Keywords

1 Introduction

Kohn–Sham density-functional theory (KS-DFT) is based on a non-interacting, single-Slater-determinant reference state having the same total density ρ as the “real” interacting system [1–3]. The KS reference orbitals ψ i satisfy the single-particle Schroedinger equation

$$ -\frac{1}{2}{\nabla}^2{\psi}_{\mathrm{i}}+{v}_{\mathrm{KS}}{\psi}_{\mathrm{i}}={e}_{\mathrm{i}}{\psi}_{\mathrm{i}} $$
(1)

where v KS is the non-interacting potential such that

$$ \rho ={\displaystyle \sum_i{n}_{\mathrm{i}}}{\psi}_{\mathrm{i}}^2,\kern0.55em {n}_{\mathrm{i}}=0,\ 1,\ \mathrm{or}\ 2 $$
(2)

equals the density of the real system. The KS potential is given by [2]

$$ {v}_{\mathrm{KS}}={v}_{\mathrm{ext}}+{v}_{\mathrm{el}}+{v}_{\mathrm{XC}},\kern0.5em {v}_{\mathrm{el}}={\displaystyle \int \frac{\rho \left({r}_2\right)}{r_{12}}{d}^3{r}_2},\kern0.5em {v}_{\mathrm{XC}}=\frac{\delta {E}_{\mathrm{XC}}}{\delta \rho } $$
(3)

where v ext is the external (one-body) potential in the real system, v el is the classical Coulomb potential arising from the electron density, and v XC is the functional derivative [3] with respect to the density of the “exchange-correlation” energy E XC defined by

$$ {E}_{\mathrm{total}}=-\frac{1}{2}{\displaystyle \sum_i{n}_{\mathrm{i}}{\displaystyle \int {\psi}_{\mathrm{i}}}{\nabla}^2{\psi}_{\mathrm{i}}}+{\displaystyle \int \rho {v}_{\mathrm{ext}}}+\frac{1}{2}{\displaystyle \iint \frac{\rho \left({r}_1\right)\rho \left({r}_2\right)}{r_{12}}}+{E}_{\mathrm{XC}} $$
(4)

i.e., that part of the total energy containing all the quantum and correlation effects.

We restrict ourselves in this work to singlet states. Thus the occupation numbers n i in (2) and (4) are 0 or 2 if the reference state is a single Slater determinant. Not all quantum systems, however, can be referred to a single determinant or “configuration”. Consider a four-electron system in a perfectly square D4h nuclear framework: square H4, for example, or the pi-electrons in square cyclobutadiene C4H4. The orbital energies are as sketched in Fig. 1, with a doubly degenerate HOMO set, ψ a and ψ b. Two equivalent singlet configurations (determinants) are possible: |ψ 2a 〉 and |ψ 2b 〉 in the figure. These interact strongly to produce the two-configuration mixtures Ψ±:

$$ {\Psi}_{\pm }=\frac{1}{\sqrt{2}}\left(\left|{\psi}_{\mathrm{a}}^2\right\rangle \pm \left|{\psi}_{\mathrm{b}}^2\right\rangle \right) $$
(5)

with Ψ− being the ground state. Neither of the individual determinants, |ψ 2a 〉 or |ψ 2b 〉, is an adequate reference state. Indeed, neither has a density having even the correct D4h symmetry of the actual ground-state density (if we constrain ourselves to real orbitals). The actual ground-state ρ is represented by

$$ \rho =2{\displaystyle \sum_{i=1}^{\mathrm{HOMO}-1}{\psi}_{\mathrm{i}}^2}+{\psi}_{\mathrm{a}}^2+{\psi}_{\mathrm{b}}^2 $$
(6)

involving occupancies other than 2 for the degenerate HOMO orbitals. In each of ψ a and ψ b, there is half an electron of spin up and half an electron of spin down.

Fig. 1
figure 1

D4h Slater determinants

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The orbital occupancies in the above example are fixed by symmetry: i.e., a doubly degenerate HOMO level yielding two symmetry-equivalent reference configurations. In general, strongly correlated systems have densities of the form

$$ \rho =2{\displaystyle \sum_{i=1}^{\mathrm{HOMO}-1}{\psi}_{\mathrm{i}}^2}+2\left(1-f\right){\psi}_{\mathrm{HOMO}}^2+2f{\psi}_{\mathrm{LUMO}}^2 $$
(7)

where f may have any value in the interval 0 ≤ f ≤ 1, depending on the HOMO-LUMO gap and the relevant interaction matrix elements. The f parameter in (7), and the corresponding fractional occupancies in (4), must be determined variationally.

