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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 s i j = r i j 2 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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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 Schrö dinger 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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The symmetry-adapted cluster-configuration interaction (SAC/SAC-CI) method is used for studying the chemiluminescence in dimethyl sulfoxide (DMSO) solution and the bioluminescence in the enzyme firefly luciferase (Luc). It is shown that the keto form of OxyL[H.sub.2] has produced red light in chemiluminescence and yellow-green in bioluminescence.

The singlet excited states and doublet ionized/electron-attached states of o-benzoquinone (o-BQ) are analyzed by using the symmetry adapted cluster (SAC)/SAC-configuration interaction (SAC-CI) method. The SAC-CI theoretical spectra have showed that it is essential to take electron correlation into account when describing the singlet and doublet excited states of o-BQ.

Accurate quantum-chemical calculations of the excitation energies and the rotatory strengths of dichalcogens R-Ch-Ch-R (Ch = S, Se, Te) are performed with the symmetry adapted cluster (SAC) and SAC-configuration interaction (CI) methods. The first and second lowest bands have depended on the R-Ch-Ch-R dihedral angle and the calculated trends have agreed with two empirical rules, the [C.sub.2] rule and the quadrant rule.

Molecular structures and excited states of CpM(CO)2 (Cp= eta5-C5H5; M=Rh, Ir) and [C12Rh(CO)2] (super -1) complexes are examined using the B3LYP and the symmetry-adapted cluster (SAC) /SAC-configuration interaction (SAC-Cl) theoretical methods. The wavelength dependence of the quantum efficiencies for the C-H/Si-H bond activations and ligand photosubstitution of CpM(CO)2 can be explained by the existence of the two different types of excited states of CpM(CO)2.