Accurate Scaling Functions of the Scaled Schrödinger Equation. II. Variational Examination of the Correct Scaling Functions with the Free Complement Theory Applied to the Helium Atom.

In a previous paper [Phys. Rev. Lett. 2004, 93, 030403.], one of the authors introduced the scaled Schrödinger equation (SSE), g(H - E)ψ = 0 for atoms and molecules, where the scaling function g is the positive function of the electron-nuclear (e-n) and electron-electron (e-e) distances. The SSE is equivalent to the Schrödinger equation (SE), (H - E)ψ = 0, that governs the chemical world but does not have the divergence difficulty that occurs when we try to solve the SE to obtain the exact solution. The g function is essential not only to prevent this divergence difficulty but also to obtain the exact wave function of the SE or SSE. In paper I of this series [J. Chem. Phys. 2022, 156, 014113.], we introduced five analytical g functions that behave correctly at both the coalescence and asymptotic regions, but we examined them only for the e-e part. In this paper, we examine these correct g functions for both e-n and e-e parts by applying the free complement (complete-element) (FC) theory variationally to the He atom. However, even for the two-electron He atom, the analytical integral formulas were not obtained when we use the correct g functions for both e-n and e-e parts, except for g = 1 - exp(-γr), but we were able to perform variational FC calculations by employing numerical integration schemes. We examined not only the energy and wave function but also the H-square error (defined by eq 14 of the text), energy lower bound, and e-n and e-e cusp properties. For the energy lower bound, we applied our FC wave functions to the method proposed recently by Pollak, Martinazzo, and others and could obtain good results. With the use of the correct-group g functions, the convergence of the FC theory to the exact analytical solution of the SE or SSE became efficient, and the performance was particularly good with the g functions, r/(r + 1/γ), Ei, and 1 - exp(-γr) in this order. These results were always superior to those obtained with g = r.

Potential Energy Curves of the Low-Lying Five 1Σ+ and 1Π States of a CH+ Molecule Based on the Free Complement - Local Schrödinger Equation Theory and the Chemical Formula Theory.

The potential energy curves (PECs) of the low-lying five 1Σ+ and 1Π states (X1Σ+, C1Σ+, 31Σ+, A1Π, and D1Π states) of a CH+ molecule, an important interstellar molecule, were calculated by the free complement (FC) - local Schrödinger equation (LSE) theory with the direct local sampling scheme. The FC wave functions were constructed based on the chemical formula theory (CFT), whose local characters correspond to the covalent dissociations: C+(2P°(s2p))) + H(2S) of the X1Σ+ and A1Π states and the ionic dissociations: C(1D(s2p2)) + H+ of the C1Σ+ and D1Π states. All the calculated PECs were obtained with satisfying the chemical accuracy, i.e., error less than 1 kcal/mol, as absolute total energy of the Schrödinger equation without any energy shift. The spectroscopic data calculated from the PECs agreed well with both experimental and other accurate theoretical references. We also analyzed the wave functions using the inverse overlap weights proposed by Gallup et al. with the CFT configurations. For the X1Σ+ and A1Π states, the covalent C+(sp2) and C+(p3) configurations played important roles for bond formation. In the small internuclear distances of the C1Σ+, D1Π, and 31Σ+ states, the covalent character was also dominant as a result of the electron charge transfer from C to H+. Thus, the present FC-LSE results not only are accurate but also can provide chemical understanding according to the CFT.

Gaussian functions with odd power of r produced by the free complement theory.

