Abstract
Hybrid density functionals, such as B3LYP and PBE0, have achieved remarkable success by substantially improving over their parent methods, namely Hartree-Fock and the generalized gradient approximation, and generally outperforming the second-order Møller-Plesset perturbation theory (MP2) that is more expensive. Here, we extend the linear scheme of hybrid multiconfiguration pair-density functional theory (HMC-PDFT) by incorporating a cross-entropy ingredient to balance the description of static and dynamic correlation effects, leading to a consistent improvement on both exchange and correlation energies. The B3LYP-like translated on-top functional (tB4LYP) developed along this line not only surpasses the accuracy of its parent methods, the complete active space self-consistent field (CASSCF) and the original MC-PDFT functionals (tBLYP and tB3LYP), but also outperforms the widely used complete active space second-order perturbation theory (CASPT2). Remarkably, while remaining satisfactory for general purpose, tB4LYP shows superior accuracy for challenging cases like the Cr2 dissociation and the associated low-lying vibrational energies, the ethylene torsional rotation and the ethyne diabatic colinear dissociations, with the significantly lower computational cost than CASPT2.
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Introduction
Nowadays, density functional approximations (DFAs) have gained widespread acceptance across a diverse range of chemical and physical disciplines. They mostly belong to the non-interacting framework of generalized Kohn-Sham (GKS) density functional theory (DFT). However, these DFAs face a significant challenge in accurately capturing static (or nondynamic) correlation effects1,2,3, a limitation that significantly impedes their applications in areas such as transition metal chemistry, bond breaking processes, and electronic transitions. Much effort in the standard GKS scheme has been devoted in the pursuit of increasingly sophisticated exchange-correlation DFAs, which reside on the top rung of Jacob’s Ladder of DFT and depend on both occupied and unoccupied orbitals4,5,6,7,8,9. Given the multiconfigurational nature of static correlation effects, using density from multiconfiguration self-consistent field (MC-SCF) calculations in the wave-function theory (WFT) could potentially simplify the construction of such DFAs. This suggests that a synergistic approach combining WFT(MC-SCF) and DFT (i.e., MC + DFT) could offer a promising solution to this challenge.
Recent progress along the MC+DFT direction is highlighted by the developments of the multiconfiguration pair-density functional theory (MC-PDFT)10,11 and its analogs12,13. In MC-PDFT, the total energy is given by
where the first term is the classical energy, encompassing kinetic energy, and various types of classical Coulomb energies between nucleus-nucleus, nucleus-electron, and electron-electron, calculated by using the MC-SCF method (usually the complete active space self-consistent field, CASSCF, method)14,15. MC-PDFT features the use of the so-called on-top density functionals (the second and third terms in the left of Equation (1)), that depend on both density and on-top (ot) pair density to include exchange (X) and correlation (C) effects. The correlation of CASSCF is not used in the final energy expression of Equation (1), which tends to avoid a long-lasting problem for many MC+DFT methods, i.e. double counting of the electron correlation from the two worlds of WFT and DFT. Moreover, MC-PDFT proposed a straightforward strategy to establish the on-top DFAs by translating from the popular DFAs in the standard GKS scheme. In consequence, tLDA denotes an on-top DFA translated from local density approximation (LDA)16,17, while tBLYP refers to an on-top generalized gradient approximation (GGA) translated from BLYP18,19, and so forth. It is important to recognize that PDFT has its formal root in theorems related to on-top pair-density and on-top XC hole20,21, while PDFT-GGA such as tBLYP exactly matches that of the standard BLYP functional for systems where static-correlation effects become negligible. This physical condition is not satisfied by other MC-DFT methods without using the ingredient based on the on-top pair density.
It has been demonstrated that systems with noticeable static correlation effects are amenable to MC-PDFT, but it was also argued that the on-top DFAs inherit the errors existing in the original GKS DFAs22. As shown in Fig. 1, while UHF and B3LYP both fail at describing the bond dissociation, MC-PDFT-tBLYP (the leading “MC-PDFT-” will be omitted for brevity in the rest of the paper) is much improved, describing correctly the stretched-bond region in the molecular dissociation curves, where the static correlation effect arising from the interaction between the bonding and anti-bonding configurations becomes noticeable. However, the performance of tBLYP in the equilibrium region as compared to the dissociation limit is not satisfactory, resulting in the calculated binding energies that are underestimated by ca. 11 kcal ⋅ mol−1 for N2 and CO compared to the spectroscopic results23,24,25.
For all panels, the bond dissociation curves calculated by UHF (dashed darkslategray), B3LYP (dashed thistle), CASSCF (darkolivegreen), CASPT2 (olive), tBLYP (darkgoldenrod), tB3LYP (sandybrown), and tB4LYP (lightpink) methods are plotted together with literature curve (black), the CASSCF, CASPT2, tBLYP, tB3LYP, and tB4LYP curves are also marked with numbers 1, 2, 3, 4, and 5, respectively. a N2 (X\({}^{1}{\Sigma }_{g}^{+}\)) bond dissociation curves using CAS(10e, 8o) active space, the literature curve is from spectroscopic Rydberg-Klein-Rees (RKR) potential energy plots for N225, note that the CASPT2 and tB3LYP curves are nearly completely overlapped around the equilibrium bond length. b CO(X1Σ+) bond dissociation curves using CAS(10e, 8o) active space, the literature curve is from spectroscopic Rydberg-Klein-Rees (RKR) potential energy plots for CO24. c O2 (X\({}^{3}{\Sigma }_{g}^{-}\)) bond dissociation curves using CAS(12e, 8o) active space, the literature curve is from high-level composite ab initio approach for O261. d CN (X2Σ+) bond dissociation curves using CAS(9e, 8o) active space, the literature curve is from MRCI + Q calculations for CN62, note that the tB4LYP and literature curves are nearly completely overlapped around the equilibrium bond length. The cc-pVTZ basis sets are used for all elements. For UHF and B3LYP, the total energy of the separate atoms is set to zero while for the rest of the tested methods the total energy of the molecule calculated at r = 10 Å is set to zero. Source data are provided as a Source Data file.
Inspired by the great success of hybrid DFAs in GKS-DFT, hybrid MC-PDFT (HMC-PDFT) was proposed recently22
where the second term is the nonlocal XC energy of CASSCF but scaled by a hybrid factor of λ. Consequently, the on-top exchange and correlation functionals are scaled accordingly in the third and fourth terms. The introduction of λ in MC + DFT can be derived from a formally exact multideterminant extension of the Kohn–Sham scheme, proposed by Sharkas et al.26 While the exchange functional is of a linear dependence with respect to λ in the context of adiabatic connection formalism, there is a non-linear dependence with k = 2 for the correlation functional based on the uniform coordinate scaling relationship26,27,28,29. Mostafanejad et al.22 suggested a hybrid MC-PDFT method with a non-linear dependence with k = 2 for the on-top correlation functional. After a systematic benchmark, Pandharkar et al. found that a larger k did not offer a significant improvement in the final performance of HMC-PDFT, and thus chose the simpler linear dependence for the on-top correlation30. Based on this linear HMC-PDFT framework (k = 1), they proposed and examined the on-top hybrid DFA of tPBE0, where the hybrid factor λ was set to 0.2530.
