Solution of large configuration mixing matrices arising in partitioning technique and applications to the generalized eigenvalue problem

A method is presented that can be used (a) to determine the several lowest eigenvalues and eigenvectors of large symmetric matrices, (b) to solve the generalized eigenvalue problem associated with energy-dependent operators, that arises in computations involving energy-dependent many-body Green's functions and in the evaluation of the true parameters of the effective valence shell hamiltonian, and (c) to directly evaluate the matrices associated with resolvent operators. The applicability to large configuration mixing calculations arises when the N-electron basis functions can be easily broken down to a few dominant configurations (the primary block) and their complement. Using the partitioning technique, the effective hamiltonian within the primary block is directly evaluated. The method is extended to evaluation of the dynamical polarizability tensor, which effectively contains the contributions from all of the eigenstates of a hamiltonian matrix, without the necessity of explicitly calculating its eigenvalues and eigenvectors.     

Definitive variational principles for physical quantities

New variational principles are proposed that provide the upper and lower limits to the square of the modulus of the matrix element for a transition. The corresponding variational problems for maximum and minimum are formulated via functional of general form that are bilinear with respect to the hamiltonian. An analogous variational principle may be derived via a functional linear with respect to the hamiltonian for a transition to a state of lowest energy of the given symmetry type. In some cases the variational estimates of the square of the modulus involve passage to the limit. Errors due to inexact eigenfunctions and replacement of limiting values by specified values for the parameters are discussed.     

On the variational solution of the coupled breathing rotation-vibration of a spherical top molecule

On the basis of the solutions of the isotropic 3-D harmonic oscillator, we show how to evaluate the matrix elements for the coupled rotation-vibration of the totally symmetric breathing mode of a rotating spherical top molecule, whereby the anharmonic potential energy is expanded in a power series of r. It is shown also how to obtain the ro-vibrational spectrum when used in conjunction with the variational method to obtain eigenvalues.     

Single determinant N‐representability and the kernel energy method applied to water clusters

The Kernel energy method (KEM) is a quantum chemical calculation method that has been shown to provide accurate energies for large molecules. KEM performs calculations on subsets of a molecule (called kernels) and so the computational difficulty of KEM calculations scales more softly than full molecule methods. Although KEM provides accurate energies those energies are not required to satisfy the variational theorem. In this article, KEM is extended to provide a full molecule single‐determinant N‐representable one‐body density matrix. A kernel expansion for the one‐body density matrix analogous to the kernel expansion for energy is defined. This matrix is converted to a normalized projector by an algorithm due to Clinton. The resulting single‐determinant N‐representable density matrix maps to a quantum mechanically valid wavefunction which satisfies the variational theorem. The process is demonstrated on clusters of three to twenty water molecules. The resulting energies are more accurate than the straightforward KEM energy results and all violations of the variational theorem are resolved. The N‐representability studied in this article is applicable to the study of quantum crystallography. © 2017 Wiley Periodicals, Inc.     

A density matrix variational calculation for atomic Be

The ground-state energy of the beryllium atom is calculated using a variational procedure in which the elements of the two-body reduced density matrix (particle–particle matrix) are the variational parameters. It is shown that, for this problem and with the limited number of spin-orbitals used, the trace condition and the simultaneous nonnegativity conditions on the particle–particle, the particle–hole, and the hole–hole matrices form a complete solution to the N-representability problem. The energy obtained is – 14.61425 a.u., practically identical to the value given by a configuration interaction calculation which uses the same states. The effects of weakening the nonnegativity conditions on each of the matrices in turn were also explored.     

Relativistic molecular wavefunctions: XeF2

A new method is presented to calculate binding energies and eigenfunctions for molecules, using the relativistic Dirac hamiltonian. A numerical basis set of four component wavefunctions is obtained from atom-like Dirac-Slater wavefunctions. A discrete variational method has been developed and applied to the linear XeF₂ molecule.     

Relativistic corrections in the multiple-scattering method

We show how to introduce the Foldy-Wouthuysen relativistic corrections in the multiple-scattering method, for the determination of the electronic energy levels of molecules. The present derivation begins from a variational expression where the Foldy-Wouthuysen hamiltonian is inserted, instead of the Schrödinger hamiltonian. The resulting secular equation becomes identical with the standard multiple-scattering secular equation, if one neglects the relativistic correction terms.     

A second-quantization representation of the Epstein-Nesbet partitioning of the full hamiltonian

A second-quantization representation of the Epstein-Nesbet partitioning of the total electronic hamiltonian is suggested. In this approach, the unperturbed hamiltonian contains not only the one-particle orbital energies but also the Coulomb and corresponding exchange two-particle terms. Such a hamiltonian can advantageously be used in all branches of the many-body diagrammatic perturbation theory for simple and correct inclusion of the diagonal ladder and ring diagrams in all orders of perturbation theory.     

Model calculation of weak and medium strong linear hydrogen bonded systems

The model hamiltonian with two coupling constants between the low and high frequency vibrations of a linear AH?B system is studied. The results show that the wave functions and energy levels of this hamiltonian obtained by exact diagonalisation can be adequately approximated by the wave functions and energy levels of a harmonic “effective hamiltonian”. The experimental spectrum of (CH₃)O?HCl in the stretching region was reconstituted by means of this model.     

