We obtain conditions, which when fulfilled, permit to transform the coordinates of a dynamical system into pairs of canonical ones for some Hamiltonian system. These conditions, restricted to the class of coordinate transformations which act on each coordinate independently, are greatly simplified. However, they are surprisingly successful in defining canonical coordinates and an associated Hamiltonian for several test examples. So, a method is proposed to exploit these simple transformations in a systematic manner.

Wavefunction forms based on products of electron pairs are usually constructed as closed-shell singlets, which is insufficient when the molecular state has a nonzero spin or when the chemistry is determined by d- or f- electrons. A set of two-electron forms are considered as explicit couplings of second-quantized operators to open-shell singlets. Geminal wavefunctions are constructed, and their structure is elaborated. Numerical results for small model systems clearly demonstrate improvement over closed-shell singlet pairs.

We present a new geminal product wave function Ansatz where the geminals are not constrained to be strongly orthogonal or to be of seniority-zero. Instead, we introduce weaker orthogonality constraints between geminals that significantly lower the computational effort without sacrificing the indistinguishability of the electrons. That is to say, the electron pairs corresponding to the geminals are not fully distinguishable, and their product has yet to be antisymmetrized according to the Pauli principle to form a bona fide electronic wave function. Our geometrical constraints translate into simple equations involving the traces of products of our geminal matrices. In the simplest non-trivial model, a set of solutions is given by block-diagonal matrices where each block is 2 × 2 and consists of either a Pauli matrix or a normalized diagonal matrix multiplied by a complex parameter to be optimized. With this simplified Ansatz for geminals, the number of terms in the calculation of the matrix elements of quantum observables is considerably reduced. A proof of principle is reported and confirms that the Ansatz is more accurate than strongly orthogonal geminal products while remaining computationally affordable.

We conjecture that “Antisymmetrized Geminal Power” wave functions, and, in particular, those of extreme type in Coleman’s terminology (i.e. with all geminal coefficients equal), can always be rewritten as antisymmetrized products of geminals orthogonal to each other. We prove this conjecture in simple cases and provide numerical evidence that it holds true in more complicated examples. Establishing the validity of this conjecture is important as it questions the physical interpretation of AGP wavefunctions.

The question of classicality is addressed in relation with the shape of the nuclear skeleton of molecular systems. As the most natural environment, the electrons of the molecule are considered as continuously monitoring agents for the nuclei. For this picture, an elementary formalism of decoherence theory is developed and numerical results are presented for few-particle systems. The numerical examples suggest that the electron-nucleus Coulomb interaction is sufficient for inducing a blurred shape with strong quantum coherences in compounds of the lightest elements, H2, D2, T2, and HeH+.

We consider an 1-electron model Hamiltonian, whose potential energy corresponds to the Coulomb potential of an infinite wire with charge Z distributed according to a Gaussian function. The time independent Schrödinger equation for this Hamiltonian is solved perturbationally in the asymptotic limit of small amplitude vibration (Gaussian function width close to zero). We propose to use the naturally polarized functions so-obtained, as orbital basis sets for quantum chemical calculations. In particular, they should be well suited to perform electron-nucleus mean field configuration interaction calculations. Since the free-parameters of the model have the remarkable property to factorize the perturbative corrections to the eigenfunctions, these corrective part in factor can be simply added as additional functions to standard basis sets, leaving it to the molecular orbital calculation to optimize the free parameters within molecular orbital coefficients.

The construction of integrity bases for invariant and covariant polynomials built from a set of three dimensional vectors under the SO(3) and O(3) symmetries is presented. This paper is a follow–up to our previous work that dealt with a set of two dimensional vectors under the action of the SO(2) and O(2) groups [G. Dhont and B. I. Zhilinskiı́, J. Phys. A: Math. Theor., 46, 455202 (2013)]. The expressions of the Molien generating functions as one rational function are a useful guide to build integrity bases for the rings of invariants and the free modules of covariants. The structure of the non–free modules of covariants is more complex. In this case, we write the Molien generating function as a sum of rational functions and show that its symbolic interpretation leads to the concept of generalized integrity basis. The integrity bases and generalized integrity bases for O(3) are deduced from the SO(3) ones. The results are useful in quantum chemistry to describe the potential energy or multipole moment hypersurfaces of molecules. In particular, the generalized integrity bases that are required for the description of the electric and magnetic quadrupole moment hypersurfaces of tetratomic molecules are given for the first time.

