A possible way out of this dilemma, that forms the basis of almost every present-day application of (approximate) DFT calculations, was suggested by Kohn and Sham.106 Instead of considering the kinetic energy of the true system of interacting electrons, they proposed to calculate the kinetic energy of a reference system of noninteracting electrons with the same electron density instead. This then already accounts for the largest part of the kinetic energy, and only a small remainder has to be approximated. The KS approach still allows for the formulation of an exact theory, which will be outlined in this section.
In KS-DFT, one considers two different quantum-mechanical systems at the same time: The true molecular system of interacting electrons and a reference system of noninteracting electrons. The link between these two systems is established by requiring that their electron densities ρ(r) and ρs(r) are equal (see Fig. 2). Their wavefunctions, however, will in general be different. For open-shell molecules, different options exist for introducing such a reference system: the first option is to require only that the electron densities of the interacting and noninteracting systems agree. This leads to a spin-restricted KS-DFT formulation. The second option is to require that in addition to the total electron densities, also the spin densities of the two systems agree. This results in a spin-unrestricted formulation of KS-DFT. Of course, these two options are equivalent for closed-shell systems (i.e., for singlet states with S = 0).
Note that any version of KS-DFT relies on the assumption that such a noninteracting reference system with the same electron density (and possibly also the same spin density) as the interacting system exists. In practice, it is always assumed that this so-called vs-representability condition is fulfilled, even though this is not guaranteed and several counter-examples are known.84, 107–111 For a detailed discussion of these subtle issues, see, for example, Refs.85.
Comparison of restricted and unrestricted formulation
The restricted and the unrestricted formulation of KS-DFT are based on different definitions of the noninteracting reference system for open-shell systems. In the spin-restricted case, the reference system is chosen such that its total electron density ρs(r) agrees with the one of the fully interacting system, while its spin density Qs(r) usually differs from the one of the interacting system. Conversely in the spin-unrestricted case, the reference system is defined such that both its total electron density and its spin density agree with those of the fully interacting system.
These different definitions of the noninteracting reference system have implications for the treatment of spin in KS-DFT. In the spin-restricted case, the wavefunction Ψs of the noninteracting reference system can always be chosen as an eigenfunction of Sˆ2. Nevertheless, the corresponding eigenvalue does not necessarily agree with the one obtained for the true interacting system. However, it is possible to require this equality with an additional constraint on the noninteracting reference system. In the spin-unrestricted case, the wavefunction Ψ of the reference system is not an eigenfunction of Sˆ2, that is, it is spin-contaminated. This is a direct consequence of the requirement that the correct spin density is obtained. Thus, the expectation value of Sˆ2 becomes a complicated functional of the electron density.115, 116 Of course, the exact ground state density will still correspond to an interacting wavefunction that is an eigenfunction of Sˆ2.
In both the restricted and unrestricted case, the wavefunction of the noninteracting reference system is an eigenfunction of Ŝz. Only in the spin-unrestricted case, it is guaranteed that the corresponding eigenvalue MS is the same as for the fully interacting system, but also in the spin-restricted case it can be chosen accordingly. These differences between the restricted and unrestricted formulation are summarized in Table 1. One important observation is that it is impossible to set up a KS-DFT formalism such that for the noninteracting reference system one obtains both the correct spin density and a wavefunction that is an eigenfunction of Sˆ2 (see also the discussion of this issue in Refs.88, 112).
