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Keywords:

  • relativistic quantum chemistry;
  • magnetic interactions;
  • spin;
  • density functional theory;
  • open-shell;
  • transition metal chemistry

Abstract

  1. Top of page
  2. Abstract
  3. Introduction
  4. Spin in Nonrelativistic Quantum Chemistry
  5. Spin in Hohenberg–Kohn DFT
  6. Spin in Kohn–Sham DFT
  7. Spin in Relativistic DFT
  8. Future Directions for Spin-DFT
  9. Appendix: Gordon Decomposition of the Current Density
  10. Biographical Information
  11. Biographical Information

The accurate description of open-shell molecules, in particular of transition metal complexes and clusters, is still an important challenge for quantum chemistry. Although density-functional theory (DFT) is widely applied in this area, the sometimes severe limitations of its currently available approximate realizations often preclude its application as a predictive theory. Here, we review the foundations of DFT applied to open-shell systems, both within the nonrelativistic and the relativistic framework. In particular, we provide an in-depth discussion of the exact theory, with a focus on the role of the spin density and possibilities for targeting specific spin states. It turns out that different options exist for setting up Kohn–Sham DFT schemes for open-shell systems, which imply different definitions of the exchange–correlation energy functional and lead to different exact conditions on this functional. Finally, we suggest possible directions for future developments. © 2012 Wiley Periodicals, Inc.


Introduction

  1. Top of page
  2. Abstract
  3. Introduction
  4. Spin in Nonrelativistic Quantum Chemistry
  5. Spin in Hohenberg–Kohn DFT
  6. Spin in Kohn–Sham DFT
  7. Spin in Relativistic DFT
  8. Future Directions for Spin-DFT
  9. Appendix: Gordon Decomposition of the Current Density
  10. Biographical Information
  11. Biographical Information

Open-shell molecules such as, for example, radicals or transition metal complexes and clusters, feature a measurable magnetic moment that originates from their electronic structure. In fact, the electronic spin gives rise to a magnetic moment that makes such molecular systems functional for various purposes. For instance, organic radicals can be used as spin probes in biomolecules1 and are of interest as building blocks for molecular spintronics devices,2, 3 single-molecule magnets have the potential to act as molecular qubits for quantum information processing,4 and open-shell transition metal compounds serve as catalytic centers in (bio-)inorganic chemistry,5–8 where a change in the spin state can be an essential step in the catalytic cycle.9

Consequently, a first-principles theory that is useful for descriptive and analytic purposes and that has the potential to be a predictive tool in theoretical studies on such chemical systems must consider the spin properties of the electronic structure. Although for closed-shell systems, quantum chemical methods—both wavefunction theory for accurate calculations on small molecules10 and density-functional theory (DFT) for studies on complex chemical systems11—offer such predictive tools, the situation is less satisfactory for open-shell systems, in particular for transition metal complexes and clusters.12

With wavefunction-based methods, a multireference treatment is in general mandatory for open-shell systems. In particular, the complete active space self-consistent field method, usually in combination with second-order perturbation theory (CASPT2), has been employed to study transition metal complexes (for examples, see Refs.13–18). However, the factorial scaling with the size of the active space puts rather severe limits on the size of the active space, which prevents most applications to polynuclear transition metal complexes and clusters. Novel approaches, such as the density matrix renormalization group algorithm19, 20 and its generalizations21 might make it possible to overcome this limitation, although the molecular sizes that can be studied are clearly much smaller compared to those accessible to DFT methods.

Therefore, DFT is usually the method of choice in theoretical studies of transition-metal catalysis as well as molecular and spectroscopic properties of open-shell molecular systems.22–25 Despite much success, it has also become clear that for open-shell systems, DFT with the currently available approximate functionals shows a number of shortcomings. In addition to inaccuracies in predicting energies, geometries, and molecular properties (for a case study, see, e.g., Ref.26 and for overviews, see, e.g., Refs.11, 27, 28), a severe limitation are unsystematic errors in the prediction of the relative energies of different spin states.29–36 Moreover, the spin density—which serves as an additional fundamental quantity in the spin-DFT formalism commonly employed for open-shell systems—is qualitatively incorrect in some cases.15, 37–39 To make things even worse, the treatment of low-spin states usually requires the use of a broken-symmetry description,40–43 which provides an unphysical spin density by construction (see, e.g., Refs. [24,44] for a discussion). This precludes the simple prediction of spectroscopic properties depending on the spin density (for schemes to address this difficulty, see e.g., Refs.45–48).

Consequently, the development of better approximate DFT methods for open-shell systems is currently still one of the most important and challenging topics in theoretical chemistry.12, 28, 49 To make progress is this area, it is important to understand the exact theory underlying DFT for open-shell systems. Although for the closed-shell case, exhaustive presentations of this theory exists in several textbooks,50–53 this is not the case for open-shell systems, which are often only mentioned in passing in these accounts. Here, we attempt to close this gap by reviewing the foundations of DFT for open-shell systems. In our presentation, we will pay particular attention to the role of the spin-density in DFT and to possibilities for targeting different spin states within the exact theory.

Even though we will not discuss the currently available approximations in detail, we believe that for the future development of better approximations, it is crucial to know which exact theory is to be approximated. This is also a prerequisite for deriving exact conditions on the approximate functionals, for setting up model systems that can be treated exactly, and for obtaining benchmark results from accurate wave-function theory calculations.

This work is organized as follows. First, we introduce spin in the context of nonrelativistic quantum chemistry. This is followed by a discussion of Hohenberg–Kohn (HK) DFT, highlighting the role of the spin density and of spin states for open-shell systems. Next, the treatment of spin in the Kohn–Sham (KS) framework of DFT is reviewed. It turns out that different options exist for deriving KS DFT for open-shell systems, which are discussed and compared in detail. For completeness, we then discuss DFT within the relativistic framework where spin is no longer a good quantum number. Finally, some possible future directions for DFT applied to open-shell systems are outlined.

Spin in Nonrelativistic Quantum Chemistry

  1. Top of page
  2. Abstract
  3. Introduction
  4. Spin in Nonrelativistic Quantum Chemistry
  5. Spin in Hohenberg–Kohn DFT
  6. Spin in Kohn–Sham DFT
  7. Spin in Relativistic DFT
  8. Future Directions for Spin-DFT
  9. Appendix: Gordon Decomposition of the Current Density
  10. Biographical Information
  11. Biographical Information

Spin structure of the one-electron wavefunction

The nonrelativistic quantum-mechanical equation of motion for a single electron in an external electrostatic potential equation image is provided by the time-dependent Schrödinger equation (SE), which in Gaussian units reads,

  • equation image(1)

with the kinetic energy operator equation image, where equation image is the momentum operator and me and qe are the mass and the charge of the electron, respectively. Stationary states can then be obtained from the time-independent Schrödinger equation,

  • equation image(2)

In a nonrelativistic framework, spin is introduced in an ad hoc fashion by using a two-component representation for the wavefunction,54, 55 i.e.,

  • equation image(3)

For a rigorous introduction of spin in quantum chemistry, it is necessary to start from relativistic quantum mechanics, where spin is naturally included in the Dirac equation. This will be discussed later on in “Spin in Relativistic DFT”.

In the nonrelativistic two-component picture, spin-independent operators—such as the one-electron Hamiltonian equation image in Eqs. (1) and (2)—act on both of these components, that is, they are proportional to the equation image unit matrix equation image. The two-component structure of the wavefunction is only probed by operators expressed in terms of the Pauli matrices,

  • equation image(4)

In particular, the operator corresponding to the electron spin is

  • equation image(5)

The three components of this spin operator fulfill the same commutation relations as those of the angular momentum operator equation image, i.e.,

  • equation image(6)

which is the basis for considering spin as an intrinsic angular momentum vector equation image. For the squared magnitude of the electron spin, one obtains the diagonal operator

  • equation image(7)

and as for the angular momentum, this operator commutes with each component of the spin (i.e., equation image for equation image).

In nonrelativistic quantum mechanics, one postulates that the spin operator equation image is related to an intrinsic magnetic moment of the electron.54–57 This spin magnetic moment is described by the operator

  • equation image(8)

where equation image is the Bohr magneton, c is the speed of light in vacuum, and the factor two is the electron g-factor when neglecting quantum electrodynamical effects.

Instead of explicitly writing two-component wavefunctions and equation image matrix operators, in quantum chemistry it is common to use a different notation, which will turn out to be particularly convenient for handling many-electron systems. Namely, as a shorthand notation, one introduces the orthonormal spin functions equation image and equation image, which depend on a spin variable s.54, 56, 58 This spin variable can only assume the values equation image and equation image, and the spin functions are defined such that

  • equation image(9)
  • equation image(10)

Then, the wavefunction of Eq. (3) can be expressed as

  • equation image(11)

with the first component given by equation image and the second component given by equation image. It is important to realize that the spin functions α and β are merely a way of expressing two-component wavefunctions, in which the spin variable s has the role of labeling the different components. Spin-independent operators are then given by a one-component operator acting only on the parts of the wavefunction that depend on the spatial coordinate r, while the spin operators equation image, equation image, equation image, and equation image act only on the parts depending on the spin variable s.

