Molecular Electronic Wave Functions Research Articles

King-Plot Analysis of Isotope Shifts in Simple Diatomic Molecules

We demonstrate that the isotope shift in isotopomers of diatomic molecules, where the nucleus of one of its constituent atoms is replaced by another isotope, can be expressed as the sum of a field shift and a mass shift, similar to the atomic case. We show that a linear relation holds between atomic and molecular isotopes shifts, thus extending the King-plot analysis to molecular isotope shifts. Optical isotope shifts in YbF and ZrO and infrared isotope shifts in SnH are analyzed with a molecular King-plot approach, utilizing Yb$^{+}$ and Zr$^{+}$ ionic isotope shifts and charge radii of Sn obtained with non-optical methods. The changes in the mean-squared nuclear charge radii $\delta \langle r^2 \rangle$ of $^{170-174,176}$Yb and $^{90-92,94,96}$Zr extracted from the molecular transitions are found to be in excellent agreement with the values from the spectroscopy of Yb$^{+}$ and Zr$^{+}$, respectively. On the contrary, in the case of the vibrational-rotational transition in SnH, no sensitivity to the nuclear volume could be deduced within the experimental resolution, which makes it unsuitable for the extraction of nuclear charge radii but provides insights into the molecular electronic wave function not accessible via other methods. The new opportunities offered by the molecular King-plot analysis for research in nuclear structure and molecular physics are discussed.

Atoms and bonds in molecules as synergisms of interactions between electrons and nuclei.

The molecular electronic wave functions are expressed in quasi-atomic form. The corresponding global energy expression exhibits the unified resolution of the various modes of chemical bonding in terms of physical interactions. The bonding patterns in several molecules is elucidated.

Atoms and interatomic bonding synergism inherent in molecular electronic wave functions.

The chemical model of matter consists of atoms held together by bonds. The success of this model implies that the physical interactions of the electrons and nuclei in molecules combine into compound interactions that create the bonding. In the quantum mechanical description, the modified atoms in molecules and the bonding synergism are contained in the molecular electronic wave function. So far, only part of this information has been recovered from the wave function. Notably, the atoms have remained unidentified in the wave function. One reason is that conventional energy decomposition analyses formulate separate model wave functions, independent of the actual wave function, to represent "prepared atoms" and preconceived interactions and, then, intuitively catenate the parts. In the present work, the embedded modified atoms and the inherent physical synergisms between them are determined by a unified derivation entirely from the actual molecular valence space wave function. By means of a series of intrinsic orbital and configurational transformations of the wave function, the energy of formation of a molecule is additively resolved in terms of intra-atomic energy changes, interference energies, and quasi-classical, non-classical, and charge-transfer Coulombic interactions. The analysis furnishes an algorithm for the quantitative resolution of the energy of formation, which enables analyses elucidating reaction energies.

Independent Sources of Information-Theoretic Descriptors of Electronic States

The need for resultant measures of the Information-Theoretic (IT) content of molecular electronic wavefunctions, combining the information contributions due to the probability and phase/current distributions, is reemphasized. Complementary measures of the state entropy (disorder) and information (order) contents are reexamined, the continuities of wavefunction components are summarized, and the probability acceleration concept is used to determine the current and information sources. The experimental elimination of the state uncertainties is discussed and limitations in this information-acquirement process imposed by the Heisenberg indeterminacy principle are commented upon.

The Structure of Complexes of Carbonylruthenium(II) with Octaethylporphyrin

Porphyrin complexes of ruthenium with carbon monoxide and other ligands, especially with nitrogen-containing ones, are convenient models for studying natural energy centers. An X-ray diffraction analysis of single crystals of the complexes has shown the structural features determining their properties. In the complex of carbonylruthenium(II) with β-octaethylporphyrin, the ruthenium atom is firmly held by four nitrogen porphyrin atoms, while carbon monoxide acts as an axial ligand. Ruthenium(II) porphyrinates with coordinated axial ligands form similar crystalline structures of island type in the triclinic syngony belonging to spatial group P1. A five-coordinate ruthenium carbonyl complex of porphyrin, which is originally obtained in the reaction of β-octaethylporphyrin with triruthenium dodecacarbonyl, easily coordinates even weak ligands, such as ethanol, that easily can be exchanged by pyridine. The coordination polyhedron of the ruthenium atom can be described as a distorted tetragonal bipyramid. Comparison of the structure of ruthenium porphyrinate complexes has clearly demonstrated the peculiarities of manifestation of the trans influence of ligands. Carbon monoxide pulls the ruthenium atom out of a plane formed by four nitrogen atoms of porphyrin, causing a significant elongation (by about 10%) and the corresponding weakening of the bond of ruthenium with a trans ligand that occupies the sixth coordination position. Similarly, trans ligands affect the Ru–CO bond according to the coordination strength of the ligand. The quantum-chemical calculation of porphyrin complexes of ruthenium has been performed. The mechanism of the trans influence of ligands is illustrated by the analysis of molecular electronic wave functions by the method of natural localized orbitals. Identification of structural patterns and their relationship with the properties of complexes contributes to understanding the functioning of similar natural complexes of iron, including heme.

A new near-linear scaling, efficient and accurate, open-shell domain-based local pair natural orbital coupled cluster singles and doubles theory.

The Coupled-Cluster expansion, truncated after single and double excitations (CCSD), provides accurate and reliable molecular electronic wave functions and energies for many molecular systems around their equilibrium geometries. However, the high computational cost, which is well-known to scale as O(N6) with system size N, has limited its practical application to small systems consisting of not more than approximately 20-30 atoms. To overcome these limitations, low-order scaling approximations to CCSD have been intensively investigated over the past few years. In our previous work, we have shown that by combining the pair natural orbital (PNO) approach and the concept of orbital domains it is possible to achieve fully linear scaling CC implementations (DLPNO-CCSD and DLPNO-CCSD(T)) that recover around 99.9% of the total correlation energy [C. Riplinger et al., J. Chem. Phys. 144, 024109 (2016)]. The production level implementations of the DLPNO-CCSD and DLPNO-CCSD(T) methods were shown to be applicable to realistic systems composed of a few hundred atoms in a routine, black-box fashion on relatively modest hardware. In 2011, a reduced-scaling CCSD approach for high-spin open-shell unrestricted Hartree-Fock reference