Table 1. Comparison of the spin-restricted and spin-unrestricted formulations of KS-DFT. “Correct” indicates that the quantity calculated for the noninteracting reference system agrees with the corresponding one of the fully interacting system.
| ||Spin-restricted KS-DFT||Spin-unrestricted KS-DFT|
| is eigenfunction of ?||Yes||No|
| is eigenfunction of ?||Yes||Yes|
The different definitions of the noninteracting reference system in the restricted and unrestricted formulations of KS-DFT also imply different definitions of the noninteracting kinetic energy, exchange–correlation energy, and exchange–correlation potential. These definitions are collected in Table 2. First of all, the use of different reference systems leads to different definitions of the noninteracting kinetic energy. In the spin-restricted case, Ts[ρ] is defined as the kinetic energy of a system of noninteracting electrons with the total electron density ρ(r) and is independent of the spin density Q(r). In contrast, in the spin-unrestricted case T[ρ, Q] is defined as the kinetic energy of a system of noninteracting electrons with the total electron density ρ(r) and the spin density Q(r). These differ by the “unrestricted” contribution to the noninteracting kinetic energy,
Table 2. Definition of the noninteracting kinetic energy, exchange–correlation energy, and exchange–correlation potential in the spin-restricted and spin-unrestricted formulations of KS-DFT.
| ||Spin-restricted KS-DFT||Spin-unrestricted KS-DFT|
|Noninteracting kinetic energy|| || |
|Decomposition of HK functional|| || |
|Exchange–correlation energy|| || |
|Exchange–correlation potential|| || |
| || || |
Only if the spin density vanishes, the restricted and unrestricted definitions of the noninteracting kinetic energy are identical, that is, Tu[ρ, Q = 0] = 0.
Because of these different definitions of the noninteracting kinetic energy, a different decomposition of the HK functional is introduced in the restricted and unrestricted formalisms, respectively, which in turn leads to different definitions of the exchange–correlation energy. These are related by
One important difference between the two formalisms is that in spin-restricted KS-DFT, the exchange–correlation energy Exc[ρ, Q] is the only contribution to the HK functional that depends on the spin density, whereas in the spin-unrestricted theory both the exchange–correlation energy E[ρ, Q] and the noninteracting kinetic energy T[ρ, Q] depend on the spin density. Therefore, the fractional spin condition of Eq. (55), formulated for the HK functional earlier, leads to different exact conditions for the exchange–correlation functional. In the spin-restricted case, the fractional spin condition applies directly to the exchange–correlation energy,
whereas in the spin-unrestricted case, it applies to the sum of the exchange–correlation energy and the noninteracting kinetic energy,
Finally, the exchange–correlation potential (and thus also the resulting KS potential) differs in the two formalisms. In spin-restricted KS-DFT, the exchange–correlation potential vxc[ρ] depends only on the total electron density and acts on electrons of both spin. Conversely, in spin-unrestricted KS-DFT the exchange–correlation potential is different for α- and β-electrons, that is, it has two distinct components. The component of the exchange–correlation potential acting on the total electron density is given by,
Here, the first term is the exchange–correlation potential in the spin-restricted formalism, while the second term is given by the functional derivative of Tu[ρ, Q]. It appears because of the different definitions of the exchange–correlation energy in the restricted and unrestricted theories. The component of the exchange–correlation potential acting on the spin density is given by
For the ground-state electron and spin densities, the first term vanishes according to Eq. (76), and the above expression for v[ρ, Q](r) reduces to the Euler–Lagrange equation for the spin density [cf. Eqs. (92) and (93)].
In summary, the KS potential in the spin-unrestricted case differs from the spin-restricted KS potential vxc[ρ, Q](r) by (a) an additional component v[ρ, Q](r)sz acting on the spin density, and (b) a correction to the spin-independent potential vu[ρ, Q](r). Thus, starting from a spin-restricted reference system with the total electron density ρ(r) and with a spin density that differs from the spin density Qs(r) of the interacting system, the KS potential is modified such that its spin density becomes equal to the one of the fully interacting system Q(r). To achieve this, the spin potential v[ρ, Q](r)σz has to be introduced. However, with the total potential kept fixed, this would lead to a change of the total electron density, and to keep ρ(r) unchanged, the correction vu[ρ, Q](r) is needed.
Spin states in KS-DFT
So far, we have only considered a spin-state independent theory, that is, with the exact exchange–correlation functionals, the spin-restricted and spin-unrestricted KS-DFT formalism discussed above will lead to the ground-state, irrespective of its spin symmetry. As discussed earlier, targeting the lowest state of a given spin symmetry (i.e., with a specific eigenvalue S(S + 1) of Sˆ2) requires a spin-state specific HK functional F[ρ] as defined in Eq. (58).