B3LYP is another popular hybrid DFA with numerous successful applications in quantum chemistry31. Analogous to tPBE0, we can transfer B3LYP to an on-top hybrid DFA in the linear HMC-PDFT framework, namely tB3LYP. The total energy is then calculated as
with the composite on-top exchange and correlation functionals defined as
Here three empirical parameters (c1 = 0.20, c2 = 0.72, c3 = 0.81) are chosen the same as those used in the standard hybrid B3LYP without additional optimization18,31. \({E}_{{{\rm{ot}}},{{\rm{x}}}}^{{{\rm{tS}}}}\) is the on-top exchange functional translated from the Slater-type LDA16 and \(\Delta {E}_{{{\rm{ot}}},{{\rm{x}}}}^{{{\rm{tB88}}}}\) is translated from the GGA correction in the form of Becke8818. \({E}_{{{\rm{ot}}},{{\rm{c}}}}^{{{\rm{tVWN}}}}\) and \(\Delta {E}_{{{\rm{ot}}},{{\rm{c}}}}^{{{\rm{tLYP}}}}\) are translated from the local Vosko-Wilk-Nusair (VWN) correlation17 and the correction of Lee-Yang-Parr (LYP) correlation approximation19, respectively. For consistency, we use the same VWN version (v3) as adopted in the original B3LYP implementation31.
In line with the previous studies22,30, we observed that the hybrid strategy keeps the good description of MC-PDFT on the stretched-bond region for the molecular dissociations (see tB3LYP curves on Fig. 1), while improves MC-PDFT on the description of the spin-flip atomic excitation energies (see Table 1 for the errors of tB3LYP and tBLYP on SFAEE11). Unfortunately, the corresponding improvement for the binding energies is not significant. As shown in Fig. 1, the errors of tB3LYP are not much different from those of tBLYP on the equilibrium region of the dissociation curves.
As shown by the linear HMC-PDFT formula (Equation (2) with k = 1), the total correlation energy is normalized, where the on-top correlation DFA with a scaled factor of (1 − λ) is complemented by a λ portion of the CASSCF correlation. Of special note is the dynamic correlation, which is almost absent in the CASSCF correlation due to the fact that a small active space is encouraged in the context of (H)MC-PDFT for the sake of computational efficiency. We therefore argue that increasing hybrid factor λ would improve the exchange energy, which at the same time would notably deteriorate the correlation energy, in particular, the dynamic part. Hence, we propose here that, instead of double counting as commonly believed, the inevitable deterioration of the correlation energy is the key problem for the linear HMC-PDFT, preventing a consistent improvement in both exchange and correlation energies.
In this work, we develop a strategy to separate the exchange and correlation parts of EWFT,xc, leading to a new linear HMC-PDFT scheme with a consistent improvement in describing both exchange and correlation energies. Note that, the fundamental success of GKS-DFT is the existence of a non-interacting pure-state vs-representable ground-state density, which is challenged by systems with noticeable static-correlation effects. We hereby choose to start with the interacting v-representability of the ground-state density
where {χi} are the natural orbitals with their occupation numbers denoted as {ni}. It has been well-documented that1,2,3,32,33,34,35,36,37, for the systems with increasing static correlation effects, the natural orbitals possess a wider distribution of electrons with more occupation numbers to be fractional. In other words, the static correlation is closely related to the degree of fractional occupations in the natural orbitals38,39,40. Collins38 has conjectured that the correlation energy of an N—representable system was negatively proportional to the information measure of the one-particle density (Equation (5)), i.e., the Jaynes information entropy \({S}_{{{\rm{Jaynes}}}}=-{\sum }_{i=1}^{\infty }{n}_{i}\ln {n}_{i}\)41. Numerical evidences were shown by calculations of small molecules39,42,43,44. By assuming a Fermi distribution of the electronic occupation numbers, Wang and coworkers have come up with a self-consistent method where a cross-entropy \({S}_{{{\rm{cross}}}}=-\mathop{\sum }_{i=1}^{\infty }[{n}_{i}\ln {n}_{i}+(1-{n}_{i})\ln (1-{n}_{i})]\) cumulant functional was used for calculating the correlation energy45,46,47. Chai obtained the same cross-entropy term in the derivation of the thermally-assisted-occupation DFT48. Following this rationale, we introduce here a one-parameter approximation to the CASSCF correlation energy in the form of the cross-entropy contribution.
where {ni} is the set of occupation numbers for the nature orbitals produced by the corresponding CASSCF wavefunction. The empirical parameter θ is assumed to be system-independent at this stage, and it is subject to optimizations for different on-top DFAs. Based on this approximation, a new HMC-PDFT formula with an extra parameter θ can be established, which in its first version, reads as
In this hybrid scheme, the cross-entropy term removes the unwanted part in the CASSCF correlation and recovers a full piece of on-top correlation DFA, ensuring a more balanced description of the static and dynamic correlation effects to achieve a better description of both exchange and correlation. Transferring B3LYP in this new HMC-PDFT scheme, we obtain a four-parameter on-top hybrid DFA, namely tB4LYP
Compared with tB3LYP (Equation (3)), tB4LYP has one more empirical global parameter θ = 5 mHartree, determined by the lowest mean absolute errors (MAEs) on the benchmark data sets shown in Supplementary Table S1 in the Supplementary Information.
Moreover, in addition to a better description of the correlation effect, this new linear HMC-PDFT scheme possesses another promising theoretical advantage that the energy expression of tB4LYP reduces to the B3LYP form for a closed-shell-singlet Slater determinant if one uses the same spin-restricted single Slater determinant. It thus guarantees a largely good performance of tB4LYP on the systems mainly driven by dynamic correlation effects. For comparison, this advantage does not hold for tB3LYP in the original linear HMC-PDFT scheme (Equation (3)).
