Solving the Schrödinger equation of atoms and molecules using one- and two-electron integrals only

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.

In a previous paper, formulas for two-electron integrals over Cartesian Gaussian functions occurring in the Hylleraas-CI method have been given. This paper reports on the extension of those formula to three- and four-electron integrals and presents test results obtained by using the two-electron integrals for H2 With only 2 p-type polarization functions on each atom, the ground state energy calculated for H2 is only about 20 cm−1 higher than the exact value.

Atomic self-consistent-field program by the basis set expansion method: Columbus version

Graphical representations of three-particle wave functions obtained by direct solution of the Schr\odinger equation with the correlation-function hyperspherical-harmonic (CFHH) method are obtained and analyzed for ground and excited states of the helium atom and for the ground state of the \ensuremath{\mu}dd mesomolecular ion. The inclusion of adequate singular and cluster-correlation behavior is shown to be of crucial importance for a proper description of the wave function. In the CFHH method the wave function is a product of a correlation function and of a smooth factor expanded into hyperspherical-harmonic (HH) functions. While the HH expansion by itself is not able to reproduce a correct form of the wave function, the inclusion of the correlations results in its proper description even for low values of the maximal global momenta ${\mathit{K}}_{\mathit{m}}$.

A method is presented for the determination, with relatively small effort, of analytic one-electron wave functions for an electron in an approximately self-consistent screened Coulomb field. The method enables one to avoid direct numerical solution of the Schr\"odinger equation. Instead, for each state one has to adjust two parameters to make a certain expression for the screening function reasonably approximate that which follows from the Thomas-Fermi ion for the atom in question. When this is done, one has automatically a simple, analytic expression for the eigenfunction and the corresponding energy eigenvalue for that state. In particular, pairs of parameters have been obtained for states of several alkali atoms, including all of the normally filled states of sodium, the $6s$ state of potassium and most of the important excited states of cesium. The analytic wave functions thus obtained are simple enough so that the expectation values of many quantities can be easily calculated by analytic methods, a few of which are illustrated. Thus, the normalization constants are calculated for three of the states of cesium. Moreover, calculations for the fine structure and hyperfine structure energy splittings of cesium are carried out as a test of the goodness of the wave functions. Good agreement is obtained with experiment and with previous calculations.

We address the issue of the relation between the canonical functional Schr\"odinger representation in quantum field theory and the approach of precanonical field quantization proposed by the author, which requires neither a distinguished time variable nor infinite-dimensional spaces of field configurations. We argue that the standard functional derivative Schr\"odinger equation can be derived from the precanonical Dirac-like covariant generalization of the Schr\"odinger equation under the formal limiting transition $\gamma^0\varkappa\rightarrow\delta(\mathbf{0})$, where the constant $\varkappa$ naturally appears within the precanonical quantization as the inverse of a small "elementary volume" of space. We obtain a formal explicit expression of the Schr\"odinger wave functional as a continuous product of the Dirac algebra valued precanonical wave functions, which are defined on the finite-dimensional covariant configuration space of the field variables and space-time variables.

SchrÖdinger Wave Functional in Quantum Yang–Mills Theory from Precanonical Quantization

Doubly excited ridge states of atoms are states in which two electrons reach high and comparable excitations. Our goal is to calculate sequences for some of these states by using a wave function that treats the pair of electrons as a single entity in solving the two-electron Schr\"odinger equation in hyperspherical coordinates: R, \ensuremath{\alpha}, ${\mathrm{\ensuremath{\theta}}}_{12}$. For double escape of slow electrons, the so-called Wannier theory predicts that the wave function will be concentrated in the region \ensuremath{\alpha}=\ensuremath{\pi}/4, ${\mathrm{\ensuremath{\theta}}}_{12}$=\ensuremath{\pi} (that is, ${\mathbf{r}}_{1}$=-${\mathbf{r}}_{2}$) and we expect something similar for the ridge states. By expanding the Schr\"odinger equation around these points and retaining the first nontrivial quadratic dependences in \ensuremath{\alpha} and ${\mathrm{\ensuremath{\theta}}}_{12}$, we seek a solution in which the form of the wave function in these two variables is analytically determined as in the Wannier theory. The R dependence of this wave function is then handled numerically and differs from the double-escape solution only in the exponential decay appropriate for bound states. For L=0 a polynomial in R is included, giving the proper number of nodes for the solution. Numerical eigenvalues for autoionizing ridge states are calculated and compared with available results.

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.

