Non-orthogonal Wavefunctions Research Articles

Orbital optimization in nonorthogonal multiconfigurational self-consistent field applied to the study of conical intersections and avoided crossings.

Nonorthogonal approaches to electronic structure methods have recently received renewed attention, with the hope that new forms of nonorthogonal wavefunction Ansätze may circumvent the computational bottleneck of orthogonal-based methods. The basis in which nonorthogonal configuration interaction is performed defines the compactness of the wavefunction description and hence the efficiency of the method. Within a molecular orbital approach, nonorthogonal configuration interaction is defined by a "different orbitals for different configurations" picture, with different methods being defined by their choice of determinant basis functions. However, identification of a suitable determinant basis is complicated, in practice, by (i) exponential scaling of the determinant space from which a suitable basis must be extracted, (ii) possible linear dependencies in the determinant basis, and (iii) inconsistent behavior in the determinant basis, such as disappearing or coalescing solutions, as a result of external perturbations, such as geometry change. An approach that avoids the aforementioned issues is to allow for basis determinant optimization starting from an arbitrarily constructed initial determinant set. In this work, we derive the equations required for performing such an optimization, extending previous work by accounting for changes in the orthogonality level (defined as the dimension of the orbital overlap kernel between two determinants) as a result of orbital perturbations. The performance of the resulting wavefunction for studying avoided crossings and conical intersections where strong correlation plays an important role is examined.

Sparse non-orthogonal wave function expansions from the extension of the generalized Pauli constraints to the two-electron reduced density matrix.

Generalized Pauli constraints (GPCs) impose constraints in the form of inequalities on the natural orbital occupation numbers of the one electron reduced density matrix (1-RDM), defining the set of pure N-representable 1-RDMs, or 1-RDMs that can be derived from an N-electron wave function. Saturation of these constraints is termed "pinning" and implies a significant simplification of the N-electron wave function as the number of Slater determinants required to fully describe the system is reduced. Recent research has shown pinning to occur for the ground states of atoms and molecules with N = 3 and r = 6, where N is the number of electrons and r is the number of spin orbitals. For N = 4 and r = 8, however, pinning occurs not to the GPCs but rather to inequalities defining the pure N-representable two-electron reduced density matrices (2-RDMs). Using these more general inequalities, we derive a wave function ansatz for a system with four electrons in eight spin orbitals. We apply the ansatz to the isoelectronic series of the carbon atom and the dissociation of linear H4 where the correlation energies are recovered to fractions of a kcal/mol. These results provide a foundation for further developments in wave function and RDM theories based on "pinned" solutions, and elucidate a fundamental physical basis for the emergence of non-orthogonal bases in electronic systems of N ≥ 4.

Configurational forces in electronic structure calculations using Kohn-Sham density functional theory

We derive the expressions for configurational forces in Kohn-Sham density functional theory, which correspond to the generalized variational force computed as the derivative of the Kohn-Sham energy functional with respect to the position of a material point $\textbf{x}$. These configurational forces that result from the inner variations of the Kohn-Sham energy functional provide a unified framework to compute atomic forces as well as stress tensor for geometry optimization. Importantly, owing to the variational nature of the formulation, these configurational forces inherently account for the Pulay corrections. The formulation presented in this work treats both pseudopotential and all-electron calculations in single framework, and employs a local variational real-space formulation of Kohn-Sham DFT expressed in terms of the non-orthogonal wavefunctions that is amenable to reduced-order scaling techniques. We demonstrate the accuracy and performance of the proposed configurational force approach on benchmark all-electron and pseudopotential calculations conducted using higher-order finite-element discretization. To this end, we examine the rates of convergence of the finite-element discretization in the computed forces and stresses for various materials systems, and, further, verify the accuracy from finite-differencing the energy. Wherever applicable, we also compare the forces and stresses with those obtained from Kohn-Sham DFT calculations employing plane-wave basis (pseudopotential calculations) and Gaussian basis (all-electron calculations). Finally, we verify the accuracy of the forces on large materials systems involving a metallic aluminum nanocluster containing 666 atoms and an alkane chain containing 902 atoms, where the Kohn-Sham electronic ground state is computed using a reduced-order scaling subspace projection technique (P. Motamarri and V. Gavini, Phys. Rev. B 90, 115127).