Results and discussion
In order to benchmark its numerical accuracy in practical systems, we performed tB4LYP calculations on five molecular dissociation curves for the ground-states of N2 (\({}^{1}{\Sigma }_{g}^{+}\)), CO(1Σ+), O2 (\({}^{3}{\Sigma }_{g}^{-}\)), CN (2Σ+), and Cr2 (\({}^{1}{\Sigma }_{g}^{+}\)). For transition metals compounds, we specifically calculated the spectroscopic constants of five dimers, i.e., Sc2 (X\({}^{5}{\Sigma }_{u}^{-}\)), Ti2(X3Δg), V2 (X\({}^{3}{\Sigma }_{g}^{-}\)), Ni2 (X1Γg), and Cu2 (X\({}^{1}{\Sigma }_{g}^{+}\)), all of which have been well-characterized ground states. The performance of tB4LYP is also benchmarked against three data sets which include the MGDBE9 data set of 9 main-group diatomic binding energies, the TMDBE8 data set of 8 transition-metal diatomic binding energies, and the SFAEE11 data set of 11 spin-flip atomic excitation energies. The geometries and literature values of MGDBE9 and TMDBE8 are taken from the Minnesota database49,50. The SFAEE11 data set is the same as those used in benchmarking the original linear HMC-PDFT method30. In addition, the torsional rotation of the ethylene molecule and the triple bond dissociation process of the ethyne molecule have been studied. The structures along the torsional angle change of ethylene are not optimized. We used the geometries provided by Ref. 51 (Equation (9)). The dissociation of ethyne follows a diabatic colinear path (Equation (10)) with fixed C-H bond lengths of 1.058 Å. Furthermore, the bond dissociation energies with respect to relaxed C-H(4Σ−) bond length at 1.083 Å52 are calculated in comparison with the experiments53,54,55. To further validate the practical advantages of the tB4LYP functional, we conducted extensive benchmarking using well-established general-purpose thermochemistry data sets. We select 9 subsets from the GMTKN55 database56,57, which comprise a total of 185 thermochemical benchmarks ranging from basic properties such as the ionization potentials and the electron affinities, to reaction energies, barrier heights, and noncovalent interactions.
Concurrently, we also performed the same calculations with spin-unrestricted Hartree-Fock (UHF), spin-unrestricted GKS-B3LYP (B3LYP), tBLYP, tB3LYP, CASSCF, and the CAS second-order perturbation theory (CASPT2)58,59. In Fig. 1, 2, and 3, the “Literature” curves shown in black are used as references, which are either fitted from spectroscopic data (N2, CO, and Cr2)24,25,60 or calculated by higher-level ab initio methods (O2, CN, ethylene and ethyne)51,61,62,63. For ethylene, a composite method with the multi-reference Fock-space coupled cluster (FSCC)64 energy has been used as reference (see also Equation (11)). Additional details of the computations such as the basis sets65,66,67 and the active space definitions of the calculated molecules can be found in the Supplementary Information. The DIRAC program68,69 was utilized for the FSCC calculations and all the rest of the calculations were performed with a locally modified version of the OpenMolcas program package70.
The bond dissociation curves calculated by CASSCF (darkolivegreen), CASPT2 (olive), tBLYP (darkgoldenrod), tB3LYP (sandybrown), and tB4LYP (lightpink) methods are plotted together with literature curve (black), the curves are also marked with numbers 1, 2, 3, 4, and 5, respectively. The calculated curves are extrapolated from the cc-pVTZ-DK and cc-pVQZ-DK results (CBS/TQ).The active space definition is CAS(12e, 22o) approached by the Density Matrix Renormalization Group (DMRG) method95,96,97,98,99,100,101. The literature curve is an RKR potential energy curve obtained from a fit to experimental vibrational levels60. The dissociation limits (i.e., the calculated binding energies De) are 5.72, 48.52, 16.40, 13.67, 27.18, and 35.97 kcal ⋅ mol−1 for CASSCF, CASPT2, tBLYP, tB3LYP, tB4LYP, and literature. The literature value is from experimentally determined dissociation energy109 carlibrated with ZPE60. Other benchmark literature values include an experimental value of 33.41 kcal ⋅ mol−1 110, an MR-AQCC/CBS value of 31.13 kcal ⋅ mol−1 71, a RASPT2/CBS value of 43.58 kcal ⋅ mol−1 72, and a DMRG-N-electron valence state perturbation theory (DMRG-NEVPT2)/CBS value of 33.44 kcal ⋅ mol−1 111.The total energy of the molecule calculated at r = 20 Å is set to zero. Source data are provided as a Source Data file.
Diatomic dissociation in main-group chemistry
Let us first focus on the potential energy curves (PEC) of four diatomic molecules shown in Fig. 1 for N2, CO, O2, and CN. It is clear that single determinantal methods, such as UHF and B3LYP, struggle to accurately describe molecules in the stretched-bond regions, where multi-reference or static correlation effect is particularly significant. In contrast, the MC-PDFT method, as exemplified by tBLYP, excels in scenarios dominated by static correlation effects. The innovative linear HMC-PDFT scheme proposed here (Equation (7)) upholds this advantage. The tB4LYP method (Equation (8)), despite its relatively small active spaces, accurately captures the correct description of the bond-stretched region. This holds true not only for the closed-shell homonuclear and heteronuclear diatomics of N2 and CO, but also for the open-shell homonuclear and heteronuclear diatomics of O2 and CN.
Figure 1 clearly (also see Supplementary Table S2) shows that tBLYP(tB3LYP) consistently underestimates the binding energies, yielding errors for N2, CO, O2, and CN by ca. 11(9) kcal ⋅ mol−1, 12(8) kcal ⋅ mol−1, 3(5) kcal ⋅ mol−1, and 13(10) kcal ⋅ mol−1, respectively. The numbers in parentheses clearly show that tB3LYP does not improve much over tBLYP. Such an underestimation tendency, however, is completely corrected by the new HMC-PDFT method of tB4LYP. Despite slightly overestimating the binding energies of N2, CO, and O2 by ca. 5 kcal ⋅ mol−1, 2 kcal ⋅ mol−1, and 2 kcal ⋅ mol−1, respectively, tB4LYP clearly outperforms tBLYP (tB3LYP), CASSCF, and even CASPT2. Note that CASPT2 is sometimes used as reference to benchmark other methods in the description of static correlation, since more elaborate methods are often out of reach22.
To better quantify the performance of tB4LYP on these PECs, we calculated the equilibrium bond length re, the harmonic frequency ωe, and the first order anharmonicity constant ωeχe. These values, together with the aforementioned binding energies, were complied in Supplementary Table S2, as the accuracy of these calculated values is crucial for a correct interpretation of the experimental results from the vibrational spectroscopy. The poor performance of UHF and B3LYP in the stretched-bond region is often attributed to the inherent limitation of single-reference methods in handling the static correlations, although these methods are not considered to be problematic in the equilibrium region. Our results support the fact that B3LYP can accurately predict the binding energies of the four molecules under consideration, even surpassing many multireference-based methods as examined in this study. However, B3LYP tends to overestimate the harmonic frequency, suggesting that the static-correlation error intrinsic to single-reference methods also influences the shape of the dissociation curve in the equilibrium region. Meanwhile, while tB4LYP outperforms tBLYP, tB3LYP, and even CASPT2 overall, all these multireference-based methods are capable of producing these spectroscopic quantities with reasonably good accuracy. Given the relatively small active spaces adopted in the calculations, CASSCF, on the other hand, is inferior for the lack of dynamic correlation.
