theories

Discussion with Prof. Walter Kohn

MAIN ACHIEVENENTS OF THEORIES

Tsuneda has developed density functional theory for chemistry targeting on the development of theories reproducing chemistry comprehensively. [Density Functional Theory in Chemistry (Springer Nature, 2014)]

1. Physically-established minimum-parameter exchange & correlation functionals and regional self-interaction correction

Conventional development of DFT functionals has primarily focused on reproducing properties using numerous semiempirical parameters. However, the use of such parameters often reduces the physical interpretability of the functionals and introduces spurious local minima in potential energy surfaces. Tsuneda and coworkers addressed this issue by developing exchange-correlation functionals with a minimal number of semiempirical parameters. Building on these developments, a regional self-interaction correction method was also introduced based on the regions of electronic motion derived from their work.

Fig. The shape of OP correlation hole.

The one-parameter progressive (OP) correlation functional was developed using only a single semiempirical parameter. Without explicitly considering fundamental conditions, this functional is derived from a correlation wavefunction that satisfies the spin-polarized correlation cusp condition. The correlation hole is formed in conjunction with the exchange functional used, with its size determined by the single parameter. Applications to atoms have demonstrated that the OP functional provides accurate correlation energies. Surprisingly, despite not being explicitly designed to satisfy fundamental conditions, the functional was found to adhere to most of these conditions. [J. Chem. Phys., 110, 10664 - 10678, 1999; ibid. 111, 5656 - 5667, 1999; Chem. Phys. Lett., 268, 510-520, 1997]

Fig. The dependence of the density matrix expansion function on electron-electron distance.

Parameter-free (PF) exchange functional was developed without any semiempirical parameters. A key feature of the PF functional is that it is directly derived from the density matrix expansion function around the Fermi momentum and is determined using the kinetic energy density at each point in molecular space. Similar to the OP functional, the PF functional is found to satisfy many fundamental conditions and provides highly accurate exchange energies, all without relying on adjustable parameters. [Phys. Rev. B, 62, 15527 - 15531, 2000]

Fig. The transversing connection & self-interaction relation between kinetic, exchange and correlation energies.

Based on the PF exchange and OP correlation functionals, the transversing connection between the fundamental conditions of kinetic, exchange & correlation energies was elucidated. Additionally, a framework is proposed for classifying electronic motions into regions of free electrons, self-interaction, or long-range interaction. [J. Chem. Phys., 114, 6505 - 6513, 2001; ``Density Functional Theory in Quantum Chemistry'']

Fig. Self-interaction region (White) of electronic motions in formaldehyde molecule.

In self-interaction regions, where electrons do not interact with other electrons of the same spin, the transverse connection is not applicable, and an alternative relationship emerges through the self-interaction density matrix. The regional self-interaction correction (RSIC) was developed based on this alternative relationship. Applications of RSIC have shown that it significantly improves the accuracy of reaction barrier predictions in GGA calculations. [J. Comput. Chem., 24, 1592 - 1598, 2003]

2. Long-range correction: A solution to many DFT challenges

With the rise of DFT-based calculations in quantum chemistry during the late 1990s, various issues were reported in calculating chemical properties and reaction mechanisms. Long-range correction emerged as a solution that simultaneously addressed many of these problems. Today, this correction is incorporated into major functionals, commonly referred to as "range-separated" functionals, and is widely and often unconsciously used in quantum chemistry calculations.

Fig. The range separation of the two electron interaction operator into short- (yellow) & long-range (red) regions.

Long-range correction (LC) was developed specifically for exchange functionals. By the early 2000s, exchange functionals based on density, its gradient, or hybrids incorporating Hartree-Fock (HF) exchange were found to exhibit significant shortcomings in quantum chemistry calculations. LC provides a physically grounded solution by dividing the electron-electron interaction operator into short-range and long-range components and correcting the long-range exchange part using the Hartree-Fock exchange integral for the long-range region. Applications of LC have demonstrated that it resolves most major issues in DFT calculations arising from the limitations of traditional exchange functionals. As a result, LC is now implemented in the official versions of many quantum chemistry software packages, such as Gaussian16. Furthermore, LC has paved the way for the development of numerous LC-based functionals, such as ωB97XD and CAM-B3LYP, which have become dominant tools in modern DFT calculations. [J. Chem. Phys., 115, 3540 - 3544, 2001]

Fig. The percentage errors in the calculated binding energies for various types of complexes: dispersion complexes (1–8), stacking complexes (9–15), dipole-induced dipole complexes (16–20), dipole-dipole complexes (21–25), and hydrogen bond complexes (26–32).