The nonrelativistic one-electron Hamiltonian equation image is spin-independent and hence commutes with all spin operators, in particular equation image. Therefore, its eigenfunctions can be chosen as eigenfunctions of equation image and of equation image. The eigenfunctions of equation image are given by equation image and equation image and the wavefunctions of an electron which are also eigenfunctions of equation image are thus of the form

  • equation image(12)

For any spatial part of the wavefunction, these two different total wavefunctions are possible, that is, each eigenvalue of the nonrelativistic one-electron Hamiltonian is twofold degenerate. The first is identified with an α- or “spin-up” electron ( equation image), whereas the second one corresponds to a β- or “spin-down” electron ( equation image). Of course, also linear combinations of these two degenerate eigenfunctions are solutions of the Schrödinger equation. Nevertheless, for a single electron any eigenfunction can be expressed as a product of a spatial part and a spin part.

Spin structure of the many-electron wavefunction

Within the two-component picture introduced in the previous section, the one-electron Hilbert space equation image is spanned by all admissible one-electron wavefunctions (which have two components related to the spin of the electron). For an N-electron system, the wavefunction is an element of the corresponding N-electron Hilbert space equation image, which is the tensor product space of the Hilbert spaces of each electron, i.e.,

  • equation image(13)

where the number given in parentheses designates the corresponding electron. As the one-particle wavefunctions have two components, the N-electron wavefunctions have 2N-components and operators have the dimension 2N × 2N. For a more detailed discussion of the tensor structure of the many-electron wavefunction, see for example, Refs.19–21 and chapter 8.4 in Ref.57.

As an alternative to explicitly handling many-component wavefunctions, it is again convenient to introduce spin coordinates si that can be used to distinguish the different components.56, 59, 60 Then, the N-electron wavefunction depends on N spin coordinates in addition to the N spatial coordinates,

  • equation image(14)

where xi = (ri, si) denotes the combination of spatial and spin coordinates. Each spin coordinate can assume the values −1/2 and +1/2. Each possible combination of values for these spin variables corresponds to one component of the many-electron wavefunction. In total, 2N different combinations of values are possible, that are thus used to label the 2N components of the many-electron wavefunction.

The nonrelativistic Hamiltonian describing N electrons in an external electrostatic potential vext(r) is (in Gaussian units) given by,

  • equation image(15)

where the Laplace operator Δi acts on the coordinate of the ith electron and rij = |rirj| is the distance between electrons i and j. This Hamiltonian does not contain any spin-dependent terms and only acts on the spatial coordinates.

The operator of the total spin Sˆ of a many-electron system is obtained58, 56 by summing the spins of the individual electrons equation image. In particular, one has for the z-component of the total spin operator,

  • equation image(16)

and for the square of the total spin

  • equation image(17)

where the first term emerges because any many-electron wavefunction is an eigenfunction of sˆ2(si) with eigenvalue equation image (i.e., electrons are spin-1/2 particles). Note that Sˆ2 couples different electrons, that is, it is a two-electron operator.61

Both Sˆ2 and Ŝz commute with the Hamiltonian of Eq. (15) and with each other, i.e.,

  • equation image(18)

Therefore, the eigenfunctions Ψ of the Hamiltonian can always be chosen as eigenfunctions of Ŝz and Ŝ2 with,

  • equation image(19)
  • equation image(20)

In general, eigenfunctions of the Hamiltonian belonging to different eigenvalues of Sˆ2 have different energies, while for each energy eigenvalue there are always 2S + 1 degenerate eigenfunctions differing in MS. As all three components of Sˆ commute with the Hamiltonian, any choice of the quantization axis is possible and will lead to identical results.

Finally, the spin structure of the many-electron wavefunction59 is also determined by the Pauli principle.62, 63 It requires that the wavefunction is antisymmetric (i.e., it has to change sign upon exchange of two electrons). This can be expressed with the permutation operator,

  • equation image(21)

as equation image. Here, equation image exchanges both the spatial and the spin coordinates of electrons i and j. To express this requirement in a different form, one can introduce the antisymmetrizer equation image defined as

  • equation image(22)

where the permutation operators equation image are ordered such that even numbers p are assigned to those that are generated by an even number of pair permutations, and odd numbers denote those generated by an odd number of pair permutations. Then, the requirement that the wavefunction is antisymmetric with respect to any pair permutation is equivalent to requiring equation image that is, Ψ has to be an eigenfunction of equation image.

As this Hamiltonian does not contain terms that couple the spatial and the spin coordinates, one could naively expect that—as in the one-electron case—the many-electron wavefunction can always be written as a product of a part depending on the spatial coordinates and of a part depending on the spin coordinates. However, for systems with more than two electrons this is in general not the case. The antisymmetrizer equation image then contains a sum of permutation operators each acting on both spatial and spin coordinates. Hence, it couples spin and spatial coordinates so that its eigenfunctions cannot be expressed as a product of a spatial and a spin part. Thus, the structure of the many-electron wavefunction with respect to the exchange of spatial coordinates is dependent on the spin structure.64–66 This is the most important consequence of the presence of spin in a nonrelativistic theory.

Spin structure of the electron density and spin density

The (total) electron density ρ(r) describes the probability density for finding any electron of a many-electron system at position r. It can be calculated from the wavefunction as

  • equation image(23)

that is, by integrating the squared absolute value of the wavefunction over all but one spatial coordinate. By writing out the integration over the corresponding spin variable explicitly,

  • equation image(24)
  • equation image(25)
  • equation image(26)

one notices that it is a sum of components ρα(r) and ρβ(r) that can be interpreted as the probability densities of finding an α- or a β-spin electron.60, 56 Their integrals,

  • equation image(27)

give the number of α- and β-electrons.

It is then natural to define the spin density, which gives the excess of α-electrons at a given point, as

  • equation image(28)

Here, the Pauli matrix σz(s1) operates on the first spin coordinate only. Just as the electron density, which is probed by X-ray diffraction experiments, the spin density is an observable. The spin density Q(RI) at the position of a nucleus I is accessible from electron paramagnetic resonance experiments, where it determines the nuclear hyperfine coupling constants.56, 67, 68 Similarly, the spin density at a nucleus can give rise to shifts in paramagnetic nuclear magnetic resonance.68–71 Full spatially resolved spin densities can be determined in neutron scattering experiments.72–77

The expectation value of a multiplicative one-electron operator such as equation image can be calculated directly from the electron density,

  • equation image(29)

that is, the full wavefunction is not needed. Similarly, expectation values of spin-dependent operators expressed only in terms of σz can be obtained from the spin density. In particular, the expectation value of Ŝz is given by

  • equation image(30)

and for eigenfunctions of Ŝz, one has

  • equation image(31)

The contribution of the electron spin magnetic moments to the interaction of a molecule with an inhomogeneous external magnetic field Bext(r), the so-called spin Zeeman interaction, is determined by the operator56

  • equation image(32)

Hence, for an inhomogeneous magnetic field in z-direction, that is, Bext(r) = (0, 0, Bz(r)), the expectation value of the spin Zeeman interaction can be evaluated directly from the spin density as

  • equation image(33)

If one considers an eigenfunction of Sˆz with eigenvalue equation image, this eigenvalue is (2S + 1)-fold degenerate and one can construct a set of 2S + 1 eigenstates of Ŝz with eigenvalues equation image, where equation image. The total electron densities equation image and the spin densities equation image of these Ŝz eigenstates are related to each other60, 78, 79: All 2S + 1 states share the same total electron density,

  • equation image(34)

and the spin densities are given by

  • equation image(35)

where equation image and equation image are the total electron density and spin density of the state with highest MS, respectively. Hence, the spin densities have the same functional form and are connected by a simple scaling. It immediately follows that equation image and that the spin density vanishes for states with MS = 0.

Spin in Hohenberg–Kohn DFT

  1. Top of page
  2. Abstract
  3. Introduction
  4. Spin in Nonrelativistic Quantum Chemistry
  5. Spin in Hohenberg–Kohn DFT
  6. Spin in Kohn–Sham DFT
  7. Spin in Relativistic DFT
  8. Future Directions for Spin-DFT
  9. Appendix: Gordon Decomposition of the Current Density
  10. Biographical Information
  11. Biographical Information

Traditionally, quantum chemistry sets out to calculate approximations to the many-electron wavefunction Ψ of a molecule in its ground state by minimizing the energy expectation value with respect to Ψ, under the constraint that Ψ represents a normalized and antisymmetric N-electron wavefunction, i.e.,

  • equation image(36)

where the nonrelativistic Hamiltonian equation image within the Born–Oppenheimer approximation was given in Eq. (15). In molecular systems, the external potential vext(r) is given by the Coulomb potential of the nuclei, that is, equation image equation image, where the sum runs over all nuclei with charges ZI at positions RI. Thus, the nonrelativistic molecular Hamiltonian assumes the form,

  • equation image(37)

According to this structure of the Hamiltonian, the energy expectation value in the above minimization is usually split up as

  • equation image(38)

However, the wavefunction itself is not directly needed for calculating these expectation values. The evaluation of the first term, corresponding to the kinetic energy, only requires the one-electron reduced density matrix (1-RDM), whereas the second term describing the electron–electron interaction can be calculated from the diagonal two-electron reduced density matrix (2-RDM). Finally, the electron–nuclear attraction energy can be evaluated directly from the multiplicative operator of the electron–nuclei Coulomb interaction and from the electron density only as,

  • equation image(39)

This can be exploited by performing the minimization with respect to the 2-RDM directly,80 but difficulties arise in enforcing that the 2-RDM corresponds to an actual antisymmetric N-electron wavefunction.