Linear-scaling subspace-iteration algorithm with optimally localized nonorthogonal wave functions for Kohn-Sham density functional theory

We present a linear-scaling method for electronic structure computations in the context of Kohn-Sham density functional theory (DFT). The method is based on a subspace iteration, and takes advantage of the nonorthogonal formulation of the Kohn-Sham functional, and the improved localization properties of nonorthogonal wave functions. A one-dimensional linear problem is presented as a benchmark for the analysis of linear-scaling algorithms for Kohn-Sham DFT. Using this one-dimensional model, we study the convergence properties of the localized subspace-iteration algorithm presented. We demonstrate the efficiency of the algorithm for practical applications by performing fully three-dimensional computations of the electronic density of alkane chains.

Electronic structure of Li 3

Non-orthogonal single- and multi-configuration ab initio calculations have been carried out on ground-state Li 3 in its minimum-energy C 2v geometry. Their results have been compared with published work and with those of SCF, frozen-core SDCI and full-valence CI calculations. The calculations confirm the feasibility of explicit basis-set optimisation, by second-order analytical methods, for correlated wavefunctions in non-linear molecules, and compare use of optimised STO basis sets with standard, high-accuracy, GTO basis sets. The results are used as basis for a discussion of the molecule’s electronic structure. The molecule’s electron density is shown to exhibit a non-nuclear maximum, both at the SCF and frozen-core full-CI levels, and with core-correlated non-orthogonal wavefunctions. An ‘Atoms-in-Molecules’ topological analysis of the electron density shows features that may be viewed as related to the occurrence of Interstitial Orbitals in non-orthogonal electronic wavefunctions for this system. The electron density difference map (molecule minus atoms) exhibits a non-nuclear maximum at roughly the same location as the molecule’s total electron density, plus three 3p-type enrichment-depletion patterns, one centred on each nucleus. Corresponding patterns are found in Li 2. Li 3’s electric dipole and electric field gradient at the nuclei are also computed. Unexpectedly, the electric dipole value is found to exhibit significant dependence on the inclusion of inner-shell out-of-plane correlation in the wavefunction.

Oscillator strengths of allowed and intercombination lines in Si II using non-orthogonal wavefunctions

The importance of valence–shell, core–valence and core–core correlation and interactions between the members of 3s2nd 2D Rydberg series and between the Rydberg series and 3s3p22D perturber state in singly ionized silicon has been examined using term-dependent non-orthogonal orbitals in the multiconfiguration Hartree–Fock approach. Large sets of spectroscopic and correlation non-orthogonal functions have been chosen to adequately describe the term dependence of wavefunctions, various correlation corrections and strong interactions in Rydberg series. The relativistic corrections are included through the one-body mass correction, Darwin and spin–orbit operators and two-body spin–other-orbit operator in the Breit–Pauli Hamiltonian. Extensive configuration-interaction wavefunctions have been used in the representation of Si II levels to calculate oscillator strengths and transition probabilities. The accuracy of present oscillator strengths is evaluated by the agreement between the length and velocity formulations combined with the agreement between the calculated and measured transition energies. The present results have been compared with previous calculations, experimental measurements and astronomical observations.

Calculation of the Cross Section of Helium Ionization by an Electron Impact with the Formation of a Helium Ion in an Excited State

The total cross sections of He and He+ ionization by an electron impact are calculated in the first Born approximation. Calculations of the matrix elements are carried out by the Fock-Dirac multiconfiguration relativistic method using an intermediate type of coupling with orthogonal functions of the initial and final states. A single-electron wave function of the continuous spectrum for an Auger electron is obtained using the Fock-Dirac single-configuration method. The results of the calculations performed with orthogonal and nonorthogonal wave functions of the initial and final states are compared. The ionization cross sections are calculated for cases in which a knock-on electron of the continuous spectrum is described by both the orthogonal and nonorthogonal wave functions with respect to the wave functions of the core electrons.