Conventional DFT has struggled to accurately reproduce van der Waals bonds, which often play a crucial role in determining the structures of large-scale systems. This failure is attributed not only to the absence of van der Waals interactions in correlation functionals but also to the lack of long-range exchange interactions. To address this issue, the ``LC+vdW method’’, which combines long-range correction (LC) with van der Waals interactions, was developed and extensively applied to weakly bonded systems. As a result, this method has been shown to successfully reproduce highly accurate descriptions of weak bonds. [J. Chem. Phys., 117, 6010 - 6015, 2002; ibid. 123, 124307(1-10), 2005; Mol. Phys. (Handy special issue), 103, 1151 - 1164, 2005; J. Chem. Phys., 126, 234114(1-12), 2007; 'π-Stacked Polymers and Molecules' (Springer, 2013)]

Fig. Comparison of the calculated hyperpolarizability of the p-quinodimethane model as a function of the diradical ｙ value (x102 a.u.) .

Conventional DFT has long faced challenges in accurately predicting higher-order response properties, often producing results that deviate significantly from experimental observations. To overcome this limitation, the coupled perturbed Kohn-Sham method based on LC-DFT was developed and applied to response property calculations. This approach has been shown to accurately reproduce nonlinear optical response properties, such as first- and second-order hyperpolarizabilities. Notably, it dramatically improves the prediction of higher-order optical response properties for systems like long-chain polyenes and diradicals, bringing the results much closer to those obtained with high-level ab initio methods. [J. Chem. Phys., 122, 234111(1-10), 2005; J. Chem. Phys., 132, 094107(1 - 11), 2010]

3. Time-dependent DFT with highest accuracy & its application to photochemical reaction simulations

One of the most significant issues resolved by LC is the underestimation of charge-transfer excitation energies in time-dependent DFT (TDDFT) calculations. Charge-transfer excitations often serve as precursors to major photochemical reactions, which made conventional TDDFT unsuitable for accurately describing these processes. By addressing this limitation, LC has completely overcome the problem, enabling precise simulations of photochemical reactions.

Fig. The dependence of calculated charge transfer excitation energies of ethylene-tetrafluoroethylene complex on the bond distance between the fragments.

TDDFT is a widely used method for efficiently and rapidly calculating excitation energies for molecules and large-scale systems. However, it faces significant challenges in accurately predicting charge-transfer excitations, Rydberg excitations, and oscillator strengths. These issues are primarily attributed to the absence of long-range exchange effects in conventional TDDFT. To address these limitations, LC-TDDFT, based on long-range corrected DFT, was developed. Applications of LC-TDDFT have demonstrated that it resolves or significantly improves these issues, effectively correcting the underestimation of charge-transfer and Rydberg excitation energies and providing more accurate oscillator strengths. [J. Chem. Phys. 120, 8425 - 8433, 2004]

Fig. Structures of (a) 4-1-pyrroryl-pyridine and (b) 4-cyano-4-methylthiodiphenylacetylene before and after excited-state geometry optimizations.

Most photochemical reactions are initially driven by long-range charge-transfer excitations. While TDDFT is widely used for simulating excited states in large-scale systems, it struggles to accurately calculate long-range charge-transfer excitations. To address this limitation, an excited state energy gradient calculation method of LC-TDDFT was developed. As a first step, the excited-state geometries and adiabatic excitation energies of small molecules were calculated. The results revealed that long-range exchange effects are essential even for accurately determining the excited-state geometries of small molecules. Additionally, this method facilitates the simulation of excited-state molecular dynamics, further expanding its applicability to photochemical reaction studies. [J. Chem. Phys., 123, 144106(1-11), 2006]

Fig. Calculated Koopmans P3/2 & P1/2 ionization energies of rare gas atoms.