The Chromium dimer: the enduring challenge in quantum chemistry
As compared to the previously discussed PECs involving only main-group elements, the ground-state \({}^{1}{\Sigma }_{g}^{+}\) of Cr2 has presented a significant challenge to electronic structure theory for decades, necessitating an accurate description of both dynamic and static correlation effects with sufficiently large active spaces and basis sets71,72. Recently, the Chan group published a pioneering computational study of the chromium dimer with spectroscopic accuracy, predicting several low-lying vibrational levels within ca. 50 cm−1 error73. Their study utilized an expensive composite method comprising very large density matrix renormalization group (DMRG) calculations (28 electrons in 76 orbitals, using up to bond dimension 28,000 and SU(2) symmetry), supplemented by multi-reference perturbation and unrestricted coupled cluster calculations with up to quintuple-ζ basis size.
While the calculated curves at the basis set levels of cc-pVT(Q)Z-DK are available in the Supplementary Information, Fig. 2 presents the bond dissociation curves of the ground state Cr2 extrapolated from these basis sets to the complete basis set (CBS) limit. Since CASSCF with a small active space contains minimal dynamic correlation, it is unsurprising that this method erroneously predicts no bonding in the inner region. In contrast, CASPT2 with the same active space overcorrects, yielding an excessively large binding energy and an overly sharp curve towards the dissociation limit.
MC-PDFT emerges as a promising alternative, offering the possibility for an efficient and balanced treatment on dynamic and static correlation effects. Figure 2 shows that the tBLYP method refines the curve shape over CASSCF and predicts a plausible equilibrium bond length, emphasizing again the importance of dynamic correlation. However, tBLYP excessively shallows the binding well and erroneously predicts a second minimum at ca. 2.5 Å, suggesting that further improvement is necessary in order to better balance the dynamic and static correlations. With the linear hybridization scheme, the tB3LYP method, unfortunately, does not improve but makes the result worse than tBLYP, exhibiting two competing equilibrium bond lengths at ca. 1.7 and 2.5 Å. Evidently, tB4LYP via HMC-PDFT provides the best performance. The tB4LYP curve maintains a gradual transition to the dissociation limit without showing a second minimum, and aligns well with the reference curve around the equilibrium region. Moreover, the tB4LYP method surpasses all other methods investigated here in terms of the binding energy (27.2 kcal ⋅ mol−1), albeit still being ca. 8 kcal ⋅ mol−1 lower than the experimental value.
Utilizing the simulated potential energy curves, we solved the vibrational Schrödinger equation, predicting seven low-lying vibrational energy levels, as shown in Table 2 (see Supplementary Table S3 for more vibrational levels). Notably, while other methods exhibit poor agreements overall, the seven vibrational energy levels calculated by tB4LYP are satisfactory, with the first three levels being in perfect agreement with the experimental results. Of course, there is still room for further improvement, as the accuracy of tB4LYP decreases for higher vibrational energy level, deviating from the reference PEC for the plateau region in the tB4LYP curve (see Fig. 2). Nevertheless, it is encouraging to see that tB4LYP, with just a fraction of the cost, yields comparable spectroscopic values to the highly expensive composite method for the low-lying vibrational levels.
The good performance of tB4LYP on Cr2 extends to other transition metal dimers with diverse spin and spatial symmetries. As evidenced in Table 3 (see Supplementary Table S4 and Fig. S8 for more details), the tB4LYP results are on par with higher-level wavefunction results, surpassing CASPT2 in terms of equilibrium bond length (re) predictions. Furthermore, the MAEs predicted by tBLYP and the linearly hybridized tB3LYP methods are three to four times larger than those of tB4LYP.
Ethylene torsional rotation and ethyne colinear dissociation
Figure 3 presents the results calculated by various methods for the ethylene torsional rotation and the ethyne colinear dissociation. Due to its significant static correlation effects, the ethylene torsion potential has been intensively studied using numerous methods.74,75,76,77,78,79 For comparison, we chose literature that utilized the same supplied benchmark geometries (Equation (9))51. It is seen that the single-reference methods, such as UHF and B3LYP, fail to describe the degenerate bonding (π) and anti-bonding (π*) orbitals at the transition state where the torsion angle is 90 degrees, resulting in unphysical cusps at the peak of the barrier. In contrast, all multireference-based methods correctly provide a smooth barrier. Compared to the MR-ccCA reference, tB4LYP seems to overestimate the barrier height by ca. 5 kcal ⋅ mol−1. However, it is known that MR-ccCA lacks an explicit treatment of dynamic correlation beyond doubles80. Using the Fock-space coupled cluster singles and doubles (FSCCSD)64 to include an explicit treatment of dynamic correlation beyond doubles by means of the “disconnected” terms, we arrived at the resulting barrier height of 71.19 kcal ⋅ mol−1 (see Equation (11) for more details), which is closer to the value of tB4LYP (73.05 kcal ⋅ mol−1) than the MR-ccCA value (68.08 kcal ⋅ mol−1).
For all panels, the bond dissociation curves calculated by UHF (dashed darkslategray), B3LYP (dashed thistle), CASSCF (darkolivegreen), CASPT2 (olive), tBLYP (darkgoldenrod), tB3LYP (sandybrown), and tB4LYP (lightpink) methods are plotted together with literature curve (black), the CASSCF, CASPT2, tBLYP, tB3LYP, and tB4LYP curves are also marked with numbers 1, 2, 3, 4, and 5, respectively. a The torsional rotation of ethylene using CAS(2e, 2o) active space, the literature curve is from theoretically calculated MR-ccCA results for ethylene51. The torsion barrier heights are 66.18, 64.48, 71.43, 70.69, 73.05, and 68.08 kcal ⋅ mol−1 for CASSCF, CASPT2, tBLYP, tB3LYP, tB4LYP and MR-ccCA, respectively. The total energy of the planar molecule is set to zero. b The dissociation of ethyne using CAS(6e, 6o) active space, the literature curve is from theoretically calculated EOMCCSD results for ethyne63. The minima of the plots are −236.24, −259.98, −275.11, −269.12, −285.81, and −293.90 kcal ⋅ mol−1 for CASSCF, CASPT2, tBLYP, tB3LYP, tB4LYP, and EOMCCSD, respectively. The dissociation curve is calculated with fixed C-H bond length at 1.058 Å. For UHF and B3LYP the total energy of the two separated CH(4Σ−) radicals is set to zero while for the rest of the methods the total energy of the ethyne molecule calculated at R(C-C) = 10 Å is set to zero. The cc-pVTZ basis sets are used for all elements. Source data are provided as a Source Data file.