Accurate calculation of oscillator strengths for Cl II lines using non-orthogonal wavefunctions

Non-orthogonal orbitals technique in the multiconfiguration Hartree-Fock approach is used to calculate oscillator strengths and transition probabilities for allowed and intercombination lines in Cl II. The relativistic corrections are included through the Breit-Pauli Hamiltonian. The Cl II wave functions show strong term dependence. The non-orthogonal orbitals are used to describe the term dependence of radial functions. Large sets of spectroscopic and correlation functions are chosen to describe adequately strong interactions in the 3s(sup 2)3p(sup 3)nl (sup 3)Po, (sup 1)Po and (sup 3)Do Rydberg series and to properly account for the important correlation and relaxation effects. The length and velocity forms of oscillator strength show good agreement for most transitions. The calculated radiative lifetime for the 3s3p(sup 5) (sup 3)Po state is in good agreement with experiment.

Computing upper and lower bounds using Monte Carlo methods

AbstractWe computed lower bounds for some ground and excited states of the helium atom using variational Monte Carlo and the Weinstein, Temple, and Stevenson formulas. For these systems, the Temple and Stevenson bounds are approximately the same and have similar standard deviations. The Weinstein bounds are much further from the actual energies and have much larger standard deviations. We also investigated the reliability of these lower bound formulas when nonorthogonal wave functions are used. The Temple formula has been shown to produce an accurate lower bound even under such circumstances, while the Weinstein and Stevenson formulas are shown to yield incorrect results. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002

Calculation of the differential cross sections of excitation and ionization of a helium atom by electrons

Equations for the amplitudes and differential cross sections of electronic excitation and ionization of a helium atom are derived in the approximation of a “frozen” ion core. The wave functions of the discrete states are chosen in the form of generalized hydrogenlike orbitals. The radial wave functions of the continuous spectrum are determined by solving the equation of motion numerically. The differential excitation cross sections of excitation of the 2p, 3p, and 4p levels and ionization of a helium atom by electrons are calculated in the energy range up to 50 eV. Estimates are obtained for the nonorthogonal wave functions in the amplitudes of the excitation and ionization processes. It is shown that the given method is more compatible with experiment than the Born method.

Generalization of the Green's-functions formalism to nonorthogonal orbitals: Application to amorphous SiO2

It is shown that the Green's-function formalism can be extended to nonorthogonal wave functions. This new technique is illustrated on the example of amorphous Si${\mathrm{O}}_{2}$ where the calculated partial densities of states are compared to experimental data.

Wave function orthogonality in coupled-channel systems

Recent work by Noble and collaborators has pointed out that wave function orthogonality in bound-state to continuum-state transitions can produce large cancellations in transition amplitudes. In single-channel examples, nonorthogonal wave functions (such as plane waves) were shown to greatly overestimate the exact results even at high energies. In this paper we examine a simple coupled-channel model which can be exactly solved for the many-channel scattering and bound wave functions. In such a model, the exact transition amplitude can be compared with standard plane-wave and distorted-wave approximations. Dispersive corrections are shown to be large and to persist even at high energies.NUCLEAR REACTIONS Orthogonality constraints on bound-continuum transitions in coupled-channel systems. Dispersive corrections to DWIA.

An alternative to orthogonalisation of wavefunctions and calculation of X-ray spectral intensities

Simplifying assumptions made while considering complex systems often lead to non-orthogonal sets of wavefunctions. A procedure is suggested for employing such non-orthogonal wavefunctions without first orthogonalising them. The method involves the use of a basis reciprocal to the basis of the non-orthogonal functions. The relative intensities of K alpha 1, K beta 1 and K beta 2 X-ray spectral lines calculated by the proposed method are in fair agreement with available experimental data.

