Phase Space View of Ensembles of Excited States
Phase Space View of Ensembles of Excited States
收稿日期: 20170811 接受日期: 20170831
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Received: 20170811 Accepted: 20170831
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TheprojectwassupportedbytheNationalResearch,DevelopmentandInnovationFundofHungary,Financedunderthe123988FundingScheme. 
The density functional theory and its extension to ensembles of excited states can be formalized as thermodynamics. However, these theories are not unique because one of their key quantities, the kinetic energy density, can be defined in several ways. Usually, the everywhere positive gradient form is applied; however, other forms are also acceptable, provided they integrate to the true kinetic energy. Recently, a kinetic energy density of the groundstate theory maximizing the information entropy has been proposed. Here, ensemble kinetic energy density, leading to extremum information entropy, is derived via constrained search. The corresponding ensemble temperature is found to be constant.
关键词：
The density functional theory and its extension to ensembles of excited states can be formalized as thermodynamics. However, these theories are not unique because one of their key quantities, the kinetic energy density, can be defined in several ways. Usually, the everywhere positive gradient form is applied; however, other forms are also acceptable, provided they integrate to the true kinetic energy. Recently, a kinetic energy density of the groundstate theory maximizing the information entropy has been proposed. Here, ensemble kinetic energy density, leading to extremum information entropy, is derived via constrained search. The corresponding ensemble temperature is found to be constant.
Keywords：
本文引用格式
NAGY Ágnes.
NAGY Ágnes.
1 Introduction
Density functional theory ^{1} is a ground state theory. It is valid for the lowestenergy state in each symmetry class ^{2, 3}. The theory was first rigorously generalized for excited states by Theophilou ^{4}. It was furher generalized by Gross, Oliveira and Kohn ^{5}. The method was used in several calculations ^{612}. The relativistic generalization of this formalism has also been done ^{13}. The optimized potential method was also extended to ensembles of excited states ^{14}. We mention by passing that theories for a single excited state also exists ^{1421}. An alternative theory, worth remarking, is timedependent density functional theory ^{22, 23}.
The groundstate density functional theory was formalized as 'thermodynamics' by Ghosh, Berkowitz and Parr ^{24}. A phasespace distribution function f(r, p) was derived by maximizing a phasespace Shannon information entropy subject to the conditions that f yields the density and the local kinetic energy density of the system. A local MaxwellBoltzmann distribution function was resulted and the concept of local temperature was introduced. This phasespace description resulted several applications ^{2529}. Extensions of the formalism have also been provided ^{3040}. A couple of years ago the local thermodynamic formalism was extended to ensembles of excited states ^{41} and ensemble local temperature was defined.
Recently the groundstate theory has been revisited ^{42}. The local temperature of the GhoshBerkowitzParr theory was defined via the kinetic energy density. However, the kinetic energy density is not uniquely defined. Usually the everywhere positive gradient form is applied, though any function that integrates to the true kinetic energy can do. It has recently been shown ^{42} that it is possible selecting the kinetic energy density so that the local temperature be a constant for the whole system under consideration. Moreover, it turned out that the kinetic energy density corresponding to the constant temperature, maximizes the information entropy.
The ensemble kinetic energy density is not defined uniquely either. In this paper the ensemble temperature and kinetic energy density resulting the maximum phasespace information entropy are presented. The extremum is obtained by the constrained search of Levy and Lieb ^{43, 44}.
The outline of this paper is as follows: In Section 2 the ensemble theory of excited states is summarized. In Section 3 the local thermodynamic formalism of ensembles is reviewed ^{41}. The constrained search is applied in Section 4 to obtain the ensemble kinetic energy density giving the maximum phasespace information entropy. Section 5 is devoted to an illustrative example and discussion.