We investigate, in this paper, the Gaussian (G) function with odd powers of r, rxaybzc exp(-αr2), called the r-Gaussian or simply the rG function. The reason we investigate this function here is that it is generated as the elements of the complement functions (cf's) when we apply the free complement (FC) theory for solving the Schrödinger equation to the initial functions composed of the Gaussian functions. This means that without the rG functions, the Gaussian set of functions cannot produce the exact solutions of the Schrödinger equation, showing the absolute importance of the rG functions in quantum chemistry. Actually, the rG functions drastically improve the wave function near the cusp region. This was shown by the applications of the present theory to the hydrogen and helium atoms. When we use the FC-sij theory, in which the inter-electron function rij is replaced with its square sij=rij2 that is integrable, we need only one- and two-electron integrals for the G and rG functions. The one-center one- and two-electron integrals of the rG functions are always available in a closed form. To calculate the integrals of the multi-centered rG functions, we proposed the rG-NG expansion method, in which an rG function is expanded by a linear combination of the G functions. The optimal exponents and coefficients of this expansion were given for N = 2, 3, 4, 5, 6, and 9. To show the accuracy and the usefulness of the rG-NG method, we applied the FC-sij theory to the hydrogen molecule.

Potential curves of the lower nine states of Li2 molecule: Accurate calculations with the free complement theory and the comparisons with the SAC/SAC-CI results.

The free-complement (FC) theory proposed for solving the Schrödinger equation of atoms and molecules highly accurately was applied to the calculations of the potential curves of the lower nine states of the Li2 molecule. The results were compared with the accurate experimental Rydberg-Klein-Rees potential curves available. They overlap completely with each other without any shift everywhere for all the states of Li2. At all the calculated points on the seven potential curves ranging between -14.83 and -15.00 hartree, the average difference was only 0.0583 kcal/mol and the maximum difference was only +0.165 kcal/mol. For the vertical excitation energies from the ground state curve to the seven excited states, the differences between theory and experiment were 0.000 645 eV in average and their maximum difference was -0.007 20 eV. The potential properties calculated with the FC theory also agreed well with the experimental values. These results show a high potentiality of the FC theory as a highly predictive quantum chemistry theory. For comparison, as an example of the Hartree-Fock based theory popular in modern quantum chemistry, we adopted the symmetry-adapted-cluster (SAC)-configuration-interaction (CI) theory using a highly flexible basis set. While the FC theory gave the absolute agreements with experiments, the SAC-CI potential curves compare reasonably well with experiments only after shifting-down of the SAC-CI curves by 5.727 kcal/mol. The differences in the excitation energies between SAC-CI and experiments were 0.004 28 eV on average, and the maximum difference was +0.109 67 eV. The SAC-CI results reported in 1985 were less accurate but still reasonable.

Accurate scaling functions of the scaled Schrödinger equation.

The scaling function g of the scaled Schrödinger equation (SSE) is generalized to obtain accurate solutions of the Schrödinger equation (SE) with the free complement (FC) theory. The electron-nuclear and electron-electron scaling functions, giA and gij, respectively, are generalized. From the relations between SE and SSE at the inter-particle distances being zero and infinity, the scaling function must satisfy the collisional (or coalescent) condition and the asymptotic condition, respectively. Based on these conditions, general scaling functions are classified into "correct" (satisfying both conditions), "reasonable" (satisfying only collisional condition), and "approximate but still useful" (not satisfying collisional condition) classes. Several analytical scaling functions are listed for each class. Popular functions riA and rij belong to the reasonable class. The qualities of many electron-electron scaling functions are examined variationally for the helium atom using the FC theory. Although the complement functions of FC theory are produced generally from both the potential and kinetic operators in the Hamiltonian, those produced from the kinetic operator were shown to be less important than those produced from the potential operator. Hence, we used only the complement functions produced from the potential operator and showed that the correct-class gij functions gave most accurate results and the reasonable-class functions were less accurate. Among the examined correct and reasonable functions, the conventional function rij was worst in accuracy, although it was still very accurate. Thus, we have many potentially accurate "correct" scaling functions for use in FC theory to solve the SEs of atoms and molecules.

Solving the Schrödinger equation of the hydrogen molecule with the free-complement variational theory: essentially exact potential curves and vibrational levels of the ground and excited states of Π symmetry.

