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, 2014)]

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

Conventional developments of DFT functionals have focused only on the reproducibility of properties using many semiempirical parameters. Semiempirical parameters diminish the physical meaning of functionals and cause spurious local minima in potential energy surfaces. Tsuneda and coworkers developed exchange & correlation functionals with keeping a minimum number of semiempirical parameters. Based on the region of electronic motions that are derived from these developments, the regional self-interaction correction is also developed.

Fig. The shape of OP correlation hole.

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

  • Parameter-free (PF) exchange functional is developed using no semiempirical parameter. The features of the PF functional is that it is directly derived from the density matrix expansion function around the Fermi momentum and it is determined using the kinetic energy density at each point of molecular space. Similar to the OP functional, this PF functional is found to satisfy many fundamental conditions and give very accurate exchange energies even without parameters. [Phys. Rev. B, 62, 15527 - 15531, 2000]

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

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

  • In self-interaction regions where electrons have no interaction with other electrons of the same spin, the transversing connection is not approved & another relation presents through the self-interaction density matrix. The regional self-interaction correction (RSIC) is developed on the basis of the latter relation. In the applications, it is found that the RSIC considerably improves some reaction barriers of GGA calculations. [J. Comput. Chem., 24, 1592 - 1598, 2003]

2. Long-range correction that simultaneously solves many DFT problems

As the increase of DFT calculations in quantum chemistry calculations in the late 90's, various problems had been reported in the calculations of chemical properties & reactions. Long-range correction simultaneously solves many of these problems. Nowadays, this correction is installed in major functionals, which are also called ``range-separation'' functionals, and is 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) is developed for exchange functionals. Up until 00's, exchange functionals of density, its gradient and the hybrid of HF exchange had been reported to give serious problems in quantum chemistry calculations. LC is a physically-established correction that divides the electron-electron interaction operator into the short- & long-range parts, & corrects for the long-range exchange part using the long-range part of the Hartree-Fock exchange integral. The applications have confirmed that LC solves most serious DFT problems coming from the failures of exchange functionals. Consequently, LC is now available in the official version of most quantum chemistry calculation programs such as Gaussian16. LC also leads to the development of many LC functionals such as ωB97XD & CAM-B3LYP, which now become a great force in DFT calculations. [J. Chem. Phys., 115, 3540 - 3544, 2001]

図．The percent errors of the calculated binding energies of dispersion complexes (1 - 8), stacking complexes (9 - 15) dipole-induced dipole complexes (16 - 20), dipole-dipole complexes (21 - 25), & hydrogen bond complexes (26 - 32).

Fig. The comparison of the dependence of the calculated hyperpolarizability of p-quinodimethanemodel on diradical ｙ value (x102 a.u.) .

  • Conventional DFT had provided poor high-order response properties that are strongly inconsistent with experimental values. Coupled perturbed Kohn-Sham method based on LC-DFT is developed & applied to response property calculations. As a result, it is found that this method accurately reproduces nonlinear optical response properties such as first- & second-order hyperpolarizabilities. In particular, this method dramatically improves the high-order optical response properties of long-chain polyenes & diradicals close to the results of 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 of the top highest accuracy & photochemical reaction simulation theory based on it

The most impactful problem that LC solves is the underestimation of charge transfer excitation energies in time-dependent DFT (TDDFT) calculations. Since charge transfer excitations are usually the precursor of most major photochemical reactions, conventional TDDFT had not targeted on these photochemical reactions. LC completely solves this problem.

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 calculating excitation energies for molecules to large-scale systems easily & fast. However, this method has serious problems in the calculations of charge transfer excitations, Rydberg excitations & oscillator strengths. Considering that these problems are attributed to the lack of long-range exchange effects, TDDFT based on LC-DFT (LC-TDDFT) is developed. Applications of LC-TDDFT show that all these problems are solved or significantly improved for the underestimation of charge transfer & Rydberg excitation energies & oscillator strengths. [J. Chem. Phys. 120, 8425 - 8433, 2004]

Fig. The structures before & after the excited state geometry optimizations of (a) 4-1-pyrroryl-pyridine & (b) 4-cyano-4-methylthiodiphenylacetylene.

  • Most photochemical reactions are initially driven by long-range charge transfer excitations. Though TDDFT is expected to perform the excited state simulations of large-scale systems, it could not calculate long-range charge transfer excitations. To solve this problem, excited state energy gradient calculation method of LC-TDDFT is developed. The excited state geometries and adiabatic excitation energies of small molecules are calculated as the first trial. As a result, it is found that long-range exchange effects are required even for small molecules in the excited state geometry calculations. This method also enables us to perform excited state molecular dynamics simulations. [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 affect the excitation energies & excited state reactions of heavy atom-containing systems. To explore the effects, relativistic spin-orbit LC-TDDFT, which incorporates two-component spin-orbit interactions in LC-TDDFT, is developed. Since the transitions between the orbitals of different electron distributions are significant in spin-orbit transitions, LC is expected to play a significant role in these transitions. As a result, the calculated results show that LC significantly affects the spin-orbit interactions for heavy atoms lower than the sixth period & orbital spinors are important even for comparatively light atoms.[J. Chem. Phys. 135, 224106(1-9), 2011]

Fig. Total CPU time & its calculation order in state-selective TDDFT calculations of water clusters.

Fig. The dependence of the calculated excitation energies of linear oligoacene molecules on the number of units.