2 Ensemble density functional theory for excited states
The Schrödinger equations of the Hamiltonian
where
are the energy eigenvalues. The ensemble energy is defined as^{5}
where
The generalized HohenbergKohntheorems for ensembles read as follows:
(ⅰ) The external potential
(ⅱ) For a trial ensemble density
and
it holds that
The ensemble functional
The ensemble KohnSham equations were also derived:
The ensemble KohnSham potential
is a functional of the ensemble density
3 A "thermodynamical" view of ensembles of excited states
Now, the thermodynamical transcription of ensembles is summarized^{41}. Consider a system of
and
There exist several distribution functions that satisfy the marginal conditions (11)–(13). One can choose a distribution function maximizing the entropy
subject to the constraints of correct density (Eq.(11)) and correct noninteracting kinetic energy (Eq.(13)).
where
One can immediately see that it can be rewritten as an ideal gas expression
where the local temperature
Then the MaxwellBoltzmann distribution function takes the form
Substituting Eq.(21) into Eq.(15) we arrive at the wellknown SackurTetrode expression of the entropy:
The distribution function can also be formulated as
where
and
The effective potential
4 Maximum information entropy with constrained search
In the previous section the information entropy was maximized with constraints (11)–(13). However, the ensemble kinetic energy density in Eq.(13) is not uniquely defined. Therefore the ensemble temperature in not unique either. Only the ensemble noninteracting kinetic energy
Now we select that ensemble temperature (or ensemble noninteracting kinetic energy density) that maximizes information entropy. We are going to find the phasespace distribution function
and
The constrained search of Levy and Lieb^{43, 44} is applied. That is, the extremum is pursued in two steps. In the first step the search is over all distribution functions that result a given ensemble temperature. In the second step the search is over all temperatures. We can immediatelly notice that the first step has already been done in the previous section. The maximum information entropy after the first step is given by the SackurTetrode expression (22). In the second step we have to find the ensemble temperature that makes the information entropy (22) with the constraint that
That is,
where
Therefore,
As the Lagrange multiplier
The distribution function obtained by the constrained search is a MaxwellBoltzmann distribution function
that is, Eq.(21) with the constant ensemble temperature (33).
5 Discussion and illustrative examples
Take the linear harmonic oscillator as the first the example. The potential is
where
The ensemble kinetic energy is
Consequently the ensemble temperature is
In the subspace theory of Theophilou ^{4}
and
respectively.
Our second example is the hydrogen atom. The eigenvalues are
where
where
where we make use of the fact that the degeneracy of a level with the principal quantum number
and
respectively.
One of our interesting results is that the ensemble temperature corresponding to the extremum phasespace information entropy is constant. This is true for any ensemble irrespective of the construction of the ensemble, that is, the weighting factors. Of course, the ensemble kinetic energy and consequently, the ensemble temperature will depend on the weighting factors, but it will always be a constant.
The ensemble kinetic energy density maximizing the phasespace information entropy is found to be proportional to the ensemble density. We emhasize here, that the knowledge of the ensemble kinetic energy density does not give us any information about the ensemble kinetic energy functional or its functional derivative.
This paper emhasizes the nonuniqueness of the local thermodynamics and shows that the maximum entropy is attained if the ensemble temperature is constant. The ambiquity of the local thermodynamics can also be considered an advantage from the point of view of practical applications. One is free to select that particular ensemble local temperature (or ensemble kinetic energy density) that is the most suitable for the given application. It might happen that the constant ensemble temperature is the most favorable in certain cases. In other cases, another ensemble kinetic energy density and the corresponding temperature are more beneficial. It means that they can provide different physical or chemical insight.
From information theoretic viewpoint the present theory has the significance that it is possible to select an ensemble kinetic energy density that is proportional to the ensemble density. It means that the ensemble kinetic energy density has almost the same information as the ensemble density. There is a difference only in the normalization: the ensemble density integrates to the number of electrons, while the ensemble kinetic energy density is normalized to the ensemble kinetic energy. That is, the maximum entropy is attained by the ensemble kinetic energy density having no new information in addition to the ensemble density.
6 Conclusions
In summary, we constructed ensembles of excited states and selected that ensemble kinetic energy density that maximizes the phasespace information entropy. The extremum was obtaned through the constrained search of Levy and Lieb. This ensemble kinetic energy density is proportional to the ensemble density and the ensemble temperature is constant.
参考文献
Density Functional Theory. In Topics in Current Chemistry; Nalewajski R., Ed.
Recent Advances in the Density Functional Methods. in Recent Advances in Computational Chemistry; Chong D. P., Ed.
Reviews of Modern Quantum Chemistry; Sen, K. D. Ed.
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