The process of ethyne triple bond breaking represents another system that has been studied intensively for its multi-reference effects53,63,81. The “head-to-head” diabatic colinear dissociation favors the first excited state CH (4Σ−) pair fragments (see Fig. 3(a) in ref. 81). The single-referenced B3LYP predicts an unphysical oscillation in the stretched-bond region, a problem that can be mitigated by employing the multi-referenced methods, including the proposed tB4LYP. The experimentally derived binding energy is 269.20 kcal ⋅ mol−1 53,54,55 (see the “Method” section for its derivation). Previous study63,81 has indicated that both EOM-CCSD and GMCPT methods tend to overestimate, leading to the calculated binding energies of ca. 293 kcal ⋅ mol−1 and 275 kcal ⋅ mol−1, respectively. The tB4LYP method exhibits a similar tendency. The resulting binding energy is 285.81 kcal ⋅ mol−1, which is closer to the EOM-CCSD value.
Considering the performance across the challenging systems studied above, tB4LYP is evidently the overall winner. The improvement of tB4LYP over tBLYP should be attributed to the new linear HMC-PDFT scheme (Equation (7)), which provides an efficient way to balance the exact exchange and the local exchange, while simultaneously without losing the accuracy for describing the static and dynamic correlations. In comparison, the tB3LYP method (Equations. (2) and (3)) that mixes in both CASSCF exchange and correlation does not show substantive improvement over tBLYP. The accuracy of tB4LYP is, hence, especially promising.
Several data sets for general purpose use
We now consider several data sets for general purpose use, including binding energies of 9 main-group diatomic molecules (MGDBE9), and 8 transition-metal diatomic molecules (TMDBE8). The mean absolute error (MAE) and the maximum absolute error (MAX) for different methods are collected in Table 1, while the individual absolute errors are plotted in Fig. 4, and the individual signed errors are provided in Supplementary Table S5 of the Supplementary Information. Due to an insufficient description of dynamic correlation, CASSCF with small active spaces is unsatisfactory for both data sets, and is not recommended for routine uses. The improvement of tBLYP over CASSCF is notable for binding energies. The corresponding MAEs of tBLYP are 8.6 kcal ⋅ mol−1 and 10.4 kcal ⋅ mol−1 for MGDBE9 and TMDBE8, respectively, which are about three times smaller than those of CASSCF. However, the performance of tBLYP is not yet comparable to CASPT2, particularly for main-group diatomic binding energies. The MAE of CASPT2 in MGDBE9 is 4.9 kcal ⋅ mol−1, about half of the MAE of tBLYP.
Individual absolute errors of the MGDBE9 and TMDBE8 data sets calculated by (DMRG-)CASSCF (darkolivegreen), (DMRG-)CASPT2 (olive), tBLYP (darkgoldenrod), tB3LYP (sandybrown), and tB4LYP (lightpink) methods are shown, the first nine (from B2 to 1SiO) belong to MGDBE9 and the rest (from Fe2 to TiCl) TMDBE8. The mean absolute errors are also provided on the rightmost entry. The dissociation energies are calculated as the energy difference between the reactant and the dissociated counterparts. Source data are provided in the Supplementary Information.
We concluded that the hybrid strategy of Equation (2) is not effective for improving the description of binding energies, in agreement with the previous work on six diatomic binding energies (DBE6)30. tB3LYP exhibits some improvements over tBLYP with MAEs of 8.3 kcal ⋅ mol−1 and 10.0 kcal ⋅ mol−1 for MGDBE9 and TMDBE8, respectively. By removing the unwanted part in the CASSCF correlation out of the hybridization scheme (Equation (7)), the tB4LYP method (Equation (8)) reduces the MAE in MGDBE9 to 2.7 kcal ⋅ mol−1, being more than 5 and 2 kcal ⋅ mol−1 better than tBLYP (tB3LYP) and CASPT2, respectively. The MAX of 5.6 kcal ⋅ mol−1 for tB4LYP occurs in B2, which is only half of the MAX for CASPT2. As having been mentioned before, the three parameters in tB4LYP are inherited from the original hybrid B3LYP without reoptimization. Such an improvement is therefore consistent.
The nine subsets–ISO34, ACONF, BHDIV10, PX13, BHROT27, DIPCS10, G21IP, G21EA, and TAUT15–from the GMTKN55 database encompass basic molecular properties such as IP, EA, and tautomerization energy, as well as more complex phenomena including proton exchange, barrier heights, intramolecular noncovalent interaction. The results of our benchmarking study are summarized in Table 4 (see Supplementary Table S6–S14 for additional details). Across all 9 subsets, tB4LYP consistently outperforms tB3LYP with a total MAE reduction of approximately 28% (0.9 kcal ⋅ mol−1). Moreover, the overall performance of tB4LYP (2.32 kcal ⋅ mol−1) is also better than the more expensive CASPT2 method (2.75 kcal ⋅ mol−1). The formal advantage of tB4LYP as discussed earlier, translates into a practical benefit, making it a robust and reliable choice for general-purpose quantum chemistry calculations.
The transition metal diatomic molecules are generally more challenging due to the involvement of d-electrons as well as complications in determining the correct ground-states and dissociation limits. As evidenced by the qualitative failure of CASSCF, the three metal dimers (see Fig. 4, Fe2, V2, and Cr2) require accurate description of the dynamic correlation alongside the static correlation. Similarly to the main-group diatomic molecules, CASPT2 outperformed tBLYP and tB3LYP but not tB4LYP. The MAX of CASPT2 occurs on Fe2 (37.1 kcal ⋅ mol−1), over three times the MAE. It was argued that introducing a proper shift on ionization potential and electron affinity (IPEA) can improve the performance of CASPT2 on binding energies, but it may introduce other issues such as shortenings of the equilibrium bond lengths72. The (H)MC-PDFT methods, especially tB4LYP, generally outperform the WFT methods. We observed the stable performance of tB4LYP on TMDBE8 with a smaller MAE of 7.6 kcal ⋅ mol−1 and MAX of 11.1 kcal ⋅ mol−1 on Cr2.
We then turn to tB4LYP’s performance on the excitation energies. The benchmark data set used in this work is SFAEE11, comprising 11 spin-flip atomic excitation energies. As shown in Table 1 (see also Supplementary Table S15 and Fig. S5 for individual errors), our results are consistent with the previous work30, showing that tBLYP (MAE = 7.1 kcal ⋅ mol−1) performs worse than CASSCF (MAE = 6.7 kcal ⋅ mol−1) on SFAEE11. The hybrid tB4LYP method reduces the MAE to 5.4 kcal ⋅ mol−1. This error can be further reduced to 4.3 kcal ⋅ mol−1, if the state averaging is considered across different spins (see Supplementary Table S16). Interestingly, the performance of tB4LYP is similar to that of tB3LYP despite their different hybrid schemes.