DFT50–53 provides the theoretical framework for calculating the energy expectation value directly from the electron density ρ(r) only. The foundations of this exact theory will be outlined in the following, focussing on its application to open-shell systems.

Hohenberg–Kohn theorems

The first HK theorem81 states that for each electron density ρ(r) that can be obtained from a ground-state wavefunction (such densities are called v-representable densities), the external potential vext(r) that yields this electron density as the ground state when employed in the Hamiltonian of Eq. (15) is unique up to a constant. Therefore, this potential is a functional of the electron density and since it completely determines the Hamiltonian, also the ground-state wavefunction—which can in turn be determined by solving the corresponding Schrödinger equation—is a functional of the electron density. Furthermore, all observables of the system, in particular the total energy in a given nuclear potential vnuc(r), can be obtained from this wavefunction. This connection between the electron density and the total energy is illustrated in Figure 1a. Therefore, there exists an energy functional E[ρ] that relates the electron density to the total energy.

thumbnail image

Figure 1. Connection between the electron density equation image and the total energy as stated by the first HK theorem.

Download figure to PowerPoint

According to the second HK theorem,81 the ground state energy E0 of a system of electrons in a given nuclear potential vnuc(r) can be determined by minimizing the total energy functional

  • equation image(40)

with

  • equation image(41)

under the constraint that ρ(r) integrates to N electrons. The electron density for which this minimum is achieved is the ground-state electron density ρ0(r). In this equation, the total energy functional has been split into a system-specific part (the first term), depending on the nuclear potential, and a system-independent part (the second term), which is called the “universal HK functional” FHK[ρ].

Following the Levy constrained-search formulation of DFT,82, 83 this universal HK functional is given by

  • equation image(42)

where equation image and equation image are the operators of the kinetic energy and of the electron–electron repulsion energy, respectively. The minimization runs over all wavefunctions Ψ that yield the target electron density ρ. From these wavefunctions, the one with the lowest expectation value of equation image is chosen. The minimization of E[ρ] will lead to the exact ground-state electron density ρ0, and the exact ground-state wavefunction Ψ0 is the one for which the minimum in Eq. (42) is obtained. This (nonrelativistic) ground-state wavefunction Ψ0 has to be an eigenfunction of Sˆ2. Therefore, it is sufficient to restrict the constrained search to wavefunctions that are eigenfunctions of Sˆ2.

Using the Levy-constrained search for defining the HK functional also extends the domain in which the above functionals are defined from v-representable densities (i.e., densities that are obtained from a ground state wavefunction) to N-representable densities (i.e., densities that are obtained from any wavefunction, not necessarily a ground state). The resulting generalization of the first HK theorem is illustrated in Figure 1b. More details on the foundations of HK-DFT and on the more general definition of the universal HK functional introduced by Lieb84 can be found in dedicated reviews on these topics.51, 85, 86

Initially, Hohenberg and Kohn explicitly excluded degeneracies in their derivation of the HK theorems.81 However, a state with S > 0 will be (2S + 1)-fold degenerate, with the different degenerate wavefunctions equation image corresponding to equation image equation image. The necessary generalization of the HK theorems is possible in a straightforward way.87 The degenerate wavefunctions equation image and all their linear combinations share the same electron density. For a given ρ, the minimum in Eq. (42) is achieved for all degenerate wavefunctions spanned by the Ŝz-eigenfunctions equation image. This is illustrated in Figure 1c. Therefore, the minimization of the total energy functional E[ρ] will still lead to a unique ground-state density ρ0. If it is necessary to obtain a unique minimizing wavefunction, the constrained search can be restricted to wavefunctions corresponding to a specific value of MS. In this case, the wavefunction is again uniquely determined by the electron density.

Spin density in Hohenberg–Kohn DFT

According to the HK theorem only the total electron density is required for obtaining the exact ground-state energy and electron density.81, 88 Therefore, irrespective of the spin state, the spin density Q(r) or the individual α- and β-electron densities ρα(r) and ρβ(r) are not required during the minimization of the total energy functional, and the ground-state spin density Q0(r) is not directly available. However, if a suitable value of MS is chosen, the wavefunction equation image is uniquely determined by the total density ρ(r), that is, equation image. From this wavefunction, the ground-state α- and β-electron densities,

  • equation image(43)
  • equation image(44)

as well as the corresponding spin density

  • equation image(45)

can be calculated. In these expressions, the superscript MS indicates that a specific value of MS has to be selected if one is interested in individual α- and β-densities or in the spin density. The chosen value of MS fixes the number of α- and β-electrons by equation image.

To generalize HK-DFT to use the individual α- and β-densities instead of the total density only, the minimization of the total energy can be rewritten as,50, 89

  • equation image(46)

and by removing the outer minimization with respect to ρ, one obtains

  • equation image(47)

This defines a universal HK functional in terms of the α- and β-electron densities,

  • equation image(48)

In contrast to the HK functional of Eq. (42), the constrained search in this spin-resolved HK functional now runs over all wavefunctions corresponding to a given pair of ρα and ρβ. By minimizing the generalized total energy functional [cf. Eq. (47)],

  • equation image(49)

under the constraint that ρα and ρβ integrate to Nα and Nβ electrons, respectively, it is then possible to obtain the ground-state α- and β-electron densities ρmath image and ρmath image. Again, in this minimization a specific value of equation image has to be selected.

Instead of using ρα and ρβ, it is also possible to use the total electron density ρ = ρα + ρβ and the spin-density Q = ρα − ρβ as variables of the spin-resolved HK functional. This gives

  • equation image(50)

Although usually—in particular in practical applications of spin-DFT—it is more common to employ the α- and β-densities as basic variables, in the following such a formulation in terms of ρ and Q will often turn out to be useful, because it allows for an easier comparison to the spin-independent functionals defined only in terms of the density ρ. We will switch between these two representations whenever suitable.

With the spin-resolved HK functional FHK[ρ, Q], it is now also possible to give a simpler prescription for obtaining the spin-density corresponding to a given total density: equation image is the spin density for which FHK[ρ, Q] is minimized, under the constraint that equation image integrates to twice the chosen value of MS, i.e.,

  • equation image(51)

Therefore, the spin-independent HK functional F[ρ] can be obtained from the spin-dependent HK functional FHK[ρ, Q] as

  • equation image(52)

where Q[ρ] is any of the spin densities that minimize FHK[ρ, Q] for the given total density ρ. Of course, as long as no specific MS is chosen, Q[ρ] is not unique, but any admissible choice must lead to the same energy.

Finally, we have to consider whether there exist extensions of the HK theorems that justify the use of ρ and Q (or of ρα and ρβ) as basic variables. For a generalization of this kind, it is not sufficient to consider wavefunctions generated by an external potential vext(r), but also wavefunctions obtained in the presence of an additional external magnetic field Bz(r) have to be taken into account. Such an extension of the HK theorems was first given by von Barth and Hedin,90 and was only recently put on more firm ground by extensions of Lieb's formulation of DFT.91, 92 Similar to the external potential, which is only known up to a constant, it is also possible to add a constant shift to the external magnetic field Bz(r) without changing the wavefunction or the (spin-)density.93, 94 This leads to a number of peculiarities related to the differentiability of the spin-dependent energy functional.95, 96 However, most of these issues do not appear if the treatment is restricted to eigenfunctions of Sz (the case of interest here)97 or can be addressed by constraining MS to a fixed value (as we are always requiring here).98, 99

Fractional spin conditions on the HK functional

For states that are not a singlet (i.e., for S > 0), there are different degenerate wavefunctions equation image. These wavefunctions all share the same electron density, but correspond to different spin densities equation image. These different spin-densities are related by [cf. Eq. (35)]

  • equation image(53)

and it follows that equation image and that one obtains equation image for even values of S. For all these degenerate spin-densities, the exact total energy functional E[ρ, Q] must yield the same value.

This statement can be generalized to linear combinations of these spin-densities. Within an ensemble formulation of spin-DFT, it can be shown100 that for any properly normalized linear combination of the spin-densities equation image, the same energy should be obtained. Therefore, one finds that

  • equation image(54)

which implies for the spin-dependent HK functional

  • equation image(55)

Such ensemble spin densities correspond to a situation with a non-integer number of α- and β-electrons (“fractional spins“). As has been pointed out by Yang and coworkers, the above constancy conditions for fractional spins, which is a property of the exact energy functional is violated by all contemporary approximations. Therefore, it was suggested that many problems appearing in practical DFT calculations with such approximations might be connected to this violation.49, 101, 102

The use of fractional spins makes it possible to further simplify the relation between the spin-dependent and spin-independent HK functionals given in Eq. (52), because one realizes that

  • equation image(56)

This equation holds both for systems with an even number of electrons (where one can always choose MS = 0, corresponding to Q(r) = 0) and for systems with an odd number of electrons (where Q(r) = 0 is only possible if one allows for fractional spins).

