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arXiv:1212.1367

Thermodynamic extension of density-functional theory. III. Zero-temperature limit of the ensemble spin-density functional theory

Robert Balawender

Published Dec 6, 2012• 3 citations
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Abstract

In this work, the zero-temperature limit of the thermodynamic spin-density functional theory is investigated. The coarse-grained approach to the equilibrium density operator is used to describe the equilibrium state. The characteristic functions of a macrostate are introduced and their zero-temperature limits are investigated. A detailed discussion of the spin-grand-canonical ensemble in the entropy and energy representations is performed. The maps between the state function variables at 0K limit are rigorously studied for both representations. In the spin-canonical ensemble, the energy surface and the discontinuity pattern are investigated. Finally, based on the maps between the state function variables at 0K limit, the Hohenberg-Kohn theorem for the systems with non-integer electron and spin numbers at zero-temperature limit is formulated.

DOI
10.48550/arxiv.1212.1367

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In this work, the zero-temperature limit of the thermodynamic spin-density functional theory is investigated. The coarse-grained approach to the equilibrium density operator is used to describe the equilibrium state. The characteristic functions of a macrostate are introduced and their zero-temperature limits are investigated. A detailed discussion of the spin-grand-canonical ensemble in the entropy and energy representations is performed. The maps between the state function variables at 0K limit are rigorously studied for both representations. In the spin-canonical ensemble, the energy surface and the discontinuity pattern are investigated. Finally, based on the maps between the state function variables at 0K limit, the Hohenberg-Kohn theorem for the systems with non-integer electron and spin numbers at zero-temperature limit is formulated.
The formalism developed in the first paper of the series [arXiv:0901.1060] is applied to two thermodynamic systems: (i) of three global observables (the energy, the total electron number and the spin number), (ii) of one global observable (the internal electron energy) and two local (position-dependent) observables (the total electron density and the spin density). The two-component potential of the many-electron system of interest is constructed of a scalar external potential and a collinear magnetic field (coupled only with the spin operator). Various equilibrium characteristics of two systems are defined and investigated. Conditions for the equivalence between two systems (the same equilibrium density matrix demanded) are derived and thoroughly discussed. The applicability of the Hohenberg-Kohn theorem is extended to the thermodynamic spin-density functional theory. Obtained results provide a rigorous mathematical foundation for future derivation of the zero-temperature limit of this theory and determination of its chemical reactivity indices.
A system composed of identical spins and described by a quantum mechanical pure state is analyzed within the statistical framework presented in Part I of this work. We explicitly derive the typical values of the entropy, of the energy, and of the equilibrium reduced density matrix of a subsystem for the two different statistics introduced in Part I. In order to analyze their consistency with thermodynamics, these quantities of interest are evaluated in the limit of large number of components of the isolated system. The main results can be summarized as follows: typical values of the entropy and of the equilibrium reduced density matrix as functions of the internal energy in the fixed expectation energy ensemble do not satisfy the requirement of thermodynamics. On the contrary, the thermodynamical description is recovered from the random pure state ensemble (RPSE), provided that one considers systems large enough. The thermodynamic limit of the considered properties for the spin system reveals a number of important features. First canonical statistics (and thus, canonical typicality as long as the fluctuations around the average value are small) emerges without the need of assuming the microcanonical space for the global pure state. Moreover, we rigorously prove (i) the equivalence of the "global temperature," derived from the entropy equation of state, with the "local temperature" determining the canonical state of the subsystems; and (ii) the equivalence between the RPSE typical entropy and the canonical entropy for the overall system.
In a previous study, the author formulated the density functional theory in terms of functional integrals. It was valid at zero and finite temperature. It was possible to derive the Hohenberg and Kohn formulation at zero temperature and the Mermin formulation at finite temperature of the density functional theory, which states that the energy or the grand potential are functionals of the true density of the system considered. In particular, the Kohn and Sham equations are proven to appear naturally by performing a saddle-point evaluation of a specific functional integral. This result is valid at zero or finite temperature. Unfortunately, the expression of the grand potential given in our previous work differs from the usual expression found in the literature. In this short paper, we derive the common expression of the grand potential in the framework of the density functional theory by starting from the expression given in this previous work. This completes the formulation of the density functional theory using functional integrals. This work could be of interest to people working in the field of quantum Monte Carlo methods at finite temperature.
A formulation of density functional theory (DFT) is constructed as an application of the method of maximum entropy for an inhomogeneous fluid in thermal equilibrium. The use of entropy as a systematic method to generate optimal approximations is extended from the classical to the quantum domain. This process introduces a family of trial density operators that are parameterized by the particle density. The optimal density operator is that which maximizes the quantum entropy relative to the exact canonical density operator. This approach reproduces the variational principle of DFT and allows a simple proof of the Hohenberg-Kohn theorem at finite temperature. Finally, as an illustration, we discuss the Kohn-Sham approximation scheme at finite temperature.
Standard approximations for the exchange-correlation functional are known to deviate from linear dependence of the energy on the electron and spin numbers (in -space). Violation of this flat-plane condition underlies the failure of all known approximate functionals to describe band gaps in strongly correlated systems. Then crucial for further development of functionals is recognition of the behavior of the energy as a function of, and its derivatives (derivatives discontinuity pattern at 0K limit). In this Letter, the energy derivatives are analyzed thoroughly. It is shown that apart from the well-known type of discontinuity pattern, three other patterns are possible in the vicinity of the singlet state point. The zero temperature limits of energy derivatives at this point are derived, found different for various directions. Existence of all discontinuity patterns is illustrated on the example of diatomic molecules set.
The infinite-volume limit of thermodynamic functions calculated in the quantum microcanonical ensemble is shown to exist for a fairly wide class of spin systems and quantum gases. The entropy is set equal to the logarithm of the number of eigenstates in an energy interval which increases linearly with the size of the system, but is otherwise arbitrary. The limiting entropy per unit volume agrees with that calculated in the canonical formalism, and possesses certain convexity properties required for thermodynamic stability. A precise criterion, in terms of the energy spectra of large systems, is given for determining the limit of the thermodynamic entropy as the temperature approaches zero. This is not determined by the degeneracy of the ground state, contrary to the discussion of the ``third law of thermodynamics'' found in some textbooks.