N-representability of the Jastrow wave function pair density of the lowest-order

Conditions for the N-representability of the pair density (PD) are needed for the development of the PD functional theory. We derive sufficient conditions for the N-representability of the PD that is calculated from the Jastrow wave function within the lowest order. These conditions are used as the constraints on the correlation function of the Jastrow wave function. A concrete procedure to search the suitable correlation function is also presented.

where γ ′ ′ = rr rr ( ; ) SSD N N (2) 0 is the PD calculated from the SSD. In the preceding paper 17 , we have confirmed that Eq. (2) meets four kinds of necessary conditions for the N-representability of the PD, and may become "approximately N-representable" [42][43][44][45][46][47] . However, the possibility of it being N-representable has not been discussed 17 . This is an arguable problem that is concerned with whether the reproduced PD is physically reasonable or not.
The aim of this paper is to discuss the N-representability of Eq. (2) and to show the way to search the suitable correlation function | − | f r r ( ) i j . The organization of this paper is as follows. For the convenience of the subsequent discussions, we first examine the properties of the LO-Jastrow PD in the next section. Then, the sufficient conditions for the N-representability of Eq. (2), which are imposed on the correlation function, will be derived recursively. Next, concrete steps for searching the correlation function that meets these conditions are discussed. Finally, concluding remarks are given in the last section.

Results
Properties of the LO-Jastrow PD. In this section, we shall discuss the properties of the LO-Jastrow PD. To this aim, the properties of PDs that are calculated from SSDs are investigated. The cofactor expansion of Scientific REPORTs | 7: 7590 | DOI:10.1038/s41598-017-07454-8 In what follows, suppose that these spin orbitals are given as the solutions of simultaneous equations of previous work 17 , and therefore they are orthonormal to each other.
(3) denote (N 0 − 1) -electron SSDs that are defined as the minor determinants multiplied by . By the repetition of this procedure, we arrive at 2 is given by the following Jastrow wave function; where γ ′ ′ = rr rr ( ; ) N n (2) denotes the PD operator for an n-electron system. Substituting Eq. (9) into Eq. (6) and rearranging, we get Scientific REPORTs | 7: 7590 | DOI:10.1038/s41598-017-07454-8 It should be noticed that the right-hand side of Eq. (10) has a characteristic form. Concerning the N-representability of this form, the following theorem holds: Theorem. If there exists the set of single-valued, continuous, smooth and finite functions {a α (x n + 1 )} (1 ≤ α ≤ n + 1) that satisfy the conditions; then the following equations hold: N n n n n N n n 1 1 1 1 (1 ≤ α ≤ n + 1) denote the n-electron wave functions, and σ is a permutation operator upon the electron coordinates, and σ is the number of interchanges in σ.
Proof. The left-hand side of Eq. (13) seems to be related to the average of γ ′ ′ = rr rr ( ; ) N n (2) with respect to a density matrix for a mixed state. Indeed, if the density matrix for the mixed state is given by is calculated as . On the other hand, it is expected that the average ρ γ ′ ′ =ˆr r rr Tr [ ( ; ) ] n N n (2) may be given as the expectation value of γ ′ ′ = rr rr ( ; ) N n (2) with respect to a pure state for the whole system that includes the n-electron system as a subsystem 48,49 . We shall take an (n + 1)-electron system as the whole system, and suppose that the wave function for such the (n + 1)-electron system is given by Eq. (14)    In addition to Eq. (8), the existence conditions for (21) and (22), are also the parts of sufficient conditions for the N-representability of Eq. (2). We assume that the set of functions