Evaluation of molecular integrals involving continuum orbitals

We extend our prior work on the construction of variational wave functions $\ensuremath{\psi}$ that are functionals of functions $\ensuremath{\chi}:\ensuremath{\psi}=\ensuremath{\psi}[\ensuremath{\chi}]$ rather than simply being functions. In this manner, the space of variations is expanded over those of traditional variational wave functions. In this article we perform the constrained search over the functions $\ensuremath{\chi}$ chosen such that the functional $\ensuremath{\psi}[\ensuremath{\chi}]$ satisfies simultaneously the constraints of normalization and the exact expectation value of an arbitrary single- or two-particle Hermitian operator, while also leading to a rigorous upper bound to the energy. As such the wave function functional is accurate not only in the region of space in which the principal contributions to the energy arise but also in the other region of the space represented by the Hermitian operator. To demonstrate the efficacy of these ideas, we apply such a constrained search to the ground state of the negative ion of atomic hydrogen ${\mathrm{H}}^{\ensuremath{-}}$, the helium atom He, and its positive ions Li${}^{+}$ and Be${}^{2+}$. The operators $W$ whose expectations are obtained exactly are the sum of the single-particle operators $W={\ensuremath{\sum}}_{i}{r}_{i}^{n},n=\ensuremath{-}2,\ensuremath{-}1,1,2$, $W={\ensuremath{\sum}}_{i}\ensuremath{\delta}({\mathbf{r}}_{i})$, $W=\ensuremath{-}\frac{1}{2}{\ensuremath{\sum}}_{i}{\ensuremath{\nabla}}_{i}^{2}$, and the two-particle operators $W={\ensuremath{\sum}}_{n}{u}^{n},n=\ensuremath{-}2,\ensuremath{-}1,1,2$, where $u=|{\mathbf{r}}_{i}\ensuremath{-}{\mathbf{r}}_{j}|$. Comparisons with the method of Lagrangian multipliers and of other constructions of wave-function functionals are made. Finally, we present further insights into the construction of wave-function functionals by studying a previously proposed construction of functionals $\ensuremath{\psi}[\ensuremath{\chi}]$ that lead to the exact expectation of arbitrary Hermitian operators. We discover that analogous to the solutions of the Schr\odinger equation, there exist $\ensuremath{\psi}[\ensuremath{\chi}]$ that are unphysical in that they lead to singular values for the expectations. We also explain the origin of the singularity.

A b initio configuration-interaction calculations have been carried out on the six electronic states of RgO (Rg=Ne, Ar, Kr, and Xe) arising from the Rg(1S)+O(3P, 1D, 1S) separated atom limits similar to the calculations of Dunning and Hay [J. Chem. Phys. 66, 3767 (1977)]. The degree of charge transfer and, hence, the extent to which the interactions are attractive is observed to be sensitive to the method of configuration selection. The off-diagonal spin–orbit coupling matrix elements 〈3Π‖Hso‖1Σ+〉 and 〈3Σ−‖Hso‖1Σ+〉 were calculated for the lowest 3Π, 3Σ−, and 1Σ+ states of RgO as a function of internuclear distance using the configuration-interaction wave functions and the full microscopic spin–orbit Hamiltonian. All one- and two-electron integrals were included. These results are compared in detail with the spin–orbit matrix elements of Cohen, Wadt, and Hay [J. Chem. Phys. 71, 2955 (1979)] computed using equivalent wave functions and an effective spin–orbit operator composed only of one-electron, one-center terms. The good agreement between the results strongly supports the use of an effective spin–orbit operator of this type. The success of their model results from the fact that electron interaction effects can be adequately accounted for by a single adjustable parameter that can be determined from the atomic fine structure. Apparently for the RgO molecules, the numerical results are a more sensitive test of the wave functions (particularly to the extent of charge transfer) than to the exact evaluation of all terms in the full spin–orbit operator. In fact, for the heaviest system considered, XeO, the effective operator approach actually appears to be superior to the ab initio treatment.