An analysis by Schipper et al. [4] of the H2 + H2 reaction is a beautiful illustration. These authors have computed accurate CI and MRCI wavefunctions at various points on the H2 + H2 potential-energy surface and, at each geometry, have extracted the exact Kohn–Sham potential v KS and orbitals ψ i using a robust Kohn–Sham inversion procedure [5]. For geometries close to perfect squares (D4h), their inversion procedure fails to converge; i.e., occupation “holes” appear under the HOMO if single-determinant occupancies are enforced. Stable KS solutions are obtained only if fractional occupancies are allowed in (7). Geometries far from D4h admit conventional integer-occupancy KS solutions. Standard exchange-correlation GGAs (Generalized Gradient Approximations) also produce fractional occupancies near D4h geometries, but with rather poor accuracy compared to the exact Kohn–Sham results [4].

In this chapter, a recently developed [6, 7] strong-correlation density functional, “B13”, is benchmarked on the Schipper et al. [4] H2 + H2 data, and also on ethylene torsion and cyclobutadiene automerization data from Jiang et al. [8]. All three of these are challenging avoided-crossing problems involving fractional occupancies near their transition states. The B13 functional is reviewed in Sect. 2. Variational optimization of B13 fractional occupancies, and reaction profiles for our three avoided-crossing tests, are discussed in Sect. 3. Dissociation curves for the single bonds H2 and LiH are presented in Sect. 4. Concluding remarks and future prospects are discussed in Sect. 5.

2 The B13 Strong-Correlation Density Functional

In recent papers [6, 7] we have introduced a correlation-energy density functional able to describe both moderately and strongly correlated systems. “Strong” correlation arises from a small HOMO–LUMO gap, resulting in strong mixing of the configurations

$$ \left|\cdots {\psi}_{\mathrm{HOMO}}^2{\psi}_{\mathrm{LUMO}}^0\right\rangle \kern0.5em \mathrm{and}\kern0.5em \left|\cdots {\psi}_{\mathrm{HOMO}}^0{\psi}_{\mathrm{LUMO}}^2\right\rangle $$
(8)

as described above. The signature of strong correlation in KS-DFT is fractional occupancies in the minimum-energy KS density (see (7)). At the same time, Janak’s theorem [3, 9] implies that

$$ {e}_{\mathrm{HOMO}}={e}_{\mathrm{LUMO}} $$
(9)

in any such strongly-correlated, fractionally-occupied, Kohn–Sham minimum.

Strong correlation is often discussed in the context of molecular bond dissociation; i.e., the correlation energy required to dissociate molecular bonds using spin-restricted orbitals. Small HOMO–LUMO gaps and strong mixing of configurations play a major role here as well. Spin-restricted dissociation of molecules implies that, for all free atoms, the usual Hund’s-rule spin-polarized configuration has the same energy as the spin-depolarized configuration obtained by replacing every unpaired electron by half an electron of spin up and half an electron of spin down (as sketched for the carbon atom in Fig. 2). This is a stringent test of density functionals. Conventional GGAs fail this test [7] with errors of the order of 40 kcal/mol in the first three rows of the periodic table. Our B13 functional, however, is designed to capture this spin-invariance property [7] while maintaining good performance in standard thermochemical applications.

Fig. 2
figure 2

Carbon atom spin states

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B13 is based on exact exchange and is hence a pure correlation theory. The total exchange-correlation energy has the form [6, 7]

$$ {E}_{\mathrm{X}\mathrm{C}}^{\mathrm{B}13}={E}_{\mathrm{X}}^{\mathrm{exact}}+{E}_{\mathrm{C}}^{\mathrm{B}13}+\Delta {E}_{\mathrm{strong}\;\mathrm{C}}^{\mathrm{B}13} $$
(10)

where E exactX is the exactly-computed Kohn–Sham (or Hartree–Fock depending on the implementation) exchange energy, E B13C is a sum of opposite- and parallel-spin static and dynamical correlation potential energies:

$$ {E}_{\mathrm{C}}^{\mathrm{B}13}={a}_{\mathrm{stat}\;\mathrm{C}}^{\mathrm{opp}}{U}_{\mathrm{stat}\;\mathrm{C}}^{\mathrm{opp}}+{a}_{\mathrm{stat}\;\mathrm{C}}^{\mathrm{par}}{U}_{\mathrm{stat}\;\mathrm{C}}^{\mathrm{par}}+{a}_{\mathrm{dyn}\;\mathrm{C}}^{\mathrm{opp}}{U}_{\mathrm{dyn}\;\mathrm{C}}^{\mathrm{opp}}+{a}_{\mathrm{dyn}\;\mathrm{C}}^{\mathrm{par}}{U}_{\mathrm{dyn}\;\mathrm{C}}^{\mathrm{par}} $$
(11)

