Adding Temperature to the Adiabatic Connection: Hybrid Functionals for Static and Dynamic Correlation

The gist The framework formally unifies information-density matrix functional theory (i-DMFT) and thermally-assisted-occupation (TAO-)DFT as special cases, clarifying their theoretical basis and the respective prescriptions for the temperature parameter>

How it works

The paper extends the adiabatic connection mechanism from density functional theory to include a temperature dimension, giving rise to thermally enhanced hybrid functionals in which static correlation is captured via fractional occupations generated by an effective “correlation temperature” within thermal Hartree-Fock, while dynamic correlation is described by established density functional approximations>

1. The framework derives DFT as a thermodynamic theory by minimizing the trial grand potential and then performing a Legendre transform to define a convex density functional as the free energy Fλ,τ (ρ) = sup v ρdr>

2. This leads to the definition of exchange-correlation energy Eθxc = F1,0 − F0,θ − EH via an integral along a path γ in (λ, τ) parameter space between a thermal KS system and the fully interacting system>

3. The resulting approximation for the exchange-correlation functional is given by Ehyb+θxc ≈ EHF+θx + (1 − a)EDFAx + E DFAc, where 'a' ranges between 0.20 [49] and 0.25 [50, 51]>

4. This approximation neatly differentiates between static and dynamic correlation, ceding the dynamic part to a DFT functional while the first term handles static correlations with an adequate choice of effective temperature θ>

Analysis of Numerical Results

The paper demonstrates that this new hybrid method can reproduce electron density, dissociation curves, and natural occupation numbers much more faithfully than the previously proposed i-DMFT method based only on the Hartree-Fock mean-field interaction>

correlation-corrected thermally enhanced calculations almost exactly reproduce the FCI results at 1.4 and 2.41 Bohr, and reproduce the cusp at the nucleus significantly better at 8 Bohr

The inclusion of a mean-field correlation functional results in smoothly changing natural occupation numbers over the entire dissociation, even though they still deviate from the CASSCF results

The reproduction of the correct barrierless dissociation surface is obviously a sign for the qualitative superiority of the DFT+θ family of methods

Approaches for Obtaining Correlation Temperatures

A crucial difficulty remains in choosing the electronic temperature θ that enters as a product −θS in the free energy F0,θ>

The methodologies that emerged for the determination of θ so far are relatively diverse in cost and complexity, but most importantly in their proposed universality

A global θ is also not to be expected based on the results of studies of the correlation conjecture and i-DMFT

In this work, we compare some of these and newly devised approaches for the determination of θ along the dissociation of dinitrogen (14 electrons) in the 6-31G basis (18 orbitals)

The paper investigates three main recipes for determining θ:

1. Fitting the dissociation energy to a reference calculation, where this can be a low-level DFT calculation used to interpolate the dissociation curve>

2. Fitting natural occupation numbers to those of a high-level method, motivated by the fact that fractional occupations are meant to describe strong/static correlation>

3. Finding θ such that the ground-state energy of the FCI reference is recovered at every step during a deformation of the molecule>

Benchmarking Larger Systems

A benchmark was conducted on nine different molecules, including single bonds between two sp2 hybridized carbon atoms and seven differently substituted biphenyls, to test transferability>

"The results of these calculations can be taken from Tab. V. Analogous to the approaches discussed in Sections IV and VI, we obtained the dissociation energy simply as the energy difference between starting point and end point in the FCI reference"

"Thermally enhanced Hartree-Fock and B3LYP calculations can reproduce the dissociation energies obtained on the CASSCF level of theory, but a different value of θ has to be applied for every molecule in order to reproduce the correct trend of dissociation energies"

The average values for θ are 0.0865 EH with HF and 0.0309 EH with B3LYP, and the standard deviations are 0.0023 EH (2.6%) and 0.019 EH (6.2%), respectively

**"In general, such a connection between bond strength and θ is apparent as illustrated in Fig.

1. textbfDefinition of Thermally Enhanced Hybrid Functionals (Eq. 19): The improved system can capture both static and dynamic correlation simultaneously at mean-field cost by mixing thermal HF with a DFA functional, as described by the expression E hyb+θ xc ≈ aEHF+θ x + (1 − a)E DFA x + E DFA c.

2. textbfSystematic Resolution of i-DMFT Failures: The improved system can systematically resolve known failures of i-DMFT in dissociation barriers and densities by testing different schemes for determining the correlation temperature and investigating their accuracy for a benchmark set derived from biphenyl, as detailed in Section IV.

3. textbfThermodynamic Foundation for DFT (Section II): The improved system is grounded in a thermodynamic theory of DFT, allowing it to be formulated via the trial grand potential (Eq. 2) and subsequently mapped onto a convex density functional (Eq. 4), providing a rigorous basis for thermally enhanced approximations where F λ,τ (ρ) = sup v [omegaλ,τ (v) − Z v(r)ρ(r)dr.

4. textbfAdaptive Correlation Temperature Determination: The improved system can determine the correlation temperature through various schemes, including fitting to reference occupation numbers or reference energies, as shown in Fig. 7 of Section V, enabling the selection of an optimal global θ or geometry-specific values for different molecules.

5. textbfAccurate Dissociation Curve Prediction: The improved system can accurately predict dissociation curves by employing two main recipes: the fitting of the dissociation energy to a reference calculation (interpolation) or the fitting of natural occupation numbers to a high-level method, which is shown to reproduce FCI results for H2 and N2.

6. textbfCharacterization of Correlation Types: The improved system can neatly differentiate between static and dynamic correlation because the approximation neatly differentiates between static and dynamic correlation, as modern DFAs handle dynamic effects while the first term in Eq. (19), aEHF+θ x, takes over the responsibility for static correlations.

7. textbfBenchmarking Against Complex Systems: The improved system can be validated on larger systems like biphenyl by demonstrating that Thermally enhanced Hartree-Fock and B3LYP calculations can reproduce the dissociation energies obtained on the CASSCF level of theory, although it requires a different value of θ has to be applied for every molecule in order to reproduce the correct trend of dissociation energies.