Initializing a single site of a lattice scalar field theory into an arbitrary state with support throughout the quantum register requires $O({2}^{{n}_{Q}})$ entangling gates on a quantum computer with ${n}_{Q}$ qubits per site. It is conceivable that instead initializing to functions that are good approximations to states may have utility in reducing the number of required entangling gates. In the case of a single site of a noninteracting scalar field theory, initializing to a symmetric exponential wave function requires ${n}_{Q}\ensuremath{-}1$ entangling gates, the minimal number necessary to create an entangled state of all ${n}_{Q}$ qubits. This is compared with the ${2}^{{n}_{Q}\ensuremath{-}1}+{n}_{Q}\ensuremath{-}3+{\ensuremath{\delta}}_{{n}_{Q},1}$ required for a symmetric Gaussian wave function. In this work, we explore the initialization of one-site (${n}_{Q}=4$), two-site (${n}_{Q}=3$), and three-site (${n}_{Q}=3$) noninteracting scalar field theories with symmetric exponential wave functions using IBM's quantum simulators and quantum devices (Poughkeepsie and Tokyo). With the digitizations attainable with ${n}_{Q}=3,4$, these tensor-product wave functions are found to have large overlap with a Gaussian wave function and provide a suitable low-noise initialization for improvement and Somma inflation. In performing these simulations, we have employed a workflow that interleaves calibrations to mitigate systematic errors in production. The calibrations allow tolerance cuts on gate performance including the fidelity of the symmetrizing Hadamard gate, both in vacuum (${|\mathbf{0}\ensuremath{\rangle}}^{\ensuremath{\bigotimes}{n}_{Q}}$) and in medium (${n}_{Q}\ensuremath{-}1$ qubits initialized to an exponential function). The results obtained in this work are relevant to systems beyond scalar field theories, such as the deuteron radial wave function, two- and three-dimensional Cartesian-space wave functions, and nonrelativistic multinucleon systems built on a localized eigenbasis.

We extend to k \ensuremath{\ne} 0 the work in a previous paper on the solution of the Schr\"odinger equation in a periodic potential. We begin with a perturbation method, and expand the wave function and energy in powers of $k$. The equations determining the wave function to any given order in $k$ are then just inhomogeneous linear equations, which are practical to solve. We give formulas for evaluating the energy from these wave functions: For bands which are not degenerate at k = 0 these involve straightforward summations over the reciprocal lattice; for bands which are degenerate at k = 0, we show how to derive the secular determinant which gives the splitting of these bands. The elements of this determinant can also be calculated as sums, which are practical to evaluate, over the reciprocal lattice. We present a simple numerical example to show how some of the above results work out in practice. Next, we discuss tight-binding formulas in the present formalism, and the practical problem of evaluating these formulas. We give simple expressions, involving the radial derivatives of the atomic momentum space wave functions, for evaluating the energy in powers of $k$; these formulas do not make near-neighbor approximations. Then, for those cases for which neither the tight-binding approximation nor an expansion in powers of $k$ is appropriate, we discuss the direct numerical solution of the equations and give a convenient partial wave expansion for this purpose. Finally, we briefly discuss the relation to each other of the various approximations presented in this paper.

We have theoretically studied the optical absorption coefficient of quantum wires in electric and strain fields with intermixing interfaces. The potential profiles are governed by the intermixing heterojunctions, internal strain due to the lattice mismatch, the external electric field and stress. The second Fick's law describes the wires' intermixing heterojunctions due to the alloy interdiffusion. We adopt the Green's function method to solve the Poisson equation for the displacement to determine the internal strain due to the lattice mismatch. The single-band Schr\"odinger equation in the effective mass approximation is used to describe the conduction subband structures while the four-band Kohn-Luttinger Hamiltonian is used to describe the valence subband structures. In solving the Schr\"odinger equation and the Kohn-Luttinger Hamiltonian, we expand the wave functions for the electrons and holes by using linear combinations of the two-dimensional harmonic oscillator wave functions to yield two matrix equations. These matrix equations are numerically solved for their eigen-energies and their corresponding eigenfunctions. We investigate some physical properties of the unstrained quantum wire $(\mathrm{Ga}\mathrm{As}∕\mathrm{Al}\mathrm{As})$ and the strained wire $(\mathrm{Cd}\mathrm{Se}∕\mathrm{Zn}\mathrm{Se})$ as examples. When the interdiffusion becomes stronger, on one hand, the internal strain is relaxed leading to the band gap shrinkage (if the initial strain ${\ensuremath{\epsilon}}_{0}<0$) and, on the other hand, the potential profiles are deformed and the effective band gap becomes wider. Variations of the transition energies with external stress show the anticrossing effect and variations of the transition energies with external electric field show the field emission effect. The hole effective masses can be enhanced or become electronlike by applying stress to the wire. The oscillator strengths of the dipole-allowed intersubband transitions for the $y$ and $z$ polarizations are calculated to infer possible optical transitions. The optical absorption coefficients together with the joint density of states are calculated.