Relativistic spin-orbit interactions significantly influence the excitation energies and excited-state reactions of systems containing heavy atoms. To investigate these effects, a relativistic spin-orbit LC-TDDFT was developed, incorporating two-component spin-orbit interactions into LC-TDDFT. Given that transitions between orbitals with different electron distributions are critical in spin-orbit transitions, the inclusion of long-range correction (LC) is expected to play a crucial role. The calculated results reveal that LC has a substantial impact on spin-orbit interactions for heavy atoms up to the sixth period, and that orbital spinors remain important even for relatively lighter atoms. [J. Chem. Phys. 135, 224106(1-9), 2011]

Fig. Total CPU time and computational scaling in state-selective TDDFT calculations for water clusters.

To enable efficient TDDFT calculations for large-scale systems, a state-selective TDDFT algorithm was developed. This algorithm identifies and selects only the transitions likely to contribute significantly to the excitations using a perturbation-based selection process, thereby reducing computational time. Furthermore, the algorithm achieves linear-scaling performance when combined with linear-scaling DFT calculations. Applied to LC-TDDFT, this algorithm significantly accelerates calculations for charge-transfer excitations, making it particularly useful for large and complex systems. [J. Theor. Comp. Chem. (APCTCC Special Issue), 4, 265 - 280, 2005;Chem. Phys. Lett., 420, 391 - 396, 2006;J. Comput. Chem., 29, 1187 - 1197, 2008]

Fig. Dependence of the calculated excitation energies of linear oligoacene molecules on the number of repeating units.

For linearly extended long-chain systems, such as polyacetylenes and oligoacenes, spin-flip (SF) LC-TDDFT was developed to incorporate double-excitation effects into LC-TDDFT. This method was applied to investigate the dependence of calculated excitation energies on the number of repeating units in these systems. The results demonstrate that SF-LC-TDDFT reproduces excitation energies with remarkable accuracy, comparable to that of multireference theories. Even today, this method remains one of the most accurate approaches within the framework of TDDFT. [J. Comput. Chem., 37, 1451-1462, 2016.]

4. First-ever quantitative calculations of occupied & unoccupied orbital energies & comprehensive orbital energy reproduction

One of the most remarkable achievements of LC is its ability to simultaneously and accurately reproduce both occupied and unoccupied orbital energies for the first time. In quantum chemistry, orbital energies were traditionally considered unreproducible and largely meaningless properties. Recent advancements in highly accurate exchange-correlation potential functionals have enabled the precise calculation of valence occupied orbital energies. However, these functionals fail to qualitatively reproduce unoccupied orbital energies, often even getting the signs incorrect. Surprisingly, LC successfully resolves this issue, providing a comprehensive and accurate reproduction of both occupied and unoccupied orbital energies.

Fig. Dependence of the calculated two-electron integral kernels of the He atom on the distance from the nucleus.

Since the development of the Hartree-Fock equation—the first self-consistent field (SCF) equation for one-electron systems—there had been no theory capable of quantitatively reproducing orbital energies for 80 years. Remarkably, LC has been found to enable the simultaneous accurate reproduction of occupied & unoccupied orbital energies for the first time. Furthermore, LC significantly reduces the self-interaction error in the exchange integral kernel, which is the second derivative of the energy functional with respect to electron density. This self-interaction error is a major cause of poor orbital energy predictions. Additionally, LC-DFT is now recognized as the only approach capable of quantitatively reproducing the band gaps of insulators. [J. Chem. Phys., 133, 174101(1-9), 2010]

Fig. Errors in the calculated core 1s orbital energies for second-row atoms in typical molecules.

While LC-DFT provides highly accurate valence orbital energies, it tends to underestimate core orbital energies. Additionally, LC-DFT underestimates the HOMO energies of hydrogen and rare gas atoms. To address this issue, the PSRSIC method, which incorporates pseudospectral HF exchange into the self-interaction region within the RSIC framework, was combined with LC-DFT, resulting in the LC-PRSIC functional. Applications of the LC-PRSIC functional demonstrate that it significantly improves core orbital energies and the HOMO energies of hydrogen and rare gas atoms while maintaining the high accuracy of valence orbital energies. [J. Chem. Phys. 139, 064102(1-10), 2013]

5. Reaction electronic theory based on orbital energy variance: A step toward the unification of reaction theories

Integrating the quantitative orbital energies provided by LC-DFT into reaction electronic theories, such as the frontier orbital theory, has led to the development of a quantitative reaction electronic theory. Building on the observation of the initial invariance in the HOMO-LUMO gap during a Diels-Alder reaction, a reactive orbital energy theory was developed to analyze reactions based on reactive orbital energy variance. This theory establishes a one-to-one correspondence between reactive orbitals and reaction pathways. Moreover, analysis using this theory reveals that the initial charge transfers between reactive orbitals directly correspond to the arrows used in the electronic theory of organic chemistry, providing a deeper and more quantitative understanding of reaction mechanisms.

Fig. Variation of global hardness during the forward process of the ethylene + butadiene Diels-Alder reaction.

The quantitative HOMO-LUMO gaps calculated using LC-DFT are found to remain nearly constant during the initial stages of many reactions. This stability arises because reactions are typically driven by charge transfers. For instance, LC-DFT calculations of the Diels-Alder reaction reveal that the global hardness (the half of the HOMO-LUMO gap) hardly change in the initial forward process of a Diels-Alder reaction reveal that the global hardness (half of the HOMO-LUMO gap) shows minimal variation during the initial forward process. This observation suggests that the early stages of the reaction are primarily governed by charge transfer interactions. [J. Comput. Chem., 34, 379-386, 2013]

Fig. Normalized reaction diagram evaluating charge transferability based on the normalized gradient of the reactive orbital energy gap.

Using the quantitative orbital energies provided by LC-DFT, a reactive orbital energy theory was developed to identify the reactive orbitals driving chemical reactions and to determine the driving forces behind them. This theory analyzes reactions based on orbital energy variance, where reactive orbitals are defined as the pair of occupied and unoccupied orbitals exhibiting the largest variance in orbital energies. Additionally, the charge transferability is assessed by evaluating the gradient of the reactive orbital energy gap at the initial stage of the reaction: a small gradient indicates a charge transfer-driven reaction, while a large gradient suggests a reaction driven by structural deformation (dynamics). [J. Comput. Chem., 35, 1093-1100, 2014; Computation, 4, 23 (1-13), 2016]

Fig. Reactive orbital energy diagram for the ethylene + butadiene Diels-Alder reaction.

In reactive orbital energy theory (ROET) analyses, the variations in reactive orbitals provide a visualization of the electronic motions driving chemical reactions. In collaboration with Prof. Chattaraj, an authority in conceptual DFT, quantitative reaction electronic theory diagrams called ``ROET diagrams'', were developed to efficiently visualize reaction mechanisms. To investigate the impact of the theoretical level on ROET analyses, the LC+LRD method was employed to calculate orbital energies. The results indicate that ROET diagram analyses, which rely on highly accurate orbital energies, can effectively and clearly visualize the electronic motions driving reactions in detail. However, the analyses are significantly influenced by the level of theory used, due to their dependence on precise orbital energy calculations. [Phys. Chem. Chem. Phys., 20, 14211-14222, 2018]

Fig. The occupied and unoccupied reactive orbital pairs corresponding to 30 reaction pathways in the global reaction route map of glycine production.

The occupied and unoccupied reactive orbital pair identified through ROET analysis can be interpreted as an inherent property of each reaction pathway. To investigate the relationship between reactive orbital pairs and reaction pathways, ROET analysis was applied to the global reaction route map for glycine production, encompassing 30 reaction pathways. Remarkably, the analysis revealed that the reactive orbital pair is unique to each reaction pathway, with no repetition across the paths. This finding establishes a one-to-one correspondence between reactive orbital pairs & reaction pathways. Furthermore, it suggests a close connection between ROET and potential energy theory, despite their independent development. [J. Chem. Theory Comput., 17, 6901-6909, 2021]

Fig. The correspondence between ROET analysis diagrams and the electronic theory of organic chemistry.

As a nonempirical reaction electronic theory, ROET can be utilized to investigate the rule-based electronic theory diagrams commonly used in organic chemistry. Aldol and Mannich reactions, which involve C-C bond formation and can proceed under both acidic and basic conditions, serve as ideal candidates for this purpose. Applying ROET analysis to these reactions reveals that the reactive orbital pairs of the initial processes closely correspond to the arrows used in the electronic theory diagrams of organic chemistry. This finding suggests that ROET has the potential to provide a foundational framework for the electronic theory of organic chemistry. [J. Comput. Chem., 44, 2391-2403, 2023.]

Fig. Effects of long-range exchange and CCSD on total electron density and orbital density.

This study clarified the effects of long-range exchange and CCSD on total electron density and orbital density, investigating the requirements for reproducing high-accuracy electron densities. As DFT is fundamentally designed to derive potentials from electron densities, the accurate reproduction of these densities is critically important. The results showed that, for total electron density, long-range exchange tends to concentrate electrons around π orbitals, while CCSD effects primarily concentrate electrons in bonding regions. Conversely, when comparing orbital densities using Dyson orbitals, both methods exhibited significant agreement in their effects on unoccupied orbitals. [J. Comput. Chem., 44, 2391-2403, 2023.]

Fig. A strong agreement between valence electron densities observed through synchrotron X-ray diffraction and calculated using long-range corrected DFT.

Using synchrotron X-ray diffraction experiments on molecular crystals, we compared the valence electron density distribution observed experimentally with the valence electron density distribution calculated using LC-DFT. The results revealed an astonishing level of agreement. In particular, for sp3 hybrid orbitals, detailed features such as the phase and distribution were found to match closely between the experimental and theoretical data. Building on this finding, we succeeded in isolating textbook-like 2pπ orbitals by subtracting the molecular orbital density obtained via LC-DFT from the experimentally observed valence electron density. This remarkable agreement strongly suggests the potential for theoretical analyses based on experimentally derived molecular orbital data, opening new avenues for quantum chemical research. [J. Am. Chem. Soc., 146, 23825-23830, 2024.]

Fig. The role of reactive-orbital-based electrostatic forces in driving nuclear motion.

Conventional theories of chemical reactivity have not elucidated how the motion of electrons drives chemical reactions. In this study, we calculated the electrostatic forces exerted by reactive orbitals, as defined in the reactive orbital energy theory (ROET), on atomic nuclei. The results revealed that these forces propel the nuclei along the reaction direction. This force generates an intrinsic reaction coordinate on the potential energy surface. Furthermore, we demonstrated that the gradient of orbital energy determines the direction of the electrostatic force, and that the sum of these gradients corresponds to the total electrostatic force exerted by all electrons. These findings establish a direct connection between electronic theory and potential energy-based reaction analysis. [Comm. Chem., 8, 158(1-7), 2025.]

6. Development of official versions of quantum chemistry calculation programs and implementation of developed theories

The theories developed by our group have been implemented in various quantum chemistry calculation programs. In the official version of Gaussian16, one of the most widely used quantum chemistry programs, long-range corrected functionals (e.g., LC-BLYP) are available as standard features and have become a significant force in the field. DFT calculation program in the official version of the GAMESS is based on the developments by Tsuneda, Kamiya, Chiba, and Fedorov, and it includes standard implementations of long-range corrected functionals, long-range corrected TDDFT and its energy gradient calculations, as well as the OP correlation functional. Additionally, long-range corrected functionals are also available in the official versions of other major quantum chemistry programs such as NWChem, Q-Chem, and Amsterdam Density Functional (ADF). Furthermore, in the official version of the periodic system calculation program Dmol3, the OP correlation functional is available for solid-state calculations.

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