Diabatic investigation for the NaRb molecule

An extensive diabatic investigation of the NaRb species has been carried out for all excited states up to the ionic limit Na^‐Rb⁺. An ab initio calculation founded on the pseudopotential, core polarization potential operators and full configuration interaction has been used with an efficient diabatization method involving a combination of variational effective hamiltonian theory and an effective overlap matrix. Diabatic potential energy curves and electric dipole moments (permanent and transition) for all the symmetries Σ⁺, Π, and Δ have been studied for the first time. Thanks to a unitary rotation matrix, the examination of the diabatic permanent dipole moment (PDM) has shown the ionic feature clearly seen in the diabatic ¹Σ⁺ potential curves and confirming the high imprint of the Na^‐Rb⁺ ionic state in the adiabatic representation. Diabatic transition dipole moments have also been computed. Real crossings have been shown for the diabatic PDM, locating the avoided crossings between the corresponding adiabatic energy curves.     

A variational method to calculate static electronic properties

Using a coupled cluster form of the wave function, a variational method is formulated for calculation of static properties of any order. Corresponding to an appropriate perturbed hamiltonian H() including the relevant static property, a size consistent functional is set up. In a hierarchical fashion, properties of different orders may be found out using a variational method.     

A note on the partitioning technique

A new derivation of the wave operator of the partitioning technique is given. Furthermore this approach is applied to derive wave operators for the inverse hamiltonian and in general for functions of the hamiltonian.     

Brackets to the eigenvalues of the Schrödinger equation,part 1. Tridiagonal matrices

The problem of upper and lower bounds to the first few eigenvalues of a very large or infinite tridiagonal matrix H is studied. Those eigenvalues of a comparison-matrix M _n which are lower than a characteristic limit, together with the corresponding eigenvalues of the variational matrix H _n are shown to bracket exact eigenvalues of H . M _n differs from H _n only in the last off-diagonal element and is easily obtained from H . Sufficient conditions for lower bounds are based on a low estimate of the characteristic limit. For increasing dimensions n, the lower bounds approach the exact eigenvalues from below. As a numerical illustration, brackets to the known eigenvalues of the harmonic oscillator with a linear perturbation are calculated.     

Overtone spectrum in terms of normal or of equivalent modes with application to H2O

An algebraic hamiltonian which closely fits the (low and) high stretching vibrational spectrum of H₂O (and of other YXY molecules such as O₃) is shown to correspond to two weakly coupled equivalent modes.     

Bonding in C2 and Be2: Broken symmetry and correlation in DFT solutions

Summary The solution of both Hartree-Fock (HF) and Kohn-Sham (KS) equations is based on the variational principle. Exact wavefunctions would obey the same symmetry restrictions contained in the total hamiltonian. However, the variational principle does not guarantee these symmetry restrictions and the HF and KS solutions are not necessarily symmetric in spin and space. Spatial and spin symmetry broken solutions with lower energies than their restricted analogues are examined for C₂ and Be₂, in the context of the KS formalism. Comparison with UHF solutions shows that KS instabilities are far less pronounced. The main differences between HF and KS solutions are related to effects of electron correlation.     

Revisiting the foundations of the quantum theory of atoms in molecules: The subsystem variational procedure and the finite nuclear models

The role of finite nuclear models (FNMs) is scrutinized within the context of the quantum theory of atoms in molecules (QTAIMs). It is demonstrated that the newly proposed analytic‐algebraic definition of the topological atoms is consistently extendable to the cases where a FNM is employed to construct the molecular hamiltonian. The whole variational procedure is reconsidered, and the insensitivity of final results relative to the employed FNMs is explicitly demonstrated. The analysis once again clearly demonstrates that the analytic‐algebraic condition is an independent axiom that must be added to the subsystem variational procedure to construct the QTAIMs. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2010     

An algorithm for calculating atomic D states with explicitly correlated gaussian functions

An algorithm for the variational calculation of atomic D states employing n-electron explicitly correlated gaussians is developed and implemented. The algorithm includes formulas for the first derivatives of the hamiltonian and overlap matrix elements determined with respect to the gaussian nonlinear exponential parameters. The derivatives are used to form the energy gradient which is employed in the variational energy minimization. The algorithm is tested in the calculations of the two lowest D states of the lithium and beryllium atoms. For the lowest D state of Li the present result is lower than the best previously reported result.     

Spin-Orbit coupling in aromatic hydrocarbons: A semiempirical evaluation of the radiative lifetimes of naphtalene and anthracene

Singlet-triplet spin-orbit matrix elements, which govern the lowest ³B_1u ← ¹A_g transition in typical aromatic molecules like naphtalene and anthracene, are calculated with INDO molecular orbitals and the conventional spin-orbit one-electron hamiltonian. The correct order of magnitude of the triplet radiative lifetimes is obtained for the two molecules, when INDO MO coefficients are referred to a symmetrically orthogonalized basis. The possibility of using the semiempirical Hamiltonian is explored using ab initio wave-functions for a few test cases. Reasonably accurate doublet-doublet and singlet-triplet matrix elements have been computed.