Spin states in Hohenberg–Kohn DFT

In their initial formulation,81 the HK theorems were applicable only to the ground state. In particular, minimizing the total energy functional E[ρ] only yields the ground-state energy and electron density (and if the spin-dependent energy functional is employed, also the spin density). However, as was first shown by Gunnarsson and Lundqvist,103 HK-DFT can be generalized to the lowest-energy states of a given symmetry. Of particular importance is the calculation of the lowest state of a particular spin symmetry, that is, of the lowest state with a particular eigenvalue of Sˆ2. Such a generalization is most easily presented within the constrained-search formulation of DFT.

To obtain the energy and electron density of the lowest state corresponding to a given value of S, one has to define the spin-state specific energy functional,

  • equation image(57)

with the spin-state specific HK functional,

  • equation image(58)

where the constrained search now only includes wavefunctions ΨS which are eigenfunctions of Sˆ2 with the proper eigenvalue. Therefore, one obtains a different HK functional and thus a different total energy functional for each value of S.

Within spin-DFT, the simplest way of obtaining a functional that at least partly allows one to select certain spin states is by choosing an appropriate value for MS. This way, only states with SMS are accessible because for spin states corresponding to a smaller values of S, the chosen value of MS is not admissible. By minimizing the energy functional Ev[ρ, Q] under the constraint that Q(r) integrates to 2MS, the lowest-energy state with SMS is obtained, i.e.,

  • equation image(59)

Equivalently, one can, of course, also minimize Evα, ρβ] under appropriate constraints for Nα and Nβ. However, fixing MS to target a specific spin-state is not completely general since the minimization is only restricted to states with SMS, not to states with a specific S. Although it is, for instance, possible to calculate the lowest triplet (S = 1) state if the ground-state is a singlet (S = 0), it is not possible to target the lowest singlet state if the ground-state is a triplet. Therefore, to be able to calculate the lowest state of a given S, it would in general be necessary to use the true spin-state specific energy functional of Eq. (57).

Spin in Kohn–Sham DFT

  1. Top of page
  2. Abstract
  3. Introduction
  4. Spin in Nonrelativistic Quantum Chemistry
  5. Spin in Hohenberg–Kohn DFT
  6. Spin in Kohn–Sham DFT
  7. Spin in Relativistic DFT
  8. Future Directions for Spin-DFT
  9. Appendix: Gordon Decomposition of the Current Density
  10. Biographical Information
  11. Biographical Information

Although the HK formulation of DFT is exact, it is very difficult to set up computationally feasible, but nevertheless accurate approximate realizations of it. This is mainly rooted in the difficulty of approximating the kinetic-energy contribution to the HK functional as a functional of the electron density only.104, 105

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).

thumbnail image

Figure 2. Relationship between wave function, total electron density, and spin density of the system of fully interacting electrons and of the spin-restricted and spin-unrestricted KS reference systems of noninteracting electrons.

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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.

Spin-Restricted Kohn–Sham DFT

Noninteracting Reference System.

First, we consider a system of N noninteracting electrons in an external potential vs(r). Such a system is described by the Hamiltonian

  • equation image(60)

where the subscript s (for “single-particle”) is introduced to indicate that the quantities refer to a system of noninteracting electrons. Because this Hamiltonian does not contain terms that couple different electrons, an exact wavefunction is given by an antisymmetrized product of one-electron functions (i.e., by a single Slater determinant), for which the short-hand notation

  • equation image(61)

can be used. The spatial parts of the one-electron functions (orbitals) can be obtained as the solutions of the one-electron equation

  • equation image(62)

which trivially emerge from the energy eigenvalue equation equation image with equation image. As this one-electron Hamiltonian does not depend on the spin of the electron, each energy eigenvalue is twofold degenerate (i.e., each spatial orbital can be combined with an α- or a β-spin function). In particular, the spatial orbitals are identical for α- and for β-electrons, and the resulting Slater determinant is therefore spin-restricted.

In such a spin-restricted Slater determinant, each spatial orbital φi can either be doubly occupied (i.e., it appears both in combination with an α- and with a β-spin function) or it can be singly occupied with either an α- or a β -electron. For the ground state, such singly occupied orbitals can only occur for the highest occupied molecular orbital (HOMO), and more than one singly occupied orbital can only be present if the HOMO is degenerate (for a detailed discussion, see e.g., Ref.109 and chapters 3.3 and 3.4 in Ref.53).

For a single, spin-restricted Slater determinant, the electron density is given by

  • equation image(63)

where the occupation numbers fi are either 1 or 2. The spin density is determined only by the singly occupied orbitals, and can be calculated as

  • equation image(64)

where the sum only runs over the singly occupied orbitals and where si = +1 for α-spin and si = −1 for β-spin orbitals, as all doubly occupied orbital contributions drop out because of the identical spatial distribution of these α, β-pairings. Finally, the kinetic energy of a spin-restricted Slater determinant can be calculated as

  • equation image(65)

Thus, it turns out that spin-restricted Slater determinants in which the occupation numbers fi are identical share the same electron density and have the same kinetic energy. Such determinants—which can only differ in the spin of the singly occupied orbitals—are degenerate with respect to the noninteracting Hamiltonian of Eq. (60).

The noninteracting Hamiltonian equation image commutes with both Ŝz and with Sˆ2. Therefore, it is always possible to combine degenerate eigenfunctions such that they are also eigenfunctions of Ŝz and Sˆ2. A restricted Slater determinant is always an eigenfunction of Ŝz with eigenvalue equation image.60, 58 If all singly occupied orbitals are either α- or β-spin orbitals, then it is also an eigenfunction of Sˆ2 with S = |MS|. In all other cases, an eigenfunction of Sˆ2 can be constructed as a linear combination of (degenerate) determinants in which the same orbitals are singly occupied and which correspond to the same value of MS. Such linear combinations are known as configuration state functions (CSF).10 It is important to understand that for any eigenfunction of the noninteracting Hamiltonian that is an eigenfunction of Sˆ2 with eigenvalue equation image, degenerate eigenfunctions with equation image equation image (for an even number of electrons) or with equation image (for an odd number of electrons) can also be constructed. Thus, for each energy eigenvalue, there is one CSF corresponding to S = 0 (i.e., a singlet state) or equation image (i.e., a doublet state) for an even or odd number of electrons, respectively.

The HK theorems still hold for a system of noninteracting electrons. Thus, the ground-state density of equation image can be determined by minimizing the noninteracting energy functional,

  • equation image(66)

where the noninteracting kinetic-energy functional Ts[ρ] can be defined in the Levy constrained-search formalism as

  • equation image(67)

In this definition, the constrained search includes all wavefunctions Ψs that correspond to a system of noninteracting electrons with density ρ. As discussed above, this could be further restricted to singlet or doublet wavefunctions. In this definition, Ts[ρ] is independent of the spin density.

The ground-state electron density ρ0 is obtained from minimizing Es[ρ] under the constraint that the number of electrons is preserved, and the corresponding ground-state wavefunction Ψs,0 is the one for which the minimum in Eq. (67) is achieved. Again, this wavefunction could always be chosen as a singlet or a doublet for an even or odd number of electrons, respectively.

Interacting Energy Functional and Exchange–Correlation Energy.

The (spin-resolved) HK functional of the true system of interacting electrons [Eq. (50)] can now be decomposed as

  • equation image(68)

where Ts[ρ] is the noninteracting kinetic energy introduced in the previous section, J[ρ] is the classical Coulomb interaction of the electron density with itself,

  • equation image(69)

and the exchange–correlation energy Exc[ρ, Q] is defined to account for the remaining energy contributions

  • equation image(70)

This exchange–correlation functional could also be expressed in terms of the α- and β-electron densities. The noninteracting kinetic energy Ts[ρ] is different from the true kinetic energy of the fully interacting system T[ρ]. Therefore, the exchange–correlation energy also contains the difference Tc[ρ] = T[ρ] − Ts[ρ] between the kinetic energy of the interacting reference system and the kinetic energy of the noninteracting system.

With these definitions, the total energy functional of the interacting system can be expressed as

  • equation image(71)

The spin-independent analogs of the exchange–correlation and the total energy functional, Exc[ρ] and E[ρ], are recovered from these definitions when setting Q(r) = 0 [cf. Eq. (56)].

The ground-state density of the true interacting system can be determined by minimizing the total energy functional E[ρ] with respect to ρ, under the constraint that it integrates to the correct number of electrons. With the exact functionals, this will lead to the exact ground-state electron density ρ0(r). The corresponding exact wavefunction Ψ0 is the one for which the minimum in the constraint search in Eq. (50) is obtained. This ground-state wavefunction has to be an eigenfunction of Sˆ2 and Ŝz, and the corresponding value of S determines the spin multiplicity of the ground-state, whereas the different MS-states are degenerate. In contrast to the noninteracting case, the ground-state wavefunction is not necessarily a singlet or doublet wavefunction, but could have a higher spin multiplicity.

Kohn–Sham Potential.

Minimization of the noninteracting energy functional [Eq. (66)] with respect to the total density ρ, under the constraint that ρ integrates to N electrons, yields the following Euler–Lagrange equation,50

  • equation image(72)

Conversely, the total energy functional of the interacting system is given by Eq. (71), which upon minimization leads to the condition

  • equation image(73)

where equation image is the classical Coulomb potential of the electrons and equation image is the exchange–correlation potential.