For singlet excitation energies (see Supplementary Table S17), our finding confirms the initial observation by Giovanni et al.10: MC-PDFT as implemented in its standard form underperforms CASPT2 when it comes to describing such states. Table S17 reveals that the MAE of tBLYP for singlet excitations is 13.7 kcal ⋅ mol−1, which notably exceeds the MAE of CASPT2 by approximately 8 kcal ⋅ mol−1. However, we observe a significant improvement upon incorporating 20% WFT exchange and correlation into the hybrid functional, resulting in an MAE of 5.81 kcal ⋅ mol−1 for tB3LYP. Regarding the specific impact of the entropic term, our results indicate a less pronounced effect on the accuracy of singlet excitation energies. The MAE for tB4LYP is 6.46 kcal ⋅ mol−1, which is somewhat poorer than those of CASPT2 and tB3LYP. This suggests that while the inclusion of the entropic component does not lead to a further enhancement in this particular application, it still contributes to maintaining a competitive level of accuracy (all around 6 kcal ⋅ mol−1) against more expensive methods like CASPT2.
To further demonstrate the validity of the proposed HMC-PDFT scheme (Equation (7)), we translated the one-parameter GKS hybrid B1LYP (Equation (12)) to the on-top functional tB2LYP with two parameters for the hybridization λ and the cross-entropy contribution θ, respectively (Equation (13)). For a direct comparison, Equation (14) shows the relevant tB1LYP functional using the HMC-PDFT scheme proposed by Mostafanejad et al.(Equation (2)). Supplementary Fig. S4 (uppper map) provides the overal MAEs of tB1LYP and tB2LYP with varying λ for both and θ for the latter for the three data sets. Our results show that the tB1LYP method (the leftmost column in Supplementary Fig. S4) with λ = 0.1 (MAE = 8.8 kcal ⋅ mol−1) exhibits a little improvement over the tBLYP method (MAE = 8.9 kcal ⋅ mol−1). However, its accuracy deteriorates quickly with the increase of the hybrid parameter λ. In comparison, the tB2LYP with a full piece of correlation from DFT (\({E}_{{{\rm{ot}}},{{\rm{c}}}}^{{{\rm{tLYP}}}}\)) provides a consistent improvement with respect to the tB1LYP at the same λ. Moreover, the improvement (i.e., MAE(tB2LYP) − MAE(tB1LYP)) becomes more significant at larger λ.
On the other hand, the marked θ-dependence observed in the MAEs for the TMDBE8 data set, depicted in Supplementary Fig. S4 (lower map) stresses the significance of subtracting the wavefunction correlation for transition metal compounds. This clearly demonstrates the detrimental influence of the CASSCF correlation on the functional performance, highlighting the importance of the cross-entropy contribution in achieving optimal results. Moreover, the exclusion of the cross-entropy contribution in tB4LYP (i.e., Equation (8)) leads to a method, dubbed as tB3LYPfc (Equation (15)), for comparison purposes. For transition metal compounds (shown in Supplementary Table S18), the inclusion of the entropy term reduces the MAE by 1.1 kcal ⋅ mol−1. The importance of the entropy term is further demonstrated by the dissociations of multiply bonded transition metal dimers, such as V2 and Cr2, which have the highest bond orders within this class of compounds82 (see Supplementary Fig. S9). Their dissociation profiles illustrated that the improvements of the entropy contribution is as vital as considering the complete Eot,c component. In the absence of the entropy correction (as shown by tB3LYPfc), the problematic double-minima structure presents for Cr2, even when utilizing a higher-level treatment (CAS(12e, 22o)/CBS(T,Q)). This observation stresses again the significance of including the cross-entropy contribution for transition metal compounds, showing a clear advantage in terms of both accuracy and robustness, for tB4LYP compared to regular hybrid functionals such as tB3LYP.
We have also conducted a thorough analysis to evaluate how parametrization influences the performance of our tB4LYP method. The detailed results are provided in Supplementary Table S19 (see also Supplementary Fig. S6, S7, and Table S5, S15). A key aspect of this analysis involved the full optimization of the four parameters in tB4LYP, leading to the development of an optimized variant, denoted as tB4LYPo. The total MAE and population variance of tB4LYPo are 5.1 kcal ⋅ mol−1 and 14.0 (kcal ⋅ mol−1)2, respectively, exhibiting only a marginal improvement over the original tB4LYP method (5.2 and 14.2, respectively, see Supplementary Table S19). Such a modest enhancement in performance is significant as it underscores the robustness of our initial parameter selection. This finding confirms the adequacy of the parameterization inherited from the standard KS-B3LYP for providing a balanced description of exchange and dynamic correlation effects within the HMC-PDFT framework. Moreover, our results demonstrate an impressive degree of parameter transferability from KS-DFT to MC-PDFT in tB4LYP. This transferability is beneficial, as it indicates the applicability and generality of tB4LYP to more complex systems. It is particularly significant in scenarios where the use of expensive multi-configurational wavefunction methods is impractical or prohibitive.
Concerning the dependence of tB4LYP on the choice of active space, we utilize the binding energy of N2 as a paradigmatic case study. Table 5 reveals that expanding the active space leads to a consistent enhancement of CASSCF results as it captures more dynamic correlation. Based on its efficacy in accounting for static correlation effect, the recommended small active space used in (Hybrid) MC-PDFT methods is (10 active electrons, 8 active orbitals) for N2 molecule, where tBLYP exhibits an error of 11.19 kcal ⋅ mol−1. Our results confirm the earlier observation by Sharma et al.83 that the performance of tBLYP can indeed deteriorate with large active spaces. Specifically, a larger space such as (10 active electrons, 14 active orbitals) results in an ~3 kcal ⋅ mol−1 increase in error. Our research highlights that incorporating 20% WFT exchange and correlation in both tB3LYP and tB4LYP mitigates largely the errors due to increasing the size of the active space, and this trend further improves upon the inclusion of the cross-entropy term. Encouragingly, the influence of active space on the tB4LYP result is less than 0.3 kcal ⋅ mol−1, suggesting that within the recommended active space, tB4LYP maintains robust and reliable performance.
As noted, the tB4LYP method (see Equation (7) and Equation (8)) proposed in this work has a promising advantage that its energy form would reduce to that of standard B3LYP for the systems without notable static correlation effects, when the wavefunction could be well represented by a single determinant. However, it does not mean that tB4LYP results will be the same as B3LYP for these cases. It is because the reference density and orbitals are solved self-consistently for the standard B3LYP calculations; while the CASSCF wavefunction used for tB4LYP will reduce to the Hartree-Fock approximation in the limit of a single determinant, it would be appropriate here to recall the density-corrected (DC) DFT method84,85,86,87, where employing the HF density has been shown to enhance the performance of conventional DFAs in many challenge cases such as predicting reaction barriers. In this consideration, further development could involve either updating the reference wavefunction for tB4LYP or re-optimizing the empirical parameters for a spectrum of different chemical properties to better fit the CASSCF wavefunction, which opens new avenue to improving tB4LYP.