and ΔE B13strong C is a strong-correlation correction given by

$$ \Delta {E}_{\mathrm{strong}\;\mathrm{C}}^{\mathrm{B}13}={\displaystyle \sum_{n=2}^N{c}_n{\displaystyle \int {x}^n{u}_{\mathrm{C}}{d}^3r}}. $$
(12)

To an excellent first approximation, the four prefactors in (11) are all equal to each other with optimum value 0.62 (see Becke [6]). Nevertheless, greater accuracy can be achieved by fitting these independently. In (12), u C is the sum of the integrands of the four terms in (11):

$$ {u}_{\mathrm{C}}={u}_{\mathrm{stat}\;\mathrm{C}}^{\mathrm{opp}}+{u}_{\mathrm{stat}\;\mathrm{C}}^{\mathrm{par}}+{u}_{\mathrm{dyn}\;\mathrm{C}}^{\mathrm{opp}}+{u}_{\mathrm{dyn}\;\mathrm{C}}^{\mathrm{par}}, $$
(13)

namely the static + dynamical correlation potential-energy density, and x is the following ratio of the static correlation potential-energy density to the total:

$$ x=\frac{u_{\mathrm{stat}\;\mathrm{C}}^{\mathrm{opp}}+{u}_{\mathrm{stat}\;\mathrm{C}}^{\mathrm{par}}}{u_{\mathrm{C}}}. $$
(14)

This is a reasonable, local measure of strong correlation. In atoms, where strong correlation is insignificant, x → 0. In stretched, spin-restricted H2, on the other hand, the quintessential case of strong correlation, x → 1. We assume that intermediate situations are well characterized by intermediate values of x. The c n in (12) are polynomial expansion coefficients fit to the atomic spin-depolarization condition of the previous paragraph.

Our fit set in Becke [7] consisted of the “G2/97” atomization energies of Curtiss, Raghavachari, and Pople [10] plus the spin-depolarization condition on all open-shell atoms Z < 36, including transition metal atoms. Two terms were deemed optimum in the strong-correlation correction, (12). The best-fit coefficients are

$$ \begin{array}{ll}{c}_{\mathrm{stat}\;\mathrm{C}}^{\mathrm{opp}}=0.552\hfill & {c}_{\mathrm{stat}\;\mathrm{C}}^{\mathrm{par}}=0.844\hfill \\ {}{c}_{\mathrm{dyn}\;\mathrm{C}}^{\mathrm{opp}}=0.640\hfill & {c}_{\mathrm{dyn}\;\mathrm{C}}^{\mathrm{par}}=0.559\hfill \\ {}{c}_2=0.825\hfill & {c}_3=-0.380\hfill \end{array} $$
(15)

and these are used in the present work. Mean absolute B13 errors on the G2/97 atomization energies and the atomic spin-depolarization tests are 3.8 and 5.7 kcal/mol, respectively [7].

It should be noted that the B13 correlation model is based on a single Slater determinant as the reference state [6]. The model adds multi-center correlations to single-determinantal pair densities as the starting point. How then, can B13 be justified in fundamentally two-configuration problems such as avoided crossings? Consider again the 50/50 mixtures at symmetric avoided-crossing transition states [11]. Imagine replicating the system of interest by an identical system at infinite distance from the first. The degenerate orbitals ψ a and ψ b on the system (A) and its replicant (B) combine to form the degenerate “super” system orbitals

$$ {\psi}_{\mathrm{a}\pm}^{\mathrm{super}}=\frac{1}{\sqrt{2}}\left({\psi}_{\mathrm{a}}^{\mathrm{A}}\pm {\psi}_{\mathrm{a}}^{\mathrm{B}}\right)\kern0.5em \mathrm{and}\kern0.5em {\psi}_{\mathrm{b}\pm}^{\mathrm{super}}=\frac{1}{\sqrt{2}}\left({\psi}_{\mathrm{b}}^{\mathrm{A}}\pm {\psi}_{\mathrm{b}}^{\mathrm{B}}\right). $$
(16)

Now consider the single supersystem Slater determinant with occupancy

$$ \left|\cdots {\psi}_{\mathrm{a}+}^{\mathrm{super}}\left(\uparrow \downarrow \right){\psi}_{\mathrm{b}+}^{\mathrm{super}}\left(\uparrow \downarrow \right)\right\rangle . $$
(17)

In each subsystem, this determinant places one electron in ψ a (half spin-up and half spin-down) and similarly one electron in ψ b, precisely the density of (6) corresponding to a 50/50 configuration mixture.

The more general densities of (7) can be reproduced by single Slater determinants spanning supersystems of more than two replicas. Appropriate numbers of replicants and appropriate occupations of supersystem orbitals can reproduce any two-configuration density. Because B13 maximizes multi-center correlation, it delivers the ground state energy corresponding to any such two-configuration density, not an excited-state energy or an ensemble energy. Excited states supported by the HOMO–LUMO orbitals will be studied in future work.

3 Avoided-Crossing Reaction Curves

In Becke [11], B13 was tested on barrier heights of four symmetric avoided-crossing reactions: H4, ethylene double-bond twisting, and double-bond automerization in cyclobutadiene and cyclooctatetraene. At the transition states of these reactions, there is 50/50 HOMO–LUMO mixing by symmetry. Variational determination of fractional occupancies was not needed. In this chapter, however, we draw full reaction curves for the first three of these reactions. Thus fractional occupancies must be computed.

Self-consistent B13 calculations are not yet possible, though prospects for SCF-B13 are good; see Proynov et al. [12–14] for an SCF implementation, and Arbuznikov and Kaupp [15] for an OEP implementation of an earlier B13 variant known as B05 [16]. Our current calculations are “post-LDA.” All total energies are evaluated using LDA orbitals computed by the grid-based NUMOL program [17, 18]. The optimum post-LDA fractional occupancies in (7) are determined by searching in the range 0 ≤ f ≤ 1 for the minimum-energy f value. We adopt the simple strategy of minimizing the energy of a three-point quadratic interpolation with calculations performed at f = 0, 1/2, and 1. For comparison purposes, the same post-LDA three-point approach is applied to the B88 [19] + PBE [20] exchange-correlation GGA.

In our first avoided-crossing reaction, H2 + H2, we consider the nine H4 geometries in Table II of Schipper et al. [4]. Each geometry is a rectangle with the longer side denoted R and the shorter side denoted r. Table 1 lists these geometries, along with the fractional occupancies f for “exact” Kohn–Sham, for the B88 + PBE exchange-correlation GGA, and for B13. We find qualitative agreement between our B88 + PBE GGA results and the B88 + LYP [21] GGA results in Schipper et al. [4] (also recorded in Table1), but both GGA sets agree poorly with exact Kohn–Sham. There is qualitative agreement, however, between our B13 fractional occupancies and the exact KS values, although B13 appears to be somewhat too strongly correlated. Note that the region of fractional occupancies near the square transition state is smaller for the GGA than for exact KS and B13. The exact KS, GGA, and B13 energy curves are plotted in Fig. 3. The energy zero for each curve is the energy at geometry R = 10.0 and r = 1.40 bohr. The B13 reaction barrier of 141.2 kcal/mol is in fair agreement with the MRCI barrier of 147.6 kcal/mol.

Table 1 Fractional occupancies as in (7) for the H2 + H2 reaction
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Fig. 3
figure 3

Reaction profile for H2 + H2

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Our second avoided-crossing test is twisting around the double bond in ethylene. We took MR-ccCA (Multi-Reference Correlation Consistent Composite Approach) reference data from Jiang, Jeffrey, and Wilson [8] and have performed all calculations at the geometries specified in their paper. Figure 4 plots reference, B88 + PBE GGA, and B13 reaction curves. Zero energy for each method is the planar ethylene equilibrium geometry. In this case, the GGA energies always minimize at f = 0 (no HOMO–LUMO mixing) producing an unphysical cusp at the top of the GGA curve. The B13 curve, on the other hand, is smooth throughout. The B13 barrier of 77.1 kcal/mol agrees better with the MR-ccCA barrier of 68.3 kcal/mol than does the GGA barrier, 90.2 kcal/mol.

Fig. 4
figure 4

Reaction profile for ethylene twist

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Double-bond automerization in cyclobutadiene is an especially challenging avoided-crossing test. The reaction profile is plotted in Fig. 5, with MR-ccCA reference data and geometries again from Jiang, Jeffrey, and Wilson [8]. Zeroes of all curves correspond to the rectangular equilibrium geometry. The GGA curve is cuspless in this case but has a barrier more than twice too large (21.8 kcal/mol compared to 9.2 kcal/mol for MR-ccCA). The B13 barrier of 10.4 kcal/mol compares quite well with the 9.2 kcal/mol MR-ccCA barrier.

Fig. 5
figure 5

Reaction profile for cyclobutadiene automerization

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4 H2 and LiH Dissociation Curves

Spin-restricted KS-DFT dissociation curves, even employing sophisticated GGA and hybrid functionals, have asymptotes well above the exact dissociation limits. Spin-unrestricted calculations (“UKS”) can capture the correct limits. UKS computations are undesirable, however, as they break spin and space symmetries. To the best of our knowledge, no successful spin-restricted DFT dissociation curve has yet been reported in the literature. We therefore apply B13 to this fundamental problem.

In Fig. 6 the H2 dissociation curve is plotted for highly accurate Hylleraas variational reference data from Sims and Hagstrom [22] for the B88 + PBE GGA and for B13. Fractional mixing of the σ g and the σ u orbitals begins at R ~ 3.4 bohr for B13. The GGA does not mix these orbitals at all, and the GGA curve is significantly above the exact asymptote. B13, on the other hand, exhibits a good dissociation limit and, quite interestingly, an asymptotic f of 0.50 (computed at R = 30.0 bohr), in accord with the asymptotic CI wavefunction

Fig. 6
figure 6

H2 dissociation

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$$ \frac{1}{\sqrt{2}}\left(\left|{\sigma}_{\mathrm{g}}^2\right\rangle -\left|{\sigma}_{\mathrm{u}}^2\right\rangle \right). $$
(18)

That B13 is sensitive enough to mix these MOs, and with the correct mixing fraction no less, is intriguing.

The heteronuclear dissociation of LiH is even more interesting. At large separation, the HOMO of the system is the 1s orbital of the H atom, and the LUMO is the 2s orbital of the Li atom. Thus the two configurations

$$ \left|1{\mathrm{s}}_{\mathrm{Li}}^2\cdot {\mathrm{HOMO}}^2\right\rangle \kern0.5em \mathrm{and}\kern0.5em \left|1{\mathrm{s}}_{\mathrm{Li}}^2\cdot LUM{O}^2\right\rangle $$
(19)

have charges Li+H− and Li−H+ and dissociation to neutral atoms requires f = 0.50. Because LDA calculations on Li− and H− are problematic, the f = 0 and f = 1 endpoints of the quadratic minimization are unreliable at large separations. Instead, we scan over f values in steps of 0.05 in order to locate the energy minimum at each internuclear separation.

We plot LiH dissociation curves for MR-CISD (Multi-Reference Configuration Interaction with Singles and Doubles) reference data [23], the B88 + PBE GGA, and B13 in Fig. 7. All curves are zeroed at their minimum energy. The limit of the GGA dissociation curve is too high. The B13 curve has an excellent asymptotic energy, and an asymptotic f (computed at R = 80.0 bohr) of 0.45, very close to the required 0.50. The dissociation limit is roughly Li+0.1H−0.1, very close to neutral atoms. The manner in which B13 captures, to a very good approximation, the correct dissociation limit of this heteronuclear bond is fascinating. At play is a “resonance” of singlet ionic atomic states.

Fig. 7
figure 7

LiH dissociation

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5 Summary and Conclusions

This work marks the first successful application of a spin-restricted Kohn–Sham density functional, “B13”, to two-configuration mixing problems with densities as in (7). Moreover, B13 is exact-exchange-based. Therefore stretched radical systems such as H2 + and He2 +, the bane of GGA and hybrid functionals [24], are correctly accommodated.

These calculations have been post-LDA and not self-consistent. It is difficult to predict how self-consistency might change their nature, and what significance would be lent to the orbital energies. Is the HOMO–LUMO mixing in H2 and LiH an artifact of the post-LDA approach here? Might mixing be obviated in an SCF approach? SCF implementations of the predecessor “B05” functional have been published [12–15]. SCF-B13 should be possible too and could be very interesting.

In future work we will also investigate multiple bond dissociations and other strongly-correlated reactions requiring variational optimization of one or more mixing fractions. Low-lying excited states of strongly-correlated systems, and how they might depend on B13 ground-state determinations, will be explored as well.

The author gratefully acknowledges the Natural Sciences and Engineering Research Council (NSERC) of Canada for research support, the Killam Trust of Dalhousie University for salary support through the Killam Chair program, and the Atlantic Computational Excellence network (ACEnet) for computing support.