for line of Eq.(68) we transfor finally Eq.(A24) into the result (70). with he second-derivatives atrix obtained fro Eq.(A21) according to (A7) is a b c d z 1 M M the third M (A25) a M 3 M 2z 2 1 (A26) z 0 0 z1 1 2 z b 1 y 2 z1 y z1 10 y z z 0 (A27) c b (A28) 2 y z 2 z d1 d2 M d 1 1 z1 z1 y z y z z (A29) 34

35 where as it can be easily verified via coponents of atrices and vectors y z M d1 1 1 M z1 z1 y z z y z y y y z d z (A30). (A31) Since the non-constant blocks of y Eq.(A23) are only y 1 z and y z1 L L the L L 1 LL non-zero blocks of y are yl 1L 1L L and yl L 11L L. his allows evaluation of Eq.(A30) as 1 0 z1 0 L U d1 1 z 1. (A32) L 0 0U or evaluation of b we need M y z which can be obtained with the help of Eqs. (A5) and (A23) as M M M M M y 1 z z1 L L L U. (A33) inally ad d2 M z (A34) where ad a b c d1 z L L L U 2z 2z z 0 z 3 M 2 M M 2 M L 2 1 U z 0. (A35) Vanishing of blocks 1L and L 1 is to be noted. Eq.(A35) can be rewritten as 35

36 ad ad 11 L U1 ad 0L ad 0 U1 0 (A36) where 2 S ˆ (A37) ad 3 BGS M 11 ad 1 M U1 U (A38) Eqs.(59) (66) and (68) were helpful in obtaining the final expressions. he result Eq.(71) corresponds to (A34) with Eqs. (A36) (A38) and (A31). APPENDIX B: PROPERIES O A POSIIVE (NEGAIVE) DEINIE MARIX or copleteness of our paper we recall shortly well known facts 76 about definiteness of a real syetric atrix f. It is called positive definite if x fx 0 (B1) for all nonzero real vectors x. his property holds if and only if there exists a real atrix g g det 0 such that f g g. (B2) Another necessary and sufficient condition for a real syetric atrix to be positive definite is that all its eigenvalues are positive. he atrix -1 f the inverse of a positive definite atrix f is also positive definite. he definition of positive definiteness is uivalent to the ruireent that the deterinants associated with all upper-left square subatrices are positive. 36

37 Any diagonal block of a positive definite atrix is also positive definite. o verify this the full set of indices is arbitrarily partitioned into three subsets: A B C (any one of A B C can be epty set). or the proof we chose the following vector x 0 x 0 where x is a nonzero vector while 0 and 0 A B C B A C denote zero vectors within the index subsets A and C. ro the assued inuality (B1) follows AA AB AC A A xb C fba fbb fbc xb xb fbbxb f f f 0 f f f 0 CA CB CC C (B3) thus the diagonal block f BB is positive definite. When the relation > is replaced by in Eq.(B1) the atrix f is called positive seidefinite. A real syetric atrix f is negative definite (seidefinite) if f is positive definite (seidefinite). Its properties follow directly fro the last replaceent. If a scalar real function f of a vector x is twice-differentiable then: (i) fro positive definitness of its second derivative atrix f x [see Eq.(A3)] follows strict convexity of f x [see Eq.(22) for this property of as a function of ] (ii) fro strict convexity of f x follows positive seidefinitness of f x. APPENDIX C: LEGENDRE RANSORMAION Since any therodynaic fns. are defined in the present paper via Legendre transforation of soe basic fn. we recall shortly the definition of this transforation 77 and discuss its basic properties. Given a scalar function f xa y B of two vector variables A x y B (their diensionalities can be different). he Legendre transfor g u y of A B f x y A B 37

38 introduced in connection with the change of variables the apping which is induced by the generating property of f x y f xa y xa is defined by B A B x y u y A B A B ua (C1) g u y f x y u x. (C2) A B A B A A When the variables ua y B are taken as independent the vector x A on the right-hand side of Eq.(C2) is a fn. of u y. Steing fro the reverse apping (assued to exist) A B ua yb xa y B this fn. is the solution of Eq.(C1) w.r.t. A he Legendre transfor g ua y ua B x while the derivative w.r.t. y B is coon for g and f : A A B x at given u y. g u y shows the following generating property (C3) A B g ua yb f xa yb y y B B (C4) provided ua and x A are related as in Eq.(C1) or (C3). Using the above definitions the Legendre transfor hx y of by the generating property (C3) is found to be A B A B A A A B g u y induced h x y g u y x u. (C5) ro coparison of Egs. (C5) and (C2) follows iediately that f x y. hus the functions f x y and A B (C1) and (C3) are Legendre-transforation uivalent. A B A B A h x y is identical with g u y showing the generating properties A B B 38

39 Using for the first derivatives vectors the notation of Appendix A like A B A f x y x f x y we suarize appings connected with the considered A A B Legendre transforations as x y u y f x y y (C6) A B A B A A B B u y x y g u y y. (C7) A B A B A A B B heir superposition leads to identities f g u y y u u y (C8) A A A B B A A B g f x y y x x y. (C9) A A A B B A A B Eq.(C4) can be rewritten in two explicit versions: g u y f g u y y u y (C10) B A B B A A B B A B f x y g f x y y x y. (C11) B A B B A A B B A B APPENDIX D: IDENIIES DUE O VARIABLES RANSORMAIONS Various Massieu fns. defined in Sec. II and Sec. III are utually related with appropriate Legendre transforations (see able I). he accopanying appings (transforations) y x of the independent-variables vectors are displayed scheatically in able II. he subscript x of y x characterizes the doain of the ap while the superscript y the range (codoain) of the ap. As shown in able II x or y ay ean M 0. Various variables doains z are related by the sae aps x y y x. (D1) 39

40 Here or ay ean anyone of three vectors: L U o ol ou o L U. We observe in able 2 that doain points ay be connected by various paths (series of x y appings) e.g. y x x x z y z y. his leads to specific identities satisfied by four vector functions that are involved in appings: o in M 0 Eq.(21) M and M Eq.(35) MM o in M M 0 Eq.(24) and MM M 0 M Eq.(46) o U L U in M M M Eq.(39) and o o in M 0 L L U Eq.(49). Obtained earlier two identities Eqs. (26) (27) correspond to return paths between M 0 and MM i.e. M 0 M M M M M 0 o o and between and. he return paths between and M and vice versa result in (D2) U L U U M o M M o o o o (D3) U L U L U U L U while the sae involving and M give (D4) L L U L M o o o o o M M o o o o. (D5) L L L U U L L U Siilarly one can obtain identities corresponding to longer paths. However they all happen to be uivalent to already knows identities steing fro the Legendre transforations connecting with M and with M naely Eqs.(44) (45) (54) (55).We rewrite the here in ters of the apping functions with the help of Eqs.(36) (40) (47) and (50) and reebering that M M : o M o o M o o M (D6) L L U L L U L U L U o o o M (D7) L L U L 40