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

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.

Using Collocation to Solve the Schrödinger Equation

We review the collocation approach to the solution of the Schrödinger equation and its uses in applications. Interrelations between collocation and other methods are highlighted. We also stress advantages and disadvantages of the rectangular collocation formulation. Using collocation makes it possible to use any, e.g. optimized, coordinates and basis functions, including nonintegrable basis functions, and provides a straightforward way of dealing with singularities in the potential. In addition, we stress that using collocation facilitates tuning the shape of basis functions and the placement of points, both of which can be done with machine-learning methods. Applications to electronic and vibrational problems are reviewed focusing on calculations for molecules on surfaces for which there are few variational calculations. Collocation has advantages when potential energy surfaces are unavailable, in particular, for molecule-surface systems, and for systems for which standard direct product quadrature grids, often used with variational methods, are costly.

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, g_iA and g_ij, 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 r_iA and r_ij 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 g_ij functions gave most accurate results and the reasonable-class functions were less accurate. Among the examined correct and reasonable functions, the conventional function r_ij 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.

Exponentially correlated Hylleraas-configuration interaction non-relativistic energy of the 1S ground state of the helium atom

A generalization of the Hylleraas-Configuration Interaction method (Hy-CI) first proposed by Wang, et al., the Exponentially Correlated Hylleraas-Configuration Interaction method (E-Hy-CI) in which the single r _ij of an Hy-CI wave function is generalized to a form of the generic type r i j ν i j e - ω i j r i j , is explored. This type of correlation, suggested by Hirshfelder in 1960, has the right behavior both in the vicinity of the r_ij cusp as r_ij goes to 0 and as r_ij goes to infinity; this work explores whether wave functions containing both linear and exponential r _ij factors converge more rapidly than either one alone. The method of calculation of the two-electron E-Hy-CI kinetic energy and electron repulsion integrals in a stable and efficient way using recursion relations is discussed, and the relevant formulas are given. The convergence of the E-Hy-CI wave function expansion is compared with that of the Hy-CI wave function without exponential correlation factors, demonstrating both convergence acceleration and an improvement in the accuracy for the same basis. This makes the application of the E-Hy-CI method to systems with N > 4, for which this formalism with at most a single r i j ν i j e - ω i j r i j factor per term leads to solvable integrals, very promising. E-Hy-CI method variational calculations with up to 10080 expansion terms are reported for the ground ¹ S state of the neutral helium atom, with a resultant nonrelativistic energy of -2.9037 2437 7034 1195 9831 1084 hartree for the best expansion.

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.

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.

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.