Discussions
Concrete steps for constructing the N-representable LO-Jastrow PD. In the preceding section, the sufficient conditions for the N-representability of the LO-Jastrow PD are derived. In this section, we consider the concrete steps for searching the correlation function that meets these conditions or checking its existence.
1. First, we give a trial form of the correlation function. Using this, simultaneous equations for the N 0 -electron system are solved in a self-consistent way 17 . 2. Let us consider the SSD that consists of the resultant spin orbitals for the simultaneous equations. The SSD can generally be expanded using the cofactor. The SSD for the N 0 -electron system is expanded along the N 0 th row, then we get the N 0 number of SSDs for the (N 0 − 1)-electron system, i.e., Φ ≤ N 0 ). Successively, each of the SSDs for the (N 0 − 1)-electron system is expanded along the (N 0 − 1)th row, and then the (N 0 − 1) number of SSDs for the (N 0 − 2)-electron system, i.e., , can be obtained for each i. After that and later, the cofactor expansions are likewise repeated, and we finally arrive at the SSDs for the two-electron system, i.e.,Φ ⋅⋅⋅⋅ x x ( , ) that are satisfied with Eqs. (21) and (22 that are satisfied with Eqs. (11) and (12), together with modifying the correlation function. If we successfully find the correlation function that meets the conditions (11) and (12) for any n(≤N 0 − 1), the LO-Jastrow PD of the N 0 -electron system becomes N-representable.
As easily inferred, the above steps are feasible only for the small-electron systems from the practical viewpoint. However, it should be noted that the correlation function that makes the LO-Jastrow PD N-representable may, in principle, be found along the above steps, though there is a possibility that the suitable correlation function may not exist in some system.
Other possibilities to obtain antisymmetric wave functions. In the above-mentioned concrete steps, antisymmetric (n + 1)-electrons wave functions are built up from given antisymmetric n-electrons wave functions via Eq. (14). In this subsection, we show that there are other possibilities to obtain antisymmetric (n+1)-electrons wave functions without using Eq. (14).
Instead of Eq. (14), we can use the following expression for Ψ where a α (x i ) and b α (x n + 1 ) denotes functions that should be determined. If these functions satisfy the following conditions; 1 becomes an antisymmetric wave function and yields the PD just given by the left-hand side of Eq. (13). Therefore, if we use the expression Eq. (28), then we have to search a set of functions a α (x i ) and b α (x n + 1 ) that satisfy Eqs. (29)-(31) instead of Eqs. (11) and (12)  , i.e., we suppose where a α (x i , x n + 1 ) denote a function that should be determined. If a α (x i , x n + 1 ) satisfies the following conditions;  then Eq. (32) becomes an antisymmetric (n + 1)-electrons wave function and yields the PD just given by the left-hand side of Eq. (13). Therefore, if we use the expression Eq. (32), then what we have to do in the concrete steps is to search a set of functions a α (x i , x n+1 ) that satisfy Eqs (33) and (34) instead of searching a set of functions that satisfy Eqs (11) and (12).
Thus, various expressions for Ψ ⋅ ⋅ ⋅ = + + x x ( , , ) N n n 1 1 1 can be adopted. This means that the present method can provide various prescriptions to construct the N-representable LO-Jastrow PD.

Concluding remarks
In this paper, the sufficient conditions for the N-representability of the LO-Jastrow PD are discussed. Using the properties of the LO-Jastrow PD, we derive the sufficient conditions that are imposed on the correlation function of the Jastrow wave function. As shown in the previous section, additional steps to search the suitable correlation function, which satisfies the sufficient conditions, are attached to the computational scheme proposed previously 17 . Although the number of steps rapidly increases with that of electrons, the concrete steps that are presented in the previous section are feasible for a small-electron system. Of course, there is a possibility that the suitable correlation function cannot be found out. In this case, as mention in the previous section, we can adopt other expressions for Ψ ⋅ ⋅ ⋅ = + + x x ( , , ) N n n 1 1 1 , so that we may possibly find out a suitable correlation function. Otherwise, as mentioned in the previous paper, LO-Jastrow PDs are approximately N-representable in a sense that they satisfy four kinds of necessary conditions 17,[42][43][44][45][46][47] .