In summary, we have proposed a new linear hybrid scheme for MC-PDFT (HMC-PDFT). By introducing a cross-entropy approximation to tune the CASSCF correlation, the HMC-PDFT scheme ensures a consistent improvement in describing both exchange and correlation energies. We then developed an on-top exchange-correlation functional, tB4LYP, translated from the successful hybrid B3LYP in GKS-DFT. Our benchmarks show that tB4LYP provides a balanced and accurate description for various properties, with an overall MAE that surpasses the parent methods of tBLYP, tB3LYP, CASSCF, and even CASPT2. Notably, similarly to the conventional PDFT methods88,89,90, the extra cost of tB4LYP is negligible compared to CASSCF, and is much lower than that of CASPT2 (see Supplementary Table S20), especially when large active space is employed. We anticipate the tB4LYP method to be a useful alternative to CASPT2, especially when CASPT2 becomes too expensive to be practically feasible.
Methods
Dunning’s correlation consistent basis sets at triple-ζ and quadruple-ζ quality, denoted as cc-pVT(Q)Z, were used for all the main group elements65,66. For transition metals, the scalar relativistic effect was included via the second order Douglas–Kroll (DK) Hamiltonian and its associated DK basis sets (cc-pVTZ-DK and cc-pVQZ-DK)67. The basis set extrapolation formula is EN = E∞ + cN−3 91. The DIRAC program68,69 were utilized for the FSCC calculations, the PySCF package92,93,94 was used for the wavefunction stability analyzes for the UHF and UKS-DFT calculations on a few important points of the potential energy surfaces of molecules from Fig. 1 and 3 (see Supplementary Table S21), and all the rest of the calculations were performed with a locally modified version of the OpenMolcas program package70. Unless otherwise stated in the context, the integral cutoff threshold for the atomic basis sets is 10−16, the grid used for numerical quadrature in DFT calculations is defined as ULTRAFINE according to the OpenMolcas definition. The convergence thresholds for CASSCF are 10−9, 10−5, and 10−5 for energy, orbital rotation matrix, and energy gradient, respectively. For the CASPT2 part of the Cr2 potential energy curves, an imaginary shift of 0.2 a.u. is applied, and the DMRG-CASSCF convergence thresholds are 10−8, 5 × 10−5, and 5 × 10−5 for energy, orbital rotation matrix, and energy gradient, respectively.
Diatomics
The potential energy curves of the ground-states of N2 (\({}^{1}{\Sigma }_{g}^{+}\)), CO(1Σ+), O2 (\({}^{3}{\Sigma }_{g}^{-}\)), CN (2Σ+), Sc2 (X\({}^{5}{\Sigma }_{u}^{-}\)), Ti2(X3Δg), V2 (X\({}^{3}{\Sigma }_{g}^{-}\)), Cr2 (X\({}^{1}{\Sigma }_{g}^{+}\)), Ni2 (X1Γg), and Cu2 (X\({}^{1}{\Sigma }_{g}^{+}\)) molecules were plotted and compared with other methods including the MC-PDFT (tBLYP) method, the HMC-PDFT (tB3LYP) method and the complete active space second-order perturbation theory (CASPT2)58,59. The reference CASSCF14,15 active space choices are small for the nonmetallic diatomics, they are CAS(10e, 8o), CAS(10e, 8o), CAS(12e, 8o), and CAS(9e, 8o) for N2, CO, O2, and CN, respectively, where CAS(Xe, Yo) means X electrons in Y orbitals for all possible occupations. For Cr2, the active space choice is CAS(12e, 22o) used in the Density Matrix Renormalization Group (DMRG) calculations95,96,97,98,99,100,101. For other metal dimers the active spaces are minimal, they are selected to include MOs composed of only the 4s3d atomic orbitals. The same active space definitions were used for CASPT2. In CASPT2, the default ionization-potential-electron-affinity (IPEA) shift to the zeroth order Hamiltonian of 0.25 Hartree was used for all cases102. No space symmetry constraint was used and all the electrons were correlated in the PT2 procedure, unless otherwise stated.
The original potential energy plots were interpolated in cubic spline functions over 300 to 800 grid points in all the figures. For the diatomic constatns of N2 (X\({}^{1}{\Sigma }_{g}^{+}\)), CO(X1Σ+), O2 (X\({}^{3}{\Sigma }_{g}^{-}\)), and CN (X2Σ+), a diatomic potential energy curve is firstly fitted using a weighted least squares approach103, and then the numbers are obtained from second order vibrational perturbation theory, the atomic masses used for C, N, O are 12.0000000 u, 14.0030740 u, 15.9949146 u, respectively, the PSI4 package has been employed in the process104. For the vibrational spectrum of Cr2 (X\({}^{1}{\Sigma }_{g}^{+}\)), the discrete variable representation (DVR)105 method is used, the atomic mass used for Cr is 51.9405115 u.
The near equilibrium potential energy curves of Sc2 (X\({}^{5}{\Sigma }_{u}^{-}\)), Ti2(X3Δg), V2 (X\({}^{3}{\Sigma }_{g}^{-}\)), Ni2 (X1Γg), and Cu2 (X\({}^{1}{\Sigma }_{g}^{+}\)) were calculated in a 60 to 80-point mesh around the equilibrium bond length with a step size of 10−2 Å. For the vibrational constants a diatomic potential energy curve is firstly fitted using a weighted least squares approach103, and then the numbers are obtained from second order vibrational perturbation theory, the atomic masses used for Sc, Ti, V, Ni, and Cu are 44.95590828 u, 47.94794198 u, 50.94395704 u, 57.93534241 u, and 62.92959772 u. The states were identified by spin and spacial symmetry as well as the expectation values of the \({L}_{z}^{2}\) operator. The plots are shown in Supplementary Fig. S8.
Ethylene torsion and ethyne dissociation
The ethylene torsional rotation follows
with constrained C-H bond length and H-C-H bond angle at 1.086 Å and 121.2°, respectively. C2 point group symmetry was used for the torsional rotation. The ethyne diabatic colinear dissociation follows
conducted with constrained C-H bond at 1.058 Å. D2h point group symmetry was used for the dissociation. The composite method used in ethylene torsion is formulated as
where EFSCCSD uses the HF closed-shell orbitals of \({{{\rm{C}}}}_{2}{{{\rm{H}}}}_{4}^{2+}\) and the (0, 2) sector to calculate the energies, the number of active particle orbitals are 2 in each spinor. EOR is the orbital relaxation effect compensated by an ad hoc MRCI procedure. It is taken as EOR = EMRCI − EUMRCI, where EUMRCI is the MRCI energy using the HF closed-shell orbitals of \({{{\rm{C}}}}_{2}{{{\rm{H}}}}_{4}^{2+}\) unrelaxed. The last term is the core-valence correlation energy contribution. It is taken as ECV = ECV MRCI − EMRCI, where ECV MRCI is the MRCI energy with all the electrons being correlated (including the core 1s electrons). It is worth mentioning that the FSCCSD singles includes a portion of the orbital relaxation effect, thus this approach poses the risk of overcounting the orbital relaxation effect. Valence cc-pVTZ basis sets have been used for the FSCCSD and orbital relaxation contributions, whereas the core-valence cc-pwCVTZ106 sets were used for the core-valence contribution. The individual contributions are EFSCCSD = 67.01 kcal ⋅ mol−1, EOR = 3.90 kcal ⋅ mol−1, and ECV = 0.28 kcal ⋅ mol−1.