As the definition of the noninteracting kinetic energy Ts[ρ] is the same in both minimizations, and because we require that the total densities obtained for the interacting and noninteracting system agree, we obtain for the KS potential

  • equation image(74)

The ground-state electron density of the fully interacting system can thus be determined by solving the Schrödinger equation of a noninteracting system [i.e., with the Hamiltonian of Eq. (60) including the external potential vs[ρ] given by Eq. (74)], and the orbitals of this noninteracting system can be obtained from the KS equations

  • equation image(75)

Thus, the ground-state electron density of the true interacting system is obtained from the wavefunction Ψs,0 of the noninteracting reference system. However, this ground-state wavefunction Ψs,0 of the noninteracting reference system does not agree with the ground-state wavefunction Ψ0 of the interacting system. Moreover, Ψs,0 can always be chosen as a singlet or doublet state (i.e., S = 0 or equation image), whereas Ψ0 can correspond to any value of S. Therefore, the spin multiplicities of the ground-states of the noninteracting and of the interacting system can be different.

Spin Density.

For S > 0, the ground state of the true interacting system is degenerate and there is a set of eigenfunctions of Ŝz with different eigenvalues MS. These wavefunctions equation image have different spin-densities equation image that are related by Eq. (35). By construction, the ground-state wavefunction Ψs,0 of the noninteracting reference system and the ground-state wavefunction Ψ0 of the fully interacting system only share the same electron density. However, the corresponding spin densities Q(r) and Qs(r) are in general not equal and the true ground state spin density cannot be calculated from Ψs,0.

From Eq. (64) it is obvious that Q(r) and Qs(r) have to be different: for the spin-restricted noninteracting reference system, the spin density is determined only by the singly occupied orbitals and will thus have the same sign at every point in space (i.e., Qs(r) > 0 for MS > 0). However, it is known both from accurate calculations and from experiment, that for the interacting system the spin density has different signs in different regions in space.112–114

To obtain the spin density in a restricted KS-DFT formulation, one has to minimize the spin-resolved HK functional FHK0, Q] defined in Eq. (50) with respect to Q(r), under the constraint that the spin density integrates to 2Ms. As the exchange–correlation energy is the only part of this functional that depends on the spin density, the minimization with respect to Q(r) leads to the condition

  • equation image(76)

where the Lagrange multiplier λ is zero because of Eq. (35). This suggests a two-step procedure for determining the spin density in restricted KS-DFT. First, the total ground-state density ρ0(r) is determined by solving the KS equations. Subsequently, the corresponding ground-state spin density Q0(r) can be calculated from the above minimization condition for a chosen value of MS.

Spin-Unrestricted Kohn–Sham DFT

Noninteracting Reference System.

The choice of a reference system of noninteracting electrons with the same total electron density as the interacting system is not the only option. Alternatively, it is also possible to envisage a reference system of noninteracting electrons that has the same α- and β-electron densities as the interacting system.90, 103, 112 In this case, a reference system with the Hamiltonian

  • equation image(77)

is used. To distinguish them from those introduced earlier for a spin-restricted reference system, the superscript “(u)” will be used for quantities referring to this spin-unrestricted reference system. Different potentials vmath image(r) = vmath image(r) + vmath image(r) and vmath image(r) = vmath image(r) − vmath image(r) for the α- and β-electrons, respectively, are now needed to allow the reference system of noninteracting electrons to have the same spin density as the interacting one. This corresponds to introducing an inhomogeneous external magnetic field in z-direction equation image that only interacts with the electronic spins (i.e., the interaction due to orbital angular momentum is ignored) [cf. Eq. (32)].

An exact solution to the corresponding Schrödinger equation has the form of a single Slater determinant, but in contrast to the spin-restricted case the spatial orbitals now differ for α- and β-electrons, i.e.,

  • equation image(78)

The spatial parts of the orbitals can be obtained from two separate sets of one-electron equations

  • equation image(79)

Both the resulting α- and β-orbitals form an orthonormal set, equation image and equation image, but α- and β-orbitals are in general not orthogonal to each other, that is, equation image.

The noninteracting Hamiltonian Hmath image still commutes with Ŝz, and any spin-unrestricted Slater determinant is an eigenfunction of Ŝz, with the eigenvalue MS being determined by the number of α- and β-electrons. However, in contrast to the spin-restricted case, the different Ŝz eigenstates are not degenerate anymore. For constructing the ground-state wavefunction, the N orbitals with the lowest orbital energies have to be occupied, which automatically fixes MS. By occupying other (α- or β-electron) orbitals, excited state wavefunctions for the noninteracting reference system corresponding to different values of MS can be obtained.

However, Hmath image in general does not commute with Sˆ2, and the ground-state wavefunction is thus not an eigenfunction of Sˆ2 anymore. Instead, the expectation value of Sˆ2 can be calculated as (assuming NαNβ)61, 115

  • equation image(80)

In the spin-restricted case, where the α- and β-orbitals are equal and therefore mutually orthogonal, the last term equals the number of doubly occupied orbitals, and one obtains equation image. In the unrestricted case, this cancellation is only partial and a larger expectation value is obtained. This is often referred to as “spin contamination.”

For an unrestricted Slater determinant, the total electron density is given by,

  • equation image(81)

and the spin density can be calculated as

  • equation image(82)

In contrast to the spin-restricted case, the spin density can now have different signs at different points in space. The kinetic energy of the unrestricted Slater determinant ϕmath image is

  • equation image(83)

and a noninteracting kinetic-energy functional can now be defined as

  • equation image(84)

or as

  • equation image(85)

In contrast to the spin-restricted case, this functional depends not only on the total electron density ρ(r) = ρα(r) + ρβ(r), but also on the spin density Q(r) = ρα(r) − ρβ(r). Usually, Tmath image[ρ, Q] yields different kinetic energies for systems that share the same total electron density, but have different spin densities.

In terms of α- and β-electron densities, the total energy functional of the noninteracting system is given by

  • equation image(86)

The ground-state α- and β-densities ρmath image(r) and ρmath image(r) of the noninteracting system can then be determined by minimizing this energy functional with respect to ρα and ρβ, under the constraint that these integrate to the correct number of α- and β-electrons.

Exchange–Correlation Energy Functional.

The spin-resolved HK functional [cf. Eq. (48)] of the true system of interacting electrons can now be decomposed as

  • equation image(87)

where the spin-resolved exchange–correlation energy is defined as

  • equation image(88)

As this functional Emath imageα, ρβ] has been defined via the spin-unrestricted reference system, it is in general different from the functional Emath imageα, ρβ] defined in the previous section. This difference arises because different definitions of the noninteracting kinetic energy are used in the two cases.

With this definition of the exchange–correlation energy, the total energy functional of the true system of interacting electrons is given by

  • equation image(89)

This total energy functional is identical to the spin-resolved version of the total energy functional derived in the spin-restricted case [cf., Eq. (71)], even though it is decomposed in a different fashion.

The ground-state α- and β-electron densities ρmath image(r) and ρmath image(r) can then be determined by minimizing this total energy functional with respect to ρα and ρβ, under the constraint that these integrate to Nα and Nβ electrons, respectively. Note that by choosing Nα and Nβ, a specific value of MS is selected. As long as this MS can be realized for the exact ground state, the resulting total ground-state density ρ0(r) = ρmath image(r) + ρmath image(r) will be independent of the choice of MS. In addition, the minimization will then yield the exact ground-state spin density equation image.

KS Potential.

The minimization of the total energy functional Emath imageα, ρβ] of the spin-unrestricted noninteracting reference system leads to these Euler–Lagrange equations

  • equation image(90)
  • equation image(91)

For the interacting system, the minimization of Eα, ρβ] with respect to the α- and β-electron densities yields

  • equation image(92)
  • equation image(93)

with the spin components of the exchange–correlation potential

  • equation image(94)

If we require that ρα(r) and ρβ(r)—and thus both the total and the spin density of the ground state—are the same for the noninteracting reference system and the true interacting system, we find that the spin components of the KS potential are given by

  • equation image(95)
  • equation image(96)

Therefore, the exact ground-state α- and β-electron densities of the true interacting system can be calculated by solving the Schrödinger equation of an auxilliary system of noninteracting electrons with the Hamiltonian of Eq. (77). The ground-state wavefunction Ψmath image of this KS reference system is given by an unrestricted Slater determinant, constructed from the orbitals obtained from the KS equations,

  • equation image(97)

Here, the equations for the α- and β-orbitals are coupled through the Coulomb and exchange–correlation potentials.

Equivalently, the KS potential can be expressed as a component that acts on the total electron density,

  • equation image(98)

and one that acts on the spin density,

  • equation image(99)

where the total and spin exchange–correlation potential are given by

  • equation image(100)

These expressions will be used in the following section for comparing with the spin-restricted theory.

Even though the electron density and spin density calculated from this Ψmath image are equal to those of the fully interacting system, it is important to realize that Ψmath image does not agree with the ground-state wavefunction Ψ0 of the true interacting system. In particular, Ψ0 can always be chosen as an eigenfunction of Sˆ2, whereas by construction, Ψmath image is not an eigenfunction of Sˆ2 for S > 0. Thus, within an exact formulation of unrestricted KS-DFT, the wavefunction of the KS reference system is always spin contaminated for S > 0.