Following a previous study of the Σ states (Phys. Chem. Chem. Phys., 2019, 21, 6327), we solved the Schrödinger equation (SE) of the hydrogen molecule in the ground and excited Π states using the free complement (FC) variational method. This method is a general method to solve the SE: the energies obtained are highly accurate and the potential energy curves dissociate correctly. The calculated energies are upper bound to the exact energies, and the wave functions at any distance are always orthogonal and Hamiltonian-orthogonal to those in the different states calculated in this study. Using the essentially exact potential energy curves, the vibrational energy levels of each state were calculated by solving the vibrational Schrödinger equation.

Light-Driven Proton, Sodium Ion, and Chloride Ion Transfer Mechanisms in Rhodopsins: SAC-CI Study.

Bacteriorhodopsin (BR) and halorhodopsin (HR) are well-known light-driven ion-pumping rhodopsins. BR transfers a proton from the intracellular medium to the extracellular medium. HR takes in chloride ion from the extracellular medium. A new light-driven sodium ion-pumping rhodopsin was discovered in 2013 by Inoue, Kandori, and co-workers ( Nat. Commun . 2013 , 4 , 1678 ). The purpose of this article is to elucidate the proton, sodium ion and chloride ion transfer mechanisms and the geometrical changes of the intermediates. The absorption maxima of three rhodopsins were calculated by the SAC/SAC-CI method using the QM/MM optimized geometries. For BR, the SAC-CI results supported the previously proposed proton-transfer mechanism; (1) the photoisomerization from all-trans to 13-cis retinal (K intermediate), (2) the relaxation of the retinal structure (L intermediate), (3) the proton transfer from the Schiff base to the counterion residue (ASP85) (M intermediate), (4) the proton transfer from the ASP96 to the Schiff base (N intermediate), and (5) the thermal isomerization from 13-cis to all-trans retinal (O intermediate). The proton releases to the extracellular medium through the ASP96, the Schiff base, the ASP85, and the GLU204 or GLU194 from the intracellular medium. Furthermore, it clarified that the guanidine group rotation of ARG82 changes the excitation energies of the L and N intermediates, but the effect is small for the resting state and the K, M, and O intermediates. The theoretical calculations suggested that the ARG82 rotation occurs in the N intermediate from the comparison between the experimental absorption spectra and the theoretical excitation energies. For the KR2, the Kandori group proposed the sodium ion transfer mechanism; (1) the photoisomerization from all-trans to 13-cis retinal (K intermediate), (2) the relaxation of the retinal structure (L intermediate), (3) the proton transfer from the Schiff base to the counterion residue (ASP116) (M intermediate), (4) the sodium ion passes through the cavity formed by the rotation of the counterion residue (ASP116) (O intermediate) and (5) the proton of the ASP116 reassociates to the Schiff base. The steps (1) to (3) are the same as ones of BR. The SAC-CI results supported the proposed sodium ion transfer mechanism and suggested that the sodium ion transfer proceeds in the O intermediate as follows; (1) the sodium ion connects with the Schiff base in the cavity formed by the ASP116 rotation, (2) at the same time that the sodium ion passes through the Schiff base, the Schiff base forms the hydrogen bond to the proton of ASP116, and (3) at the same time that the sodium ion transfers to the extracellular medium, the proton reassociates with the Schiff base from the ASP116. Furthermore, our results indicated that the retinal is not all-trans but 13-cis when the sodium ion passes through the Schiff base in the O intermediate. For the HR, since the counterion residue is replaced by the THR126, the proton dose not transfer from the Schiff base. Instead, the chloride ion transfers in the opposite direction to the proton of BR and the sodium ion of KR2. The SAC-CI results supported the previously proposed chloride ion transfer mechanism; (1) the photoisomerization from all-trans to 13-cis retinal (K intermediate), (2) the relaxation of the retinal structure (L intermediate), (3) the chloride ion passes through the Schiff base from the extracellular medium side to the intracellular medium side (N intermediate) and (4) the chloride ion transfer from the Schiff base to the intracellular medium and the thermal isomerization from 13-cis to all-trans retinal (O intermediate). Furthermore, our results suggested that the Schiff base forms bonds to the hydroxide ion instead of the chloride ion in the O intermediate. The negative ion is necessary to keep the total charge around the Schiff base in the O intermediate.