For ethyne dissociation, the dissociation energies with relaxed C-H (4Σ−) bond (1.083 Å) calculated by CASSCF, CASPT2, tBLYP, tB3LYP, tB4LYP, and EOMCC are 236.19, 260.04, 275.13, 269.17, 285.77, and ca. 290 kcal ⋅ mol−1, respectively. The EOMCC value is only roughly estimated to be ca. 290 kcal ⋅ mol−1 since all the points on the plotted curve are calculated with fixed C-H bond lengths at 1.058 Å, which is only 0.025 Å smaller than the relaxed one63. The experimental value (269.20 kcal ⋅ mol−1) is obtained from extrapolated enthalpies of formation at zero K (236.60 kcal ⋅ mol−1) of C2H2 and CH, the verticle excitation energy (33.42 kcal ⋅ mol−1) of CH (X2Π to 4Σ−), and ZPE corrections (−0.82 kcal ⋅ mol−1) to the first-order anharmonicity constants53,54,55. It is significantly smaller than tB4LYP, EOMCC, and GMCPT81 results. To make a definitive conclusion, we also did single-reference coupled cluster singles and doubles with perturbative triples [CCSD(T)] calculations with all the electrons correlated in core-valence quadruple-ζ (cc-pwCVQZ) basis sets and obtained a dissociation energy of 268.56 kcal ⋅ mol−1.
Benchmarks
The tB4LYP method was benchmarked against two sets of strongly multi-reference diatomic dissociation energies selected from the Minnesota Database 201949,50, the first includes 9 main group nonmetal diatomic (MGDBE9) molecules.
The second set is comprised of 8 transition metal (TMDBE8) molecules including three metal-metal dimers. The details of the two benchmark sets along with the choices of the active spaces can be found in Supplementary Table S5. The dissociation energies were obtained with the total energies calculated at the 10 or larger Angstroms bond lengths minus the total energies calculated at the equilibrium bond lengths. The mean absolute errors (MAEs) of all the tested methods are calculated. Additionally, 11 spin-flip atomic excitation energies (SFAEE11) were also benchmarked to test tB4LYP’s performance on electron excitation energies. The same SFAEE11 set was used in the linear hybrid MC-PDFT paper for benchmarking30. The calculated individual binding energies can differ slightly from those obtained from the dissociation curves (<0.5 kcal ⋅ mol−1). This is due to minor differences in the equilibrium bond lengths or the literature benchmark binding energy values.
The atomic states in the SFAEE11 set are determined through state-averaged CASSCF calculations without spatial symmetry. Due to the limitations of the program, the orbitals are optimized separately for different spin symmetries. A single root CASCI calculation is then conducted with the optimized orbitals from the state-averaged CASSCF, and finally the (H)MC-PDFT calculations are carried out with the CASCI results. However, since the orbitals are not optimized across different spin states, the final energy might be biased towards one specific spin state. To assess such bias, we employed PySCF92,93,94 for state-averaged CASSCF with different spin states, and the MC-PDFT (tBLYP) calculations were made available with extensions implemented by Gagliadi’s group107. We also implemented the tB4LYP method based on their code and calculated state-averaged tB4LYP numbers with the SFAEE11 set. The differences are presented in Supplementary Table S16.
Other details
The one-parameter GKS hybrid B1LYP takes the form of
To establish the on-top exchange-correlation functional based on the new HMC-PDFT formula proposed in this work (Equation(7)), we have the tB2LYP method written as
The corresponding tB1LYP using the HMC-PDFT formula porposed by Mostafanejad et al.22,30 (Equation(2)) is formulated as
The heat map provided in Supplementary Fig. S4 presents the overall MAEs of tB1LYP and tB2LYP with varying hybrid parameter of λ ∈ (0, 1) on the threes data sets of MGDBE9, TMDBE8 and SFAEE11. The results of tB1LYP are shown on the leftmost column; while those of tB2LYP are shown on the right with λ ranging from 0.1 to 1.0 and the entropy-scaling parameter θ ranging from 0.0 to 9.0 mHartrees. The upper map is for all three data sets and the lower map is for the TMDBE8 data set only.
Leaving out the cross-entropy contribution in tB4LYP, we have yet another hybrid functional called tB3LYPfc written as
which differs the tB3LYP functional from taking a full piece of DFT correlation.
The color representation of the plotted curves follows the perceptual uniformity and order108. The hex color codes are 104261, fbcdfa, 4d734d, 818332, bf9035, f19e6a, and fdb4b3 for darkslategray, thistle, darkolivegreen, olive, darkgoldenrod, sandybrown, and lightpink, respectively.
Data availability
The data needed to recreated Figs 1–3, and Supplementary Fig. S1–S9 are provided in the Source Data file. Source data are provided with this paper.
Code availability
The codes for the tB4LYP method are available via the tB4LYP branch of the OpenMolcas official gitlab page at https://gitlab.com/Molcas/OpenMolcas/-/tree/tB4LYP. A permanent copy of the code can also be downloaded via the DOI at https://doi.org/10.5281/zenodo.14410416.
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Acknowledgements
The authors acknowledge the funding support by the National Natural Science Foundation of China with the following approval codes: 22125301 (I.Y.Z.), 22393911 (X.X.), 22393912 (I.Y.Z.), 22321003 (X.X.), 22303015 (R.F.), 22233002 (I.Y.Z.), the Innovation Program for Quantum Science and Technology (2021ZD0303305), Innovative research team of high-level local universities in Shanghai and a key laboratory program of the Education Commission of Shanghai Municipality (ZDSYS14005). The authors would like to thank Prof. Donald G. Truhlar from the University of Minnesota for useful discussions. His suggestion of studying the on-top hybrid functionals translated from B1LYP leads to strong evidence for the general validity of the HMC-PDFT scheme proposed in the work. One of the authors (R.F.) thanks Prof. Xinlu Cheng from Sichuan University (CN dissociation) and Dr. Y. Sajeev from Bhabha Atomic Research Centre (ethyne dissociation) for providing valuable data for benchmark.
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I.Y.Z. and X.X. designed the study. R.F. and I.Y.Z. performed research and analyzed data. R.F., I.Y.Z., and X.X. wrote the paper.
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Feng, R., Zhang, I.Y. & Xu, X. A cross-entropy corrected hybrid multiconfiguration pair-density functional theory for complex molecular systems. Nat Commun 16, 235 (2025). https://doi.org/10.1038/s41467-024-55524-z
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DOI: https://doi.org/10.1038/s41467-024-55524-z