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 equation image 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 Ψmath image 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-DFTSpin-unrestricted KS-DFT
Correct equation image?YesYes
Correct equation image?NoYes
equation image is eigenfunction of equation image?YesNo
Correct equation image?MaybeNo
equation image is eigenfunction of equation image?YesYes
Correct equation image?MaybeYes

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 Tmath image[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-DFTSpin-unrestricted KS-DFT
Noninteracting kinetic energy equation image equation image
Decomposition of HK functional equation image equation image
Exchange–correlation energy equation image equation image
Exchange–correlation potential equation image equation image
   equation image
  • equation image(101)

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

  • equation image(102)

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 Emath image[ρ, Q] and the noninteracting kinetic energy Tmath image[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,

  • equation image(103)

whereas in the spin-unrestricted case, it applies to the sum of the exchange–correlation energy and the noninteracting kinetic energy,

  • equation image(104)

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,

  • equation image(105)

Here, the first term is the exchange–correlation potential in the spin-restricted formalism, while the second term equation image 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

  • equation image(106)

For the ground-state electron and spin densities, the first term vanishes according to Eq. (76), and the above expression for vmath image[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 vmath image[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 vmath image[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 Fmath image[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

  • equation image(107)

where equation image 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 equation image 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

  • equation image(108)

One protocol for constructing this spin-state dependent exchange–correlation functional then proceeds using the spin density equation image of the noninteracting reference system to distinguish the different spin states. To this end, Emath image[ρ] is expressed as

  • equation image(109)

Here, Emath image[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 Emath image[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 Emath image[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 Emath image[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 Emath image[ρ, Q]) instead of Emath image[ρ, Q] would be required. Of course, the fractional spin condition applies to Exc[ρ, Q] and Emath image[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 Tmath image[ρ, Q]. Consequently, the spin-state dependence only enters the exchange–correlation functional, which becomes

  • equation image(110)

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

  • equation image(111)

Here, different spin states can be distinguished based on the integral of the spin density. However, the functional Emath image[ρ, Q] does not describe the correct spin-density dependence of Emath image[ρ, Q] anymore. Neither does Emath image[ρ, Q], which is limited to spin densities corresponding to MS = S. However, the spin densities equation image corresponding to other eigenvalues of Ŝz can be obtained from the scaling relation of Eq. (35). Again, it is important to realize that the fractional spin condition does not apply to Emath image[ρ, Q].

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 Emath image[ρ, Q]), as discussed above for spin-restricted KS-DFT.

Spin in Relativistic DFT

  1. Top of page
  2. Abstract
  3. Introduction
  4. Spin in Nonrelativistic Quantum Chemistry
  5. Spin in Hohenberg–Kohn DFT
  6. Spin in Kohn–Sham DFT
  7. Spin in Relativistic DFT
  8. Future Directions for Spin-DFT
  9. Appendix: Gordon Decomposition of the Current Density
  10. Biographical Information
  11. Biographical Information

Spin and current in relativistic quantum mechanics

So far, the discussion has focused on spin in nonrelativistic DFT. For the sake of completeness, we now consider the generalization to the more fundamental relativistic regime. The relativistic theory relies on Dirac's semiclassical theory of the electron (see Ref.57 for a detailed account). This quantum theory describes the relativistic motion of the electron in a classical external electromagnetic field, represented by the scalar and vector potentials ϕext(r) and Aext(r). The equation of motion reads in this case (in Gaussian units),

  • equation image(112)

which yields an energy eigenvalue equation for determining the stationary states when the right hand side is replaced by Eψ(r,t). The Dirac Hamiltonian equation image consists of the kinetic energy operator equation image, the rest energy term equation image, and the interaction operator equation image, where c is the speed of light and equation image is the momentum operator as before. The Dirac matrices are contained in the parameters α = (αx, αy, αz) and β, which are in the standard representation,

  • equation image(113)

where σi are the the Pauli spin matrices σx, σy, and σz defined in Eq. (4). A consequence of these four-dimensional operators is that the Dirac Hamiltonian equation image is a 4×4 matrix operator with a four-component eigenvector ψ(r,t), the so-called 4-spinor. Such a four-component description naturally includes spin, which had to be introduced in an ad hoc fashion in the nonrelativistic theory.

In analogy to the nonrelativistic case, one can define the relativistic spin operator as

  • equation image(114)

This operators still obey the commutation relations of an angular momentum. However, in the relativistic case equation image and equation image do not commute with the Hamiltonian. Therefore, eigenstates of equation image cannot be chosen as eigenfunctions of equation image and equation image anymore and spin is thus not a good quantum number in relativistic theory. In spherically symmetric systems such as atoms, one can consider the total angular momentum instead. This also has the consequence that, in contrast to the nonrelativistic case, the expectation value of equation image will depend on the choice of the quantization axis.

In the case of many electrons, the relativistic wavefunction again assumes a tensor structure, that is, for N electrons, the wavefunction formally has 4N components. It has to fulfill the Pauli principle by being antisymmetric with respect to the exchange of any two electron. The relativistic many-electron Hamiltonian is then given by (neglecting projectors on positive-energy solutions for the sake of brevity),

  • equation image(115)

where equation image is the one-electron Hamiltonian of Eq. (112) acting on electron i and equation image is the operator describing the interaction between electrons i and j. The form of the exact electron–electron interaction operator can be derived from quantum electrodynamics, but usually only approximate forms are employed in practice. The simplest approximation is to employ the nonrelativistic Coulomb operator. However, the resulting Dirac–Coulomb Hamiltonian is not Lorentz invariant. This approximation is improved by the Dirac–Coulomb–Gaunt Hamiltonian, which also includes the unretarded magnetic interaction between electrons, whereas the Dirac–Coulomb–Breit Hamiltonian approximately describes the retarded electromagnetic interaction (for a detailed discussion, see chapter 8 in Ref.57).

As in nonrelativistic theory, an electron density and a current density can be defined in the relativistic many-electron theory such that they fulfill a continuity equation. This results in the definition of the electron density as57, 130

  • equation image(116)

and of the current density as

  • equation image(117)

where α1 indicates that the Dirac matrices act on the first electron and the dagger denotes the transposed and complex conjugate spinor. This definition of the current density still holds in the presence of external magnetic fields.

Relativistic current-DFT

A central aspect of relativistic theories is that all fundamental physical equations must preserve their form under Lorentz transformations from one inertial frame of reference to another one. This requires that the equations are castable in tensorial form. Hence, all quantities are joined to 4-vectors, which are basic physical quantities in any relativistic theory. For instance, the electromagnetic potentials are joined to yield the 4-potential Aμ = (ϕ, A). This is the reason why a relativistic theory has to include both the scalar and vector potentials simultaneously. The density ρ(r) is also part of a 4-vector, namely of the 4-current jμ(r). The other three components are given by the current density j(r) such that jμ = (cρ, j).

Now, a similar decomposition of the energy expectation value as in the nonrelativistic case can be performed,

  • equation image(118)

where equation image is the relativistic “kinetic energy” operator, collecting all terms of the one-electron Hamiltonian equation image containing the Dirac matrices α and β (i.e., the kinetic energy and the rest energy terms), equation image consists of the remaining one-electron terms and describes the interaction with the external electromagnetic potentials, and equation image is the electron–electron interaction operator. The second term can be calculated directly from the 4-current, without the need to know the full wavefunction, as

  • equation image(119)

where Einstein's convention of implicit summation over repeated lower and upper indices has been used. The first term is the same as in the nonrelativistic case and describes the electrostatic interaction of the electron density with the external potential, whereas the second term accounts for the interaction of the current density with the external magnetic field.

A 4-current may be understood as the source of a 4-potential which can be calculated from the relativistic generalization of the Poisson equation of electrostatics,

  • \input amssym $$\font\abc=cmmib10\def\bi#1{\hbox{\abc #1}}{\square A^\mu({\bi r}) = {4\pi\over c} j^\mu({\bi r}),}$$(120)

with the D'Alembertian operator being the Minkowski space generalization of the three-dimensional Laplacian, that is, it is defined as \input amssym $\square= {1\over c^2} {\partial^2\over \partial t^2 } - \Delta$. From the solution of this equation, one obtains for the 4-potential generated by the electronic 4-current jμ,

  • equation image(121)

The classical interaction energy J[jμ] of the 4-current jμ with itself (i.e., the relativistic analog of the classical Coulomb interaction in nonrelativistic theory) is then given by

  • equation image(122)

Here, the first term is the classical electrostatic (Coulomb) interaction J[ρ], whereas the second term accounts for the magnetic interaction of the electrons. Note that this second term should only be present if the Gaunt or Breit interaction has been included in the Hamiltonian.