Solving the Schrödinger equation with the free-complement chemical-formula theory: Variational study of the ground and excited states of Be and Li atoms.

The chemical formula theory (CFT) proposed in Paper I of this series [H. Nakatsuji et al., J. Chem. Phys. 149, 114105 (2018)] is a simple variational electronic structure theory for atoms and molecules. The CFT constructs simple, conceptually useful wave functions for the ground and excited states, simultaneously, from the ground and excited states of the constituent atoms, reflecting the spirits of the chemical formulas. The CFT wave functions are also designed to be used as the initial wave functions of the free complement (FC) theory, that is, the exact theory producing the exact wave functions of the Schrödinger accuracy. This combined theory is referred to as the FC-CFT. We aim to construct an exact wave function theory that is useful not only quantitatively but also conceptually. This paper shows the atomic applications of the CFT and the FC-CFT. For simplicity, we choose the small atoms, Be and Li, and perform variational calculations to essentially exact levels. For these elements, a simple Hylleraas CI type formulation is known to be potentially highly accurate: we realize it with the CFT and the FC-CFT. Even from the CFT levels, the excitation energies to the Rydberg excited states were calculated satisfactorily. Then, with increasing the order of the FC theory in the FC-CFT, all the absolute energies and the excitation energies of the Be and Li atoms were improved uniformly and reached rapidly to the essentially exact levels in order 3 or 4 with moderately small calculational labors.

Photoelectron spectrum of NO2 - : SAC-CI gradient study of vibrational-rotational structures.

Three-dimensional accurate potential energy surfaces around the local minima of NO2 - and NO2 were calculated with the SAC/SAC-CI analytical energy gradient method. Therefrom, the ionization photoelectron spectra of NO2 - , the equilibrium geometries and adiabatic electron affinity of NO2 , and the vibrational frequencies including harmonicity and anharmonicity of NO2 - and NO2 were obtained. The calculated electron affinity was in reasonable agreement with the experimental value. The SAC-CI photoelectron spectra of NO2 - at 350 K and 700 K including the rotational effects were calculated using the Franck-Condon approximation. The theoretical spectra reproduced well the fine experimental photoelectron spectra observed by Ervin et al. (J. Phys. Chem. 1988, 92, 5405). The results showed that the ionizations from many vibrational excited states as well as the vibrational ground state are included in the experimental photoelectron spectra especially at 700 K and that the rotational effects are important to reproduce the experimental photoelectron spectra of both temperatures. The SAC/SAC-CI theoretical results supported the analyses of the spectra by Ervin et al., except that we could show some small contributions from the asymmetric-stretching mode of NO2 - . © 2018 Wiley Periodicals, Inc.

Solving the Schrödinger equation of hydrogen molecules with the free-complement variational theory: essentially exact potential curves and vibrational levels of the ground and excited states of the Σ symmetry.

The Schrödinger equation of hydrogen molecules was solved essentially exactly and systematically for calculating the potential energy curves of the electronic ground and excited states of the 1Σg, 1Σu, 3Σg, and 3Σu symmetries. The basic theory is the variational free complement theory, which is an exact general theory for solving the Schrödinger equation of atoms and molecules. The results are essentially exact with the absolute energies being correct beyond µ-hartree digits. Furthermore, all of the present wave functions satisfy correct orthogonalities and Hamiltonian-orthogonalities to each other at every nuclear distance along the potential curve, which makes systematic analyses and discussions possible among all the calculated electronic states. It is noteworthy that these conditions were not satisfied in many of the accurate calculations of H2 reported so far. Based on the present essentially exact potential curves, we calculated and analyzed the vibrational energy levels associated with all the electronic states. Among them, the excited states having double-well potentials showed some interesting features of the vibrational states. These results are worthy of future investigations in astronomical studies.

