Pair density functional theory for excited states of Coulomb systems

Pair density functional theory is extended to excited states of Coulomb systems. It is shown that the pair density determines the Hamiltonian of the Coulomb system. A universal kinetic energy functional appropriate for the ground and all bound excited states is defined. The Euler equation can be rewritten as a two-particle auxiliary equation in which the unknown Pauli-like potential should be approximated.


Introduction
One of the most frequently used approach to the electronic structure of many electron systems is the density functional theory (DFT). A similar, though less regularly applied method is the pair density functional theory (PDFT). While in DFT the fundamental variable is the density, in PDFT the basic quantity is the pair density. The Hohenberg-Kohn theorems [1] were extended by Ziesche [2] to give firm foundation to PDFT. There are other approaches [3][4][5][6][7] for the ground state. It was shown [4,5,7] that in the ground state the pair density can be determined by solving a single two-particle auxiliary equation, that is, the problem can be reduced to a two-particle problem for an arbitrary system. Further relevant theorems and properties of PDFT can be found in [8][9][10][11][12][13][14][15]. Important extensions and applications are in [16][17][18][19][20][21][22][23].
In this paper, the excited-state density functional theory for Coulomb systems [35][36][37] is extended to pair density. A universal PDFT valid for the ground and all bound excited states of Coulomb systems is constructed. It is shown that the pair density determines the Hamiltonian of the Coulomb system and the degree of excitation. A kinetic energy functional is defined. This sole functional is sufficient to treat any Coulomb excited state. Euler equation is derived and rewritten as a two-particle Schrödinger-like equation in which the unknown Pauli-potential should be approximated.
The paper is organized as follows: the following section presents the important properties of Coulomb pair densities and formalizes the universal PDFT. The last section is devoted to discussion.

Universal pair density functional theory of Coulomb systems
The Hamiltonian of a Coulomb system has the form T and V ee are the kinetic energy and the electron-electron energy operators, while the external Coulomb potential has the form where r = |r − R | . N and M are the number of electrons and the nuclei. R and Z stand for the position and the charge of the nucleus . The pair density n is defined as where Ψ is the wave function and the integration is for all but two spatial ( r ) variables and summation for all spin ( ) variables.
Theorem 1 Let n be a Coulomb pair density. Then, n determines the external potential.
Proof Shift the origin of the coordinate system to the nucleus and average n for the polar angles r 1 The following cusp condition holds [9]: That means that the cusps of n reveal the position of the nuclei and the atomic numbers. The integral of n yields the number of electrons because Consequently, Eqs. (5) and (6) provide the parameters of the Coulomb potential (2), i.e., the Coulomb pair density n determines the external potential, the Hamiltonian and any property of the system. Kato's theorem is valid for excited states as well. (We mention in passing that n possesses a somewhat different cusp relation for very highly excited states, but we can still extract the same information from n [9].) Consequently, the knowledge of n of the Coulomb system is enough in principle to determine all of its properties. The energy is the functional of n.
Theorem 2 Let n be a Coulomb pair density (that is, an eigen pair density of Ĥ Coul of Eq. (1)). Then, n is not an eigen pair density of any other Coulomb potential.
Proof The asymptotic behavior of n is [41] where E is the total energy of the given excited state (or ground state) of the system under investigation, while E N−2 0 is the ground-state energy of the N − 2-electron system obtained after removing two electrons from the original N-electron system. Observe that E N−2 0 does not depend on the degree on excitation. Therefore, the asymptotic decay of n determines E.
The energy can be written as the sum of the electron-electron, electron-nucleon and the kinetic terms Utilizing Theorem 2 the functional can be constructed, where n is the pair density of a stationary state of any Coulomb system. This functional exists for any bound stationary state arising from Coulomb external potential. But it is unsuitable for practical use, because Coulomb pair densities should be applied to create it. In fact, no approach is available to tell if a given pair density is Coulombic or not, unless we construct the external potential, solve the Schrödinger equation and compare the given pair density with the calculated pair density. But, of course, this is not practicable. After all, there is no point solving the Schrödinger equation in PDFT. Therefore, the way proposed in DFT [35] is now extended to PDFT: T is defined for all n, not only for Coulombic n.
First a bifunctional is taken. The minimization of the kinetic energy is done over the wave functions Ψ producing the excited-state pair density n and orthogonal to the first k − 1 eigenfunctions of the Coulomb system fixed by n Coul . Presume the existence of a unique Coulomb pair density that is closest to the (non-Coulomb) pair density n. In case more than one Coulomb pair density is found in the same distance from n, the one generating the smallest T in Eq. (11), is chosen. is selected to be large enough for being at least one Coulomb pair density in the distance smaller than . Finally, T with the smallest is taken: The minimization leads to the Euler equation up to a constant.
It should be stressed that both the functional in Eq. (13), and the Euler Eq. It is interesting to note that there is a relation between v Coul p and the DFT effective potential v DFT ef f . The DFT Euler equation can be rewritten as [42] where v is the external potential, and DFT is the chemical potential. The density can be obtained from the pair density v Coul ef f (r 1 , where v KS is the Kohn-Sham potential. n can also be written as [40] and can be expressed with the amplitude Φ The relationship between v Coul p and v DFT ef f has already been derived [40]: The functional T Coul Finally, the virial theorem is presented. The derivation is detailed in ref. [6]. The virial theorem for the Weizsäckerlike functional is The virial theorem for the potential v Coul p takes the form Eq. (26) can also be considered a constraint that is useful to test an approximate functional.

Discussion
T[n, n Coul ] is a bifunctional similar to the one defined in [38].
Here T[n, n Coul ] appears only as a step in the construction of T Coul [n] . In [38], on the other hand, the kinetic bifunctional depends not only on n but on the external potential, too.
The very special properties of the Coulomb pair density makes it possible to define universal kinetic energy functional. That is, T Coul [n] is a single functional valid for the ground and all bound excited states. The Pauli energy functional T Coul p is also universal. It should be emphasized that the excited-state (or the ground-state) problem can be reduced to Eq. (15), that is, to the solution of a two-electron problem. Of course, the exact form of T Coul p is unknown and should be approximated. The advantage of the present approach is that we have only one unknown functional: T Coul p . A single functional is appropriate for any excited state. However, finding suitable approximation will not be easy, as the functional studied should be free of the N-representability problem [41,[43][44][45][46][47][48][49][50][51][52][53][54][55][56][57][58][59].
It Eq. (29)) can be rewritten as v Coul pk is the Pauli potential, the functional derivative of the Pauli energy T Coul p,k = T Coul k − T w with respect to the pair density.
In summary, it is shown that the pair density determines the Hamiltonian of the Coulomb system and the degree of excitation. Based on these findings the pair DFT is extended to excited states of Coulomb system. A kinetic energy functional T Coul [n] is defined for the ground and all bound excited + v Coul p,k (r 1 , r 2 ) n 1∕2 (r 1 , r 2 ) = k n 1∕2 (r 1 , r 2 ).
states. One functional is enough for treating any Coulomb excited state. The Euler equation can be rewritten as a twoparticle Schrödinger-like equation in which the unknown Pauli-potential should be approximated.