The analogy between Vext[jμ] and J[jμ] [Eqs. (119) and (122)] and their nonrelativistic counterparts Vext[ρ] and J[ρ] suggests that the fundamental quantity for a relativistic formulation of DFT is actually the 4-current, that is, the combination of the electronic density ρ(r) and the current density j(r). For this reason, the relativistic generalization of standard DFT has been known as current-DFT (CDFT).131–134

It can be shown that in relativistic theory, an analog of the HK theorem exists: the 4-current jμ(r) uniquely determines the external 4-potential Aμ(r) up to a gauge transformation.131 Therefore, it is possible to consider the energy E[jμ] as a functional of the 4-current jμ(r). As in the nonrelativistic case, one then proceeds by introducing a reference system of noninteracting electrons with the relativistic Hamiltonian

  • equation image(123)

The eigenfunctions of this noninteracting Hamiltonian are given by Slater determinants equation image constructed from 4-spinors equation image, which solve the relativistic one-electron KS equations

  • equation image(124)

For the resulting single Slater determinant, the electron density is given by

  • equation image(125)

an the relativistic current density reads

  • equation image(126)

The noninteracting kinetic energy equation image can then be defined as the kinetic energy equation image of such a noninteracting system with the 4-current equation image, which allows for a similar decomposition of the total energy functional as in nonrelativistic DFT, i.e.,

  • equation image(127)

Here, the exchange–correlation energy functional is defined as containing all the energy contributions not accounted for by the first three terms. In analogy to nonrelativistic KS-DFT, it can then be shown that the 4-current equation image of the true interacting system can be determined from the self-consistent solution of the relativistic KS equations for a noninteracting system,131, 135 with the 4-potential

  • equation image(128)

The functional derivatives of the exchange–correlation functional with respect to the components of the 4-current defines the exchange–correlation potential equation image, which now assumes a four-component form.

Electron density and spin density in relativistic DFT

In view of the previous discussions concerning nonrelativistic KS-DFT, we now face the following question: (i) Is it still possible to formulate a relativistic DFT in terms of the electron density only, (ii) how is the fundamental 4-current related to the density and spin density considered as fundamental quantities in nonrelativistic DFT, and (iii) how do the nonrelativistic restricted and unrestricted formulations of KS-DFT emerge from the relativistic framework?

Regarding the first question, under some additional assumptions it is indeed possible to prove a relativistic HK theorem for the electron density only135–137: In the case considered throughout this article, the external potential is the electrostatic potential of all atomic nuclei in a molecule that are at rest (Born–Oppenheimer approximation). As the nuclei are not moving, they do not create magnetic fields. In addition, we neglect any magnetic fields that stem from nuclear spins and assume that there are no additional external electromagnetic fields. As a consequence, the external electromagnetic 4-potential equation image only contains the time-independent scalar potential of the atomic nuclei, that is, equation image and equation image. This assumption is also common practice in almost every relativistic quantum chemical calculation. Then, it can be shown that the external scalar potential equation image within this specific reference frame is (up to a constant) uniquely determined by the electron density equation image only.135–137 Within such a framework, the relativistic total energy functional becomes

  • equation image(129)

Here, the current density equation image is still required, but it is now uniquely determined by the electron density (i.e., equation image), in the same way in which the spin density is determined by the electron density in nonrelativistic DFT. If a magnetic interaction between the electron is not included (i.e., the Dirac–Coulomb Hamiltonian is employed), also the analog of the Coulomb interaction reduces to a functional equation image of the density only.135 Moreover, it now becomes possible to set up theories in which not only the full 4-current is used as fundamental variable but also the parts of it that are related to the total electron density and the spin density. This will be discussed further in the following subsection.

To understand the relation of the 4-current to the spin density, it is important to realize that the definition of the current density (naturally) involves a velocity operator, which is in close analogy to classical mechanics (correspondence principle).57 The velocity operator in Dirac's theory of the electron follows from the Heisenberg equation of motion applied to the position operator and turns out to be equation image. Hence, as the Dirac α matrices contain the Pauli spin matrices σ, we see immediately that the current density j carries the spin information.130

The relation to the spin density can be made more explicit by invoking a Gordon composition of the current density which separates it into a charge- and a spin-related current.53, 131 For the one-electron case, one can carry out this decomposition by rewriting the Dirac eigenvalue equation as,

  • equation image(130)

and by splitting up the definition of the current in a somewhat artificial way as

  • equation image(131)

Equation (130) can now be used to replace equation image in the first term and equation image in the last term. As is shown in the Appendix, by exploiting the commutation relations of the Dirac matrices equation image and equation image, one arrives at53, 131 (see also pages 552–558 in Ref.138)

  • equation image(132)

where the 3-vector equation image contains the equation image Pauli matrices equation image introduced in Eq. (114). In a many-electron system described by the Dirac–Coulomb Hamiltonian, a similar decomposition of the current density can be performed (even though the derivation becomes slightly more complicated, see Appendix), and one obtains,

  • equation image(133)

The first line of this expression resembles the definition of the current density in nonrelativistic quantum mechanics, whereas the second term can be identified as arising from the electron spin. This can be made more apparent by defining the magnetization (density),

  • equation image(134)

Then, the contribution of the second term of Eq. (133) to the interaction energy with the external electromagnetic potentials [cf. Eq. (119)] becomes

  • equation image(135)

where equation image is the external magnetic field. The minus sign originates from the negative charge of the electron. This closely resembles the form of the spin Zeeman interaction in the nonrelativistic case [cf. Eq. (32)]. If an inhomogeneous magnetic field in z-direction is considered, Eq. (135) reduces to

  • equation image(136)

and by comparison with Eq. (33), we notice that equation image can be identified with the spin density Q(r) in nonrelativistic theory. However, while in the nonrelativistic case, the spin density is (in the absence of external magnetic fields) independent of the choice of the quantization axis, this is not the case in the relativistic theory, where equation image does not commute with the Hamiltonian because of the presence of spin–orbit interactions.

Relativistic spin-DFT

In the relativistic CDFT formalism discussed above, the noninteracting reference system is chosen such that it has the same 4-current equation image [i.e., the same electron density ρ(r) and the same current density j(r)] as the true interacting system. This results in a noninteracting kinetic-energy functional Ts[r, j] that can be defined as

  • equation image(137)

where the constrained search has to be restricted to positive-energy wavefunctions equation image to avoid a variational collaps. In such a formalism, the KS equations then contain a four-component exchange–correlation potential.

However, we also pointed out that in the case of molecular systems in the absence of external magnetic fields, a description relying on the electron density ρ(r) only as fundamental variable is sufficient. Therefore, it is formally also possible to develop a relativistic “density-only” KS-DFT that resembles the nonrelativistic restricted KS-DFT formalism. This can be achieved by only requiring from the noninteracting reference system that it has the same electron density as the interacting system, and consequently defining the noninteracting kinetic energy as,

  • equation image(138)

again restricting equation image to positive energy solutions. Then, the total energy functional can be decomposed as,

  • equation image(139)

where we have neglected the magnetic interactions in equation image. Note that such a decomposition implies a different definition of the exchange–correlation energy. The resulting KS equations then feature a one-component exchange–correlation potential equation image. However, in such a formalism the current density equation image and magnetization equation image of the noninteracting reference system do not agree with the true interacting system. Instead, these are again a functional of the density only, just as the spin density is in nonrelativistic restricted KS-DFT.

In between full CDFT and relativistic density-only KS-DFT, different intermediate formulations of relativistic KS-DFT are now also possible. For a relativistic system of noninteracting electrons, the magnetization is given by

  • equation image(140)

The noninteracting reference system can then be set up such that, in addition to the electron density, some parts of the magnetization density agree with those of the interacting system. For instance, we can require that the z-components of the magnetization equation image match, or we can demand that the lengths of the magnetization vector equation image at each point in space agree. The choice of what quantity is to be reproduced by a relativistic KS-DFT formalism in addition to the density is our freedom of choice. If we choose to reproduce the z-component of the magnetization density, this approach is called the collinear [“(cl)''] approach because an artificial external global quantization axis is introduced for the spin. If we require to reproduce the length of magnetization instead, this is a noncollinear [“(nc)''] approach because the magnetization is a vector field representing a magnetic dipole moment whose direction changes with position.135, 137, 139, 140, 141

Note that these different choices for the noninteracting reference system each implies a different definition of the noninteracting kinetic energy functional, equation image and equation image, and thus also of the exchange–correlation energy, equation image and equation image, respectively. For the choices mentioned here, where in addition to the electron density one additional quantity is reproduced by the KS system, the resulting exchange–correlation potential has two components. This is in close analogy to the case of nonrelativistic unrestricted KS-DFT. Therefore, approximate exchange–correlation functionals developed in the nonrelativistic domain are usually used in practical applications of such relativistic spin-DFT schemes. However, the exchange–correlation potential is defined differently in nonrelativistic unrestricted KS-DFT and in the relativistic collinear and noncollinear cases. Consequently, also different exact conditions apply to the exchange–correlation functional. The different possible choices for setting up relativistic KS-DFT are summarized in Figure 3.

thumbnail image

Figure 3. Relationship between wave function, total electron density, and the magnetization of the system of fully interacting electrons and of the KS reference systems of noninteracting electrons in different version of relativistic KS-DFT.

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Finally, we discuss how the nonrelativistic unrestricted KS-DFT formalism emerges from relativistic spin-DFT. This is most easily seen for the collinear approach, although the noncollinear one reduces to the same nonrelativistic limit as well. The z-component of the magnetization density of the KS reference system is given by

  • equation image(141)

If spin–orbit coupling is neglected, the Hamiltonian commutes with the spin operators and the KS spinors can each be expressed as (two-component) spin orbitals equation image or equation image, where equation image are 2-spinors consisting of an upper and a lower component. Then, the spin–orbit coupling free (SOfree) z-component of the magnetization resembles the nonrelativistic spin density,

  • equation image(142)

where the superscript “U“ and “L“ denote the upper and lower components, respectively. The neglect of spin–orbit coupling is, of course, an approximation that yields a scalar-relativistic Hamiltonian (which considers kinematic relativistic effects only) or even a nonrelativistic Hamiltonian if the speed of light is taken to be infinity.