Solving the Schrödinger equation of atoms and molecules: Chemical-formula theory, free-complement chemical-formula theory, and intermediate variational theory.

Chemistry is governed by the principle of quantum mechanics as expressed by the Schrödinger equation (SE) and Dirac equation (DE). The exact general theory for solving these fundamental equations is therefore a key for formulating accurately predictive theory in chemical science. The free-complement (FC) theory for solving the SE of atoms and molecules proposed by one of the authors is such a general theory. On the other hand, the working theory most widely used in chemistry is the chemical formula that refers to the molecular structural formula and chemical reaction formula, collectively. There, the central concepts are the local atomic concept, transferability, and from-atoms-to-molecule concept. Since the chemical formula is the most successful working theory in chemistry ever existed, we formulate our FC theory to have the structure reflecting the chemical formula. Our basic postulate is that as far as the SE is the principle of chemistry, its solutions for chemistry should have the structure that can be related to the chemical formulas. So, in this paper, we first formulate a theory that designs the wave function to reflect the structure of the chemical formula. We call this theory chemical formula theory (CFT). In the CFT, we place the valence ground and excited states of each atom at each position of the chemical formula of the molecule and let them interact using their free valences to form the ground and excited states of the molecule. The principle there is the variational principle so that the ground and excited states obtained satisfy the orthogonality and Hamiltonian-orthogonality relations. Then, we formulate the exact FC theory starting from the initial functions produced by the CFT. This FC theory is referred to as free-complement chemical-formula theory (FC-CFT), which is expected to describe efficiently the solution of the SE by the above reason. The FC-CFT wave function is modified from that of CFT. Since this modification is done by the exact SE, its analysis may give some insights to chemists that assist their chemistry. Thus, this theory would be not only exact but also conceptually useful. Furthermore, the intermediate theory between CFT and FC-CFT would also be useful. There, we use only integratable functions and apply the variational principle so that we refer to this theory as FC-CFT-variational (FC-CFT-V). It is an advanced theory of CFT. Since the variational method is straightforward and powerful, we can do extensive chemical studies in a reasonable accuracy. After finishing such studies, if we still need an exact level of solutions, we add the remaining functions of the FC-CFT and perform the exact calculations. Furthermore, when we deal with large and even giant molecules, the inter-exchange (iExg) theory for the antisymmetry rule introduced previously leads to a large simplification. There, the inter-exchanges between distant electron pairs fade away so that only Coulombic interactions survive. Further in giant systems, even an electrostatic description becomes possible. Then, the FC-CFT for exactly solving the SE would behave essentially to order N for large and giant molecular systems, though the pre-factor should be very large and must be minimized.

Solving the Schrödinger equation of atoms and molecules with the free-complement chemical-formula theory: First-row atoms and small molecules.

The free-complement chemical-formula theory (FC-CFT) for solving the Schrödinger equation (SE) was applied to the first-row atoms and several small molecules, limiting only to the ground state of a spin symmetry. Highly accurate results, satisfying chemical accuracy (kcal/mol accuracy for the absolute total energy), were obtained for all the cases. The local Schrödinger equation (LSE) method was applied for obtaining the solutions accurately and stably. For adapting the sampling method to quantum mechanical calculations, we developed a combined method of local sampling and Metropolis sampling. We also reported the method that leads the calculations to the accurate energies and wave functions as definite converged results with minimum ambiguities. We have also examined the possibility of the stationarity principle in the sampling method: it certainly works, though more extensive applications are necessary. From the high accuracy and the constant stability of the results, the present methodology seems to provide a useful tool for solving the SE of atoms and molecules.

Solving the Schrödinger equation of hydrogen molecule with the free complement-local Schrödinger equation method: Potential energy curves of the ground and singly excited singlet and triplet states, Σ, Π, Δ, and Φ.