Molecular structure is often considered as emerging from the decoherence effect of the environment. Electrons are part of the environment of the nuclei in a molecule. In this work, their contribution to the classical-like geometrical relationships often observed between nuclei in molecular systems is investigated. Reduced density matrix (RDM) elements are evaluated from electron-nucleus wave functions. The computational results show that the electrons play a role in the localization of the nuclei around specific geometries. Although the electronic environment alone cannot explain molecular symmetry-broken isomers, it can contribute to their dynamical stability by reducing off-diagonal RDM elements.

The electronic absorption (EA), circular dichroism (ECD), and anisotropy spectra of the L-valine zwitterion and D-glyceraldehyde are calculated by time-dependent density functional theory (TDDFT) with the M06-2X and B3LYP functionals. It is found that the absorption and ECD spectra from TDDFT/M06-2X agree well with experimental results measured from the amorphous film of L-valine. Moreover, the calculations reproduce all three major peaks observed in the experimental anisotropy spectra. For D-glyceraldehyde, the TDDFT/M06-2X calculations indicate that the excitation wavelengths of the first excited state of 32 stable conformers distribute from 288 to 322 nm, giving rise to two ECD peaks with opposite signs centered at 288 and 322 nm. The very weak absorption of the first excited state (S1) induces two high peaks in the anisotropy spectra of D-glyceraldehyde, which should be seen in future experimental studies.

A new lower bound for the maximal length of a multivector is obtained. It is much closer to the best known upper bound than previously reported lower bound estimates. The maximal length appears to be unexpectedly large for $n$-vectors, with n>2, since the few exactly known values seem to grow linearly with vector space dimension, whereas the new lower bound has a polynomial order equal to n-1 like the best known upper bound. This result has implications for quantum chemistry.

We present generalized versions of the concepts of seniority number and ionicity. These generalized numbers count respectively the partially occupied and fully occupied shells for any partition of the orbital space into shells. The Hermitian operators whose eigenspaces correspond to wave functions of definite generalized seniority or ionicity values are introduced. The generalized seniority numbers (GSNs) afford to establish refined hierarchies of configuration interaction (CI) spaces within those of fixed ordinary seniority. Such a hierarchy is illustrated on the buckminsterfullerene molecule.

We introduce a definition of electron–nucleus correlation energy by analogy with the quantum chemical definition of electronic correlation energy. The uncorrelated reference function is obtained by repeated electron–nucleus mean field configuration interaction steps until self-consistency is achieved. Electron–nucleus correlation and electronic correlation energies are compared on dihydrogen isotopologues.

We propose to use the eigenfunctions of a one-electron model Hamiltonian to perform electron-nucleus mean field configuration interaction (EN-MFCI) calculations. The potential energy of our model Hamiltonian corresponds to the Coulomb potential of an infinite wire with charge Z distributed according to a Gaussian function. The time independent equation for this Hamiltonian is solved perturbationally in the limit of small amplitude vibration (Gaussian function width close to zero).

The concept of seniority number is generalized, as well as that of seniority number operator. It affords to define new hierarchies of configuration interaction spaces. The usefulness of such a hierarchy is illustrated on the buckminsterfullerene system treated at the Huckel level of theory.

An expression for the square of the spin operator expectation value, $$\langle{S^{2}}\rangle$$ , is obtained for a general complex Hartree–Fock wave function and decomposed into four contributions: the main one whose expression is formally identical to the restricted (open-shell) Hartree–Fock expression. A spin contamination one formally analogous to that found for spin unrestricted Hartree–Fock wave functions. A noncollinearity contribution related to the fact that the wave function is not an eigenfunction of the spin-S z operator. A perpendicularity contribution related to the fact that the spin density is not constrained to be zero in the xy - plane. All these contributions are evaluated and compared for the H2O+ system. The optimization of the collinearity axis is also considered.