The two 2-spinors equation image and equation image considered here can each be reduced to one-component spin-orbitals if a unitary transformation142 is performed to decouple the upper and lower components. This then also requires a unitary transformation of the operators involved in the calculation of equation image.143 Still, taking the limit equation image yields the nonrelativistic theory and relativistic spin-DFT reduces to nonrelativistic unrestricted KS-DFT.

Future Directions for Spin-DFT

  1. Top of page
  2. Abstract
  3. Introduction
  4. Spin in Nonrelativistic Quantum Chemistry
  5. Spin in Hohenberg–Kohn DFT
  6. Spin in Kohn–Sham DFT
  7. Spin in Relativistic DFT
  8. Future Directions for Spin-DFT
  9. Appendix: Gordon Decomposition of the Current Density
  10. Biographical Information
  11. Biographical Information

As we have seen, different options exist for setting up KS-DFT for open-shell systems. In the nonrelativistic case, one has to choose between a spin-restricted and a spin-unrestricted formulation of KS-DFT. In the former case, the wavefunction of the noninteracting reference system can always be chosen as an eigenfunction of equation image, but its spin density differs from the correct one. Alternatively, in spin-unrestricted KS-DFT the noninteracting reference system has the correct spin density, but cannot be an eigenfunction of equation image. In relativistic DFT even more options in between density-only KS-DFT (with a one-component exchange–correlation potential) and full CDFT (with a four-component exchange–correlation potential) are possible. In particular, one can choose to reproduce either a single component or the magnitude of the magnetization vector in collinear and noncollinear relativistic KS-DFT (which both use a two-component exchange–correlation potential), respectively.

Which choices we make may be determined by our ability to set up a proper approximation to the corresponding exchange–correlation energy functional. However, it is important to understand that these different choices imply different definitions of the noninteracting kinetic energy and the exchange–correlation functional. Thus, different exact conditions apply to these functionals, which must be considered when developing such approximations. This is most obvious for the fractional spin condition in the nonrelativistic case. While in spin-restricted KS-DFT, the constancy condition directly applies to the exchange–correlation functional, in the spin-unrestricted case it does not hold for the exchange–correlation functional alone, but to the sum of noninteracting kinetic and exchange–correlation energy. It can be expected that in the latter case, devising approximate exchange–correlation functionals that include this condition will be significantly more difficult.

Although exact DFT should always lead to the correct ground-state—irrespective of its spin state—in practice it appears more useful to rely on a theory that is able to target different spin states separately. This should also simplify the development of approximate functionals because it becomes possible to account for the different exact conditions applying to the exchange–correlation hole for different spin states.129, 144 Note that such a spin-state specific DFT always has to operate within a spin–orbit coupling free framework, because in the fully relativistic theory equation image does not commute with the Hamiltonian and different spin states do not correspond to the lowest state of a specific symmetry anymore.

Thus, besides finding approximations that accurately account for the spin-density dependence of the (nonrelativistic) exchange–correlation functionals equation image or equation image, finding ways of including the spin-state dependence into these functionals is another important problem for open-shell systems. In common approximations, it appears that the spin-density dependence is actually used to model this spin-state dependence (i.e., the integral of the spin density is used to distinguish spin states). Spin-state and spin-density dependence of the exchange–correlation functional are intermingled in all available approximate functionals, which manifests itself in their violation of the fractional spin condition.101, 102 To make progress in the development of reliable density-functional approximations, we believe it will be essential to consider the spin-state and the spin-density dependence of the exchange–correlation functional separately.

Appendix: Gordon Decomposition of the Current Density

  1. Top of page
  2. Abstract
  3. Introduction
  4. Spin in Nonrelativistic Quantum Chemistry
  5. Spin in Hohenberg–Kohn DFT
  6. Spin in Kohn–Sham DFT
  7. Spin in Relativistic DFT
  8. Future Directions for Spin-DFT
  9. Appendix: Gordon Decomposition of the Current Density
  10. Biographical Information
  11. Biographical Information

In the one-electron case, the current density is given by

  • equation image(A1)

For the k-component, we can rewrite this definition in a somewhat artificial way as

  • equation image(A2)

where we dropped the dependence of equation image on the spatial coordinate to simplify the notation. The eigenvalue equation of the one-electron Dirac Hamiltonian equation can be rewritten to obtain an expression for equation image,

  • equation image(A3)

and by taking the transpose and complex conjugate also an expression for equation image (exploiting equation image)

  • equation image(A4)

These expressions can now be used to replace equation image in the first term and equation image in the second term of Eq. (A2) to obtain,

  • equation image(A5)

and by reordering the different terms we get

  • equation image(A6)

The last term is zero because of equation image. For the first term, we use

  • equation image(A7)

to arrive at (using the convention of implicit summation over repeated indices)

  • equation image(A8)

After regrouping the different terms, we obtain

  • equation image(A9)

where the last term is zero because equation image. Hence, we find

  • equation image(A10)

and combining the different components of j again gives

  • equation image(A11)

For an N-electron system, we start from the definition of the many-electron current density,130

  • equation image(A12)

and the wavefunction is an eigenfunction of the many-electron Dirac–Coulomb Hamiltonian

  • equation image(A13)

projected onto the electronic (positive-energy) states.

We can now rewrite the corresponding eigenvalue equation in a similar fashion as in the one-electron case to obtain for the wavefunction

  • equation image(A14)

and for its transpose and complex conjugate

  • equation image(A15)

Now, we can rewrite the k-component of integrand in Eq. (A12) as

  • equation image(A16)

After substituting Eq. (A14) for equation image in the first term and Eq. (A15) for equation image in the second term, we obtain, in addition to the term already present in the one-electron case [cf. Eq. (A11)],

  • equation image

With a multiplicative operator acting equally on all components of the wavefunction for equation image, the last term is zero because of equation image. However, if the Gaunt or Breit interaction is included, the corresponding term is not zero and gives an additional contribution to the current that is not present in the one-electron case. This term contains the spin–spin interactions.

For the remaining terms, we now consider one of the terms separately, for instance equation image, and note that

  • equation image(A17)

because of equation image. Integrating the expression in square brackets over equation image we now find

  • equation image(A18)

where for the second term, we exploited that equation image is hermitian and commutes with equation image and equation image because these act on different electrons. Combining all of these results, we find that (with the Dirac–Coulomb Hamiltonian),

  • equation image(A19)

Biographical Information

  1. Top of page
  2. Abstract
  3. Introduction
  4. Spin in Nonrelativistic Quantum Chemistry
  5. Spin in Hohenberg–Kohn DFT
  6. Spin in Kohn–Sham DFT
  7. Spin in Relativistic DFT
  8. Future Directions for Spin-DFT
  9. Appendix: Gordon Decomposition of the Current Density
  10. Biographical Information
  11. Biographical Information
Thumbnail image of

Christoph R. Jacob heads an independent “Young Scientist Group” in theoretical chemistry at the Karlsruher Institute of Technology (KIT). He studied chemistry and mathematics at Philipps University Marburg and at the University of Karlsruhe. Following a 1-year stay in the group of Peter Schwerdtfeger at Auckland University in 2004, he joined the group of Lucas Visscher und Evert-Jan Baerends at VU University Amsterdam, from where he obtained his PhD degree in 2007. He then moved on to the group of Markus Reiher at ETH Zurich, before taking up his current position at KIT in 2010. His main research interests are the development of quantum-chemical method for complex systems, which includes the development of subsystem and embedding methods as well as fundamental work in density-functional theory, and theoretical spectroscopy of large chemical systems, ranging from vibrational spectroscopy of biomolecules to optical and X-ray spectroscopy of nanostructures. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

Biographical Information

  1. Top of page
  2. Abstract
  3. Introduction
  4. Spin in Nonrelativistic Quantum Chemistry
  5. Spin in Hohenberg–Kohn DFT
  6. Spin in Kohn–Sham DFT
  7. Spin in Relativistic DFT
  8. Future Directions for Spin-DFT
  9. Appendix: Gordon Decomposition of the Current Density
  10. Biographical Information
  11. Biographical Information
Thumbnail image of

Markus Reiher is professor for theoretical chemistry at ETH Zurich since 2006. After studying chemistry, he received his PhD in theoretical chemistry from the University of Bielefeld working with Jürgen Hinze in 1998. In 2002, he finished his habilitation thesis in the group of Bernd Artur Hess at the University of Erlangen and continued as a private docent first in Erlangen and then at the University of Bonn. In 2005, he accepted an offer for a professorship in physical chemistry from the University of Jena, where he worked until he moved to ETH Zurich. His research covers many different areas in theoretical chemistry and ranges from relativistic quantum chemistry, (vibrational) spectroscopy, density functional theory, transition metal catalysis and bioinorganic chemistry to the development of new electron-correlation theories and smart algorithms for inverse quantum chemistry. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]