The free-complement (FC) theory for solving the Schrödinger equation (SE) was applied to calculate the potential energy curves of the ground and excited states of the hydrogen molecule (H2) with the 1Σg +, 1Σu +, 3Σg +, 3Σu +, 1Πg, 1Πu, 3Πg, 3Πu, 1Δg, 1Δu, 3Δg, 3Δu, 1Φg, 1Φu, 3Φg, and 3Φu symmetries (in total, 54 states). The initial functions of the FC theory were formulated based on the atomic states of the hydrogen atom and its positive and negative ions at the dissociation limits. The local Schrödinger equation (LSE) method, which is a simple sampling-type integral-free methodology, was employed instead of the ordinary variational method and highly accurate results were obtained stably and smoothly along the potential energy curves. Thus, with the FC-LSE method, we succeeded to perform the comprehensive studies of the H2 molecule from the ground to excited states belonging up to higher angular momentum symmetries and from equilibriums to dissociation limits with almost satisfying spectroscopic accuracy, i.e., 10-6 hartree order around 1 cm-1, as absolute solutions of the SE by moderately small calculations.

Accuracy of Td-DFT in the Ultraviolet and Circular Dichroism Spectra of Deoxyguanosine and Uridine.

Accuracy of the time-dependent density functional theory (Td-DFT) was examined for the ultraviolet (UV) and circular dichroism (CD) spectra of deoxyguanosine (dG) and uridine, using 11 different DFT functionals and two different basis sets. The Td-DFT results of the UV and CD spectra were strongly dependent on the functionals used. The basis-set dependence was observed only for the CD spectral calculations. For the UV spectra, the B3LYP and PBE0 functionals gave relatively good results. For the CD spectra, the B3LYP and PBE0 with 6-311G(d,p) basis gave relatively permissible result only for dG. The results of other functionals were difficult to be used for the studies of the UV and CD spectra, though the symmetry adapted cluster-configuration interaction (SAC-CI) method reproduced well the experimental spectra of these molecules. To obtain valuable information from the theoretical calculations of the UV and CD spectra, the theoretical tool must be able to reproduce correctly both of the intensities and peak positions of the UV and CD spectra. Then, we can analyze the reasons of the changes of the intensity and/or the peak position to clarify the chemistry involved. It is difficult to recommend Td-DFT as such tools of science, at least from the examinations using dG and uridine.

Similarities and Differences between RNA and DNA Double-Helical Structures in Circular Dichroism Spectroscopy: A SAC-CI Study.

The helical structures of DNA and RNA are investigated experimentally using circular dichroism (CD) spectroscopy. The signs and the shapes of the CD spectra are much different between the right- and left-handed structures as well as between DNA and RNA. The main difference lies in the sign at around 295 nm of the CD spectra: it is positive for the right-handed B-DNA and the left-handed Z-RNA but is negative for the left-handed Z-DNA and the right-handed A-RNA. We calculated the SAC-CI CD spectra of DNA and RNA using the tetramer models, which include both hydrogen-bonding and stacking interactions that are important in both DNA and RNA. The SAC-CI results reproduced the features at around 295 nm of the experimental CD spectra of each DNA and RNA, and elucidated that the strong stacking interaction between the two base pairs is the origin of the negative peaks at 295 nm of the CD spectra for both DNA and RNA. On the basis of these facts, we discuss the similarities and differences between RNA and DNA double-helical structures in the CD spectroscopy based on the ChiraSac methodology.

Electronic Transitions in Conformationally Controlled Peralkylated Hexasilanes.

The photophysical properties of oligosilanes show unique conformational dependence due to σ-electron delocalization. The excited states of the SAS, AAS, and AEA conformations of peralkylated n-hexasilanes, in which the SiSiSiSi dihedral angles are controlled into a syn (S), anti (A), or eclipsed (E) conformation, were investigated by using UV absorption, magnetic circular dichroism (MCD), and linear dichroism spectroscopy. Simultaneous Gaussian fitting of all three spectra identified a minimal set of transitions and the wavenumbers, oscillator strengths, and MCD B terms in all three compounds. The results compare well with those obtained by using the symmetry-adapted-cluster configuration interaction method and almost as well with those obtained by time-dependent density functional theory with the PBE0 functional. The conformational dependence of the transition energies and other properties of free-chain permethylated n-hexasilane, n-Si6 Me14 , was also examined as a function of dihedral angles, and the striking effects found were attributed to avoided crossings between configurations of σσ* and σπ* character.