In the spin-restricted case, this is formally possible by using the spin-state independent definition of the noninteracting kinetic energy, which results in a spin-state specific exchange–correlation functional. In practice, a different strategy is followed: The noninteracting reference system is defined such that it is described by a single Slater determinant with MS = S. This can be achieved by defining the spin-state specific noninteracting kinetic energy as
where is a Slater determinant with MS = S. Such a Slater determinant is always an eigenfunction of Sˆ2 with eigenvalue S(S + 1). Thus, it is ensured that the wavefunction of the noninteracting reference system has the same spin symmetry as the wavefunction of the true interacting system. However, as always in spin-restricted KS-DFT, for S > 0 the spin density of the noninteracting reference system will differ from the one of the fully interacting system.
With this definition of a spin-state specific noninteracting kinetic energy, the spin-state specific exchange–correlation energy is given by
Here, E[ρ, Q] is not equal to the spin-resolved exchange–correlation functional Exc[ρ, Q] defined in Eq. (70). Its dependence on Q does not describe the spin-density dependence of the exchange–correlation energy, but instead introduces the spin-state dependence. This is indicated by the superscript “(ss).” The multiplet-DFT scheme of Daul117, 118 and the restricted open-shell KS (ROKS) scheme119–124 as well as related approaches125, 126 proceed along these lines, but usually include additional ideas originating from Hartree–Fock theory.127, 128
It appears that if applied in a spin-restricted formalism, all available approximate exchange–correlation functionals have to be understood as approximations to E[ρ, Q] and not as approximations to Exc[ρ, Q]. This has important consequences for the construction of such approximate exchange–correlation functional. In particular, one has to realize that E[ρ, Q] does not fulfill the fractional spin condition and thus this condition should not be included when constructing approximations to it.
In addition, in spin-restricted KS-DFT the ground-state spin density is not directly available and has to be determined after calculating the ground-state electron density by minimizing Exc[ρ, Q] with respect to Q [cf. Eq. (76)]. In this step, one has to use Exc[ρ, Q]—which includes the correct spin-density dependence—instead of E[ρ, Q]. Thus, the construction of a different class of approximate exchange–correlation functionals that include the spin-density dependence by approximating Exc[ρ, Q] (or its spin-state specific analog E[ρ, Q]) instead of E[ρ, Q] would be required. Of course, the fractional spin condition applies to Exc[ρ, Q] and E[ρ, Q], and should also be incorporated when constructing such approximations.
In spin-unrestricted KS-DFT, it is not possible to require that the noninteracting reference system is an eigenfunction of Sˆ2.88, 112 Thus, it is not possible to define a spin-state specific analog of T[ρ, Q]. Consequently, the spin-state dependence only enters the exchange–correlation functional, which becomes
For constructing approximations to this spin-state specific exchange–correlation functional, usually a strategy similar to the one in spin-restricted KS-DFT is applied. To this end, the description is restricted to the case of MS = S, that is, only the maximal eigenvalue of Ŝz is allowed. Then, the spin-state specific exchange–correlation functional can be expressed as
The idea of using the spin density as a means to distinguish different spin states in a spin-unrestricted KS-DFT formalism is taken even further in broken-symmetry DFT,24, 40, 44 where the requirement that the spin density of the noninteracting reference system matches the correct spin density of the fully interacting system is sacrificed in favor of obtaining accurate energetics for low-spin states. Consequently, it has been suggested that in this case, the spin density in fact serves to describe the (spin-state specific) on-top pair density.129 If broken-symmetry DFT calculations are interpreted in this way, one would need to determine the spin density in a separate step from the minimization of Exc[ρ, Q] (or its spin-state specific analog E[ρ, Q]), as discussed above for spin-restricted KS-DFT.