Linked cluster theorem for non-orthogonal wavefunctions

It is proved that the first and second order perturbation energies of a system described by non-orthogonal single particle wavefunctions can be expressed as a series of ‘linked diagram’ terms.

On the Non-Orthogonality Difficulty in Ferromagnetism

In a many-electron system, one is faced to some difficulties when one-electron wave functions are not orthogonal to each other. For example, the inner product (\(\varPhi\), \(\varPhi\)) (\(\varPhi\) is a Slater determinant made from these non-orthogonal wave functions) loses its meaning as the number of electrons becomes large. We avoid this difficulty by using the expansion coefficients instead of the inner products, in order to define the matrix elements of operators. Applying this method to a magnetic problem, in which there are some doubts about the validity of the Heisenberg model \(\displaystyle \mathcal{H} = -2 \sum_{i>j} J_{i,j}S_{i} \cdot S_{j} \), we can derive the energy of the ground state and the energy spectrum of spin waves. The results are written as

Use of Non-orthogonal Wave Functions in the Treatment of Solids, with Applications to Ferromagnetism

It is proved by a rearrangement of L\"owdin's solution to the many-body problem that a vector model type energy expression is valid for a solid provided (in the simplest case of one electron per atom) the number of nearest neighbors times the overlap integral between them is small compared with unity.The approximation is also carried to include third order permutation terms, and the spin coupling via closed shells these "triple exchange" terms can produce is discussed.General expressions are given for the exchange integrals to be used in a vector model solution. These under the proper conditions reduce to the conventional integrals except for a slight modification, which helps to make the integrals negative.

The Theory of the Dielectric Constant of Ionic Crystals, I

We worked out the dielectric constants of 0 and of the alkali-halide crysral on the basis of the quantum theory of solids. We have used the variational method of Slater and Kirkwood in our energy-calculations and adopted Landshoff's method for evaluating the exchange integrals among the non-orthogonal wave functions. The relation among the vatious theories, due to Shockley, Mott and Frohlich, of the dielectric constant of ionic crystal have been clarified from the viewpoint of our theory. The numerical results for LiF are in good accordance with the experiment. § 1. Introductionl ) Many of the characteristic properties of polar crystals, i.e., the behaviour of the conduction electrons or the existence of defects in the lattice will, after all, sensitively depend on the dielectric properties of the crystal. So we see it is essentially required to elucidate the dielectric properties of the mentioned crystals in detail on the basis of the quantum theory of solids, since the available theories of dielectric properties are usually based on the rather phenomenological considerations. As a preliminary to the general formulation of the theory we shall here work quantum-mechanically the simplest case of the dielectric constant of ionic crystals in the homogeneous external fields of both high frequency and static natures. As is well-known in the classical theory, the relation between the polarizability (a) of an ion pair and the dielectric constant (xo) for high frequency field is represented by one or the other of the following two formulae according to the assumptions of the effective field I!',tI' around each ion of the crystal. Namely, we have

Intensities in the principal series of lithium

The effect of using orthogonal instead of non-orthogonal wave functions in the calculation of the oscillator-strengths (f numbers) of the first three lines of the principal series of lithium has been investigated. The Hartree self-consistent field method is employed to calculate the (1s) radial wave function, and the (2s) radial wave function is made orthogonal to the (1s) by linear combination. The orthogonal functions have been used to calculate the energies of the optical terms (2s), (2p), (3p) and (4p), and the f numbers of the first three lines of the principal series. Very satisfactory agreement with observed values is obtained for the energies (the maximum difference being 3 per cent). The agreement for the f numbers is not so good, but is better than when non-orthogonal functions are used.

Non-Orthogonal Wave Functions and Ferromagnetism

Close consideration of the influence of non-orthogonality of the electronic wave functions in a crystal shows that because of the large number of terms arising from the possible permutations of electrons, the usual procedure involving the neglect of most of the terms arising from this lack of orthogonality may be seriously in error. The dependence of the relative energies of the states of low and high multiplicity on this factor is discussed and exemplified in the cases of some simple molecular models.