Indicator of the Stacking Interaction in the DNA Double-Helical Structure: ChiraSac Study.

The double-helical structures of DNA are experimentally distinguished by the circular dichroism (CD) spectra. The CD spectra are quite different between the left- and right-handed double-helical structures of DNA. The lowest peak is negative for the left-handed Z-DNA but positive for the right-handed B-DNA. Using the Z-DNA model with a strong stacking interaction, we examined whether the CD spectra depend on the distance between the two base pairs, deoxy-guanosine (dG) and deoxy-cytidine (dC). The result showed that the feature of the SAC-CI CD spectra changes from Z-DNA to B-DNA when increasing the distance between the two base pairs. Therefore, we concluded that the stacking interaction is the origin of the lowest negative peak, being the feature of the CD spectra of Z-DNA, and at the same time that the lack of the negative peak at about 290-300 nm of the CD spectra of B-DNA is due to the weak stacking interaction in B-DNA.

Solving the Schrödinger equation of molecules by relaxing the antisymmetry rule: Inter-exchange theory.

The Schrödinger equation (SE) and the antisymmetry principle constitute the governing principle of chemistry. A general method of solving the SE was presented before as the free complement (FC) theory, which gave highly accurate solutions for small atoms and molecules. We assume here to use the FC theory starting from the local valence bond wave function. When this theory is applied to larger molecules, antisymmetrizations of electronic wave functions become time-consuming and therefore, an additional breakthrough is necessary concerning the antisymmetry principle. Usually, in molecular calculations, we first construct the wave function to satisfy the antisymmetry rule, "electronic wave functions must be prescribed to be antisymmetric for all exchanges of electrons, otherwise bosonic interference may disturb the basis of the science." Starting from determinantal wave functions is typical. Here, we give an antisymmetrization theory, called inter-exchange (iExg) theory, by dividing molecular antisymmetrizations to those within atoms and between atoms. For the electrons belonging to distant atoms in a molecule, only partial antisymmetrizations or even no antisymmetrizations are necessary, depending on the distance between the atoms. So, the above antisymmetry rule is not necessarily followed strictly to get the results of a desired accuracy. For this and other reasons, the necessary parts of the antisymmetrization operations become very small as molecules become larger, leading finally to the operation counts of lower orders of N, the number of electrons. This theory creates a natural antisymmetrization method that is useful for large molecules.

Free-complement local-Schrödinger-equation method for solving the Schrödinger equation of atoms and molecules: basic theories and features.

The free-complement (FC) method is a general method for solving the Schrödinger equation (SE): The produced wave function has the potentially exact structure as the solution of the Schrödinger equation. The variables included are determined either by using the variational principle (FC-VP) or by imposing the local Schrödinger equations (FC-LSE) at the chosen set of the sampling points. The latter method, referred to as the local Schrödinger equation (LSE) method, is integral-free and therefore applicable to any atom and molecule. The purpose of this paper is to formulate the basic theories of the LSE method and explain their basic features. First, we formulate three variants of the LSE method, the AB, HS, and H(T)Q methods, and explain their properties. Then, the natures of the LSE methods are clarified in some detail using the simple examples of the hydrogen atom and the Hooke's atom. Finally, the ideas obtained in this study are applied to solving the SE of the helium atom highly accurately with the FC-LSE method. The results are very encouraging: we could get the world's most accurate energy of the helium atom within the sampling-type methodologies, which is comparable to those obtained with the FC-VP method. Thus, the FC-LSE method is an easy and yet a powerful integral-free method for solving the Schrödinger equation of general atoms and molecules.