  • For linearly-extended long-chain systems, polyacetylenes & oligoacenes, spin-flip (SF) LC-TDDFT, which simply incorporates the double-excitation effects into LC-TDDFT, is developed and applied to explore the dependence of the calculated excitation energies on the number of the units. As a result, it is found that the SF-LC-TDDFT very accurately reproduces these excitation energies with the comparable accuracy of multireference theories. This method is TDDFT of the top highest accuracy even now.[J. Comput. Chem., 37, 1451-1462, 2016.]

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

The most surprising problem that LC solves is the first-ever simultaneous accurate reproduction of occupied & unoccupied orbital energies. In quantum chemistry, orbital energies had been accepted as unreproducible & meaningless properties. Recent development of highly-accurate exchange-correlation potential functionals fulfills the accurate valence occupied orbital energy calculations. However, these potential functionals cannot qualitatively reproduce unoccupied orbital energies even for the signs. Surprisingly, LC solves this problem.

Fig. The dependence of the calculated two-electron integral kernels of He atom on the distance from the nuclei.

  • Since the development of the Hartree-Fock equation, which is the first one-electron SCF equation, there had been no theory that can quantitatively reproduce orbital energies for 80 years. Surprisingly, it is found that LC enables the simultaneous accurate reproduction of occupied & unoccupied orbital energies for the first time ever. It is also revealed that LC dramatically reduces the self-interaction error of the exchange integral kernel, which is the second derivative of energy functional with respect to electron density. This self-interaction error causes poor orbital energies. In addition, LC-DFT is also accepted to be the only theory that can quantitatively reproduce the band gaps of insulators. [J. Chem. Phys., 133, 174101(1-9), 2010]

Fig. The errors of the calculated core 1s orbital energies of the second-row atoms of typical molecules.

  • Though LC-DFT provides very accurate valence orbital energies, it underestimates core orbital energies. LC-DFT also underestimates the HOMO energies of hydrogen & rare gas atoms. To solve this problem, PSRSIC, which incorporates the pseudospectral HF exchange into the self-interaction region in the RSIC method, is combined with LC-DFT: i.e., LC-PRSIC functional. Applying the LC-PRSIC functional shows that core orbital energies & the HOMO energies of H & rare gas atoms are all dramatically improved with keeping the accuracy of valence orbital energies. [J. Chem. Phys. 139, 064102(1-10), 2013]

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

Incorporating the quantitative orbital energies of LC-DFT to reaction electronic theories such as the frontier orbital theory leads to the development of a quantitative reaction electronic theory. Based on the finding of the initial invariance for the HOMO-LUMO gap of a Diels-Alder reaction, reactive orbital energy theory for discussing reactions on the basis of reactive orbital energy variance, is developed. It consequently becomes responsible for the finding of the one-to-one correspondence between reactive orbitals & reaction paths. Furthermore, the reactive orbital energy theory analysis reveals that the initial charge transfers between reactive orbitals correspond to the arrows in the electronic theory of organic chemistry.

Fig. The variance of global hardness in the forward process of the ethylene + butadiene Diels-Alder reaction.

Fig. Normalized reaction diagram, in which the charge transferability is evaluated as the normalized reactive orbital energy gap gradient．

  • Based on the quantitative orbital energies of LC-DFT, reactive orbital energy theory, which determines reactive orbitals driving reactions & the driving forces, is developed for discussing reactions on the basis of orbital energy variance. Reactive orbitals are the occupied & unoccupied orbital pair of the most varied orbital energies. Furthermore, the charge transferability is evaluated by the gradient of the reactive orbital energy gap at the initial reaction process: charge transfer-driven for the small gradient & structural deformation (dynamics)-driven for the large gradient. [J. Comput. Chem., 35, 1093-1100, 2014; Computation, 4, 23 (1-13), 2016]

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

  • In reactive orbital energy theory (ROET) analyses, the variation of reactive orbitals can visualize the electronic motions driving the reactions. Collaborating with Prof. Chattaraj, who is an authority of conceptual DFT, quantitative reaction electronic theory diagrams called ``ROET diagrams'' are constructed for the efficient visualization of reactions. To explore the effect of the level of the theory on the ROET analyses, the LC+LRD method is used to calculate the orbital energies. It is consequently found that the ROET diagram analyses, which require highly-accurate orbital energies, can clearly visualize the electronic motions driving the reactions in detail, though they considerably depend on the level of the theory due to the requirement of highly-accurate orbital energies. [Phys. Chem. Chem. Phys., 20, 14211-14222, 2018]

Fig. The occupied & unoccupied reactive orbital pairs for 30 reaction paths in the global reaction route map of glycine production reaction.

  • The occupied & unoccupied reactive orbital pair, that is determined by ROET analysis, are interpretable as an inherent property of each reaction path. To reveal the relation between the reactive orbital pair & reaction path, the ROET analysis is applied to the global reaction route map of glycine production reactions including 30 reaction paths. Surprisingly, it is found that the reactive orbital pair is not repeated for all reaction paths. This indicates the one-to-one correspondence between reactive orbital pairs & reaction paths & suggests that the ROET is closely related with potential energy theory, though these theories are independently developed. [J. Chem. Theory Comput., 17, 6901-6909, 2021]

Fig. The consistency between the diagrams of ROET analysis & the electronic theory of organic chemistry.

  • Since ROET is a nonempirical reaction electronic theory, it is available to explore the rule-based electronic theory diagrams of organic chemistry. Aldol & Mannich reactions of C-C bond formations are a best reactions for this purpose, because they proceed in both acidic & basic conditions. Applying the ROET analysis to these reactions, it is found that the reactive orbital pairs of the initial processes are almost identical to the arrows of the electronic theory diagrams of organic chemistry. This indicates that ROET can serve the foundation for the electronic theory of organic chemistry. [J. Comput. Chem., in press.]