Multiconfigurational Pair-Density Functional Theory Is More Complex than You May Think

Multiconfigurational pair-density functional theory (MC-PDFT) is a promising way to describe both strong and dynamic correlations in an inexpensive way. The functionals in MC-PDFT are often “translated” from standard spin density functionals. However, these translated functionals can in principle lead to “translated spin densities” with a nonzero imaginary component. Current developments so far neglect this imaginary part by simply setting it to zero. In this work, we show how this imaginary component is actually needed to reproduce the correct physical behavior in a range of cases, especially low-spin open shells. We showcase the resulting formalism on both local density approximation and generalized gradient approximation functionals and illustrate the numerical behavior by benchmarking a number of singlet–triplet splittings (ST gaps) of organic diradicals and low-lying excited states of some common organic molecules. The results demonstrate that this scheme improves existing translated functionals and gives more accurate results, even with minimal active spaces.

1 Complex arithmetic for translated functionals

General rules and Slater exchange
The trick to translate a functional to a complex formalism is to express the translated α and β densities in exponential form.As a reminder, any complex number z = a + ib can be expressed in its polar form as z = re iθ with: and In this form, it becomes easy to take power, multiplications and divisions.Applying the definition used in Eq. 5 from the main text we can proceed with the translation of densities for cases where ∆ < 0 and the "translated densities" necessarily have an imaginary component: where we again assign the "+" and "-" signs to our α and β densities, respectively.
Let us illustrate this for the Slater exchange functional.The Slater functional exchange energy can be defined as: which becomes by transforming ρ a and ρ b in terms of ρ and Π accordingly to Eq. 2 and using our definition of ∆ from Eq. 5.
Starting from Eq. S3, the power 4 3 affects r and becomes a multiplicative factor in the exponent.Then, since the α and β densities differ by the sign of the exponential, summing them gives a cosine: which is equivalent to the formula presented in Ref. 1.
Commonly, instead of spin densities, the functionals are expressed in terms of the spin polarization ζ and the spin-polarization dependent term F (ζ), which is often written as: For the complex case this can be translated as: where η = √ −∆ ρ and ζ = ±iη.

VWN3 and PBE correlation functionals
In the VWN3 2 (as well many others) correlation functional it is necessary to calculate the spin polarization factor ζ, which is dependent on ρ α and ρ β , and, therefore, needs translation.
However, the ζ dependence is of the form Using Eq.S8, this F (ζ) term is simply translated as The correlation functional in PBE 3 contains one such term as well as another of a similar form: which similarly translates to

Full Translation of PBE exchange functional
The exchange of PBE is a bit more complicated.First, following Li Manni et al, we translate the gradient of the spin-density χ α as which corresponds to neglecting the explicit dependence on the gradient of ζ.The gradient is used in a dimensionless constant s 2 α which is defined (except for a constant) as: and for complex cases we have with θ = − 2 3 arctan η and the complex conjugate for β 2 .Being in the a + ib form allows us to do additions and substractions easily, which we need to compute 1 + µs 2 κ in the PBE exchange expression, and then we can for example convert back to exponential form to take the division in κ 1+µs 2 /κ .This type of back-and-forth allows to extend any functional to cases where ∆ < 0 2 Singlet-triplet gaps using DFT orbitals and CASCI Instead of using MCSCF orbitals and densities, we can investigate the importance of orbital relaxation in MC-PDFT by using the orbitals from a Kohn-Sham DFT triplet calculation.
More specifically, in this section, we started with a triplet restricted open-shell DFT calculations as these orbitals should also be the best orbitals for the triplet MC-PDFT with the corresponding functional.We used the same orbitals for the singlet, as for a true openshell singlet, the triplet orbitals should be also good approximations of the optimal singlet state orbitals.We then obtained the one and two-particle density matrices from a CASCI calculation for both singlet and triplet cases.The results are shown in table S1.
Table S1: Singlet-triplet splitting errors in kcal mol -1 against doubly electron-attached coupled-cluster reference 4 or experimental reference 5 (O 2 only), for minimal and π active spaces when using KS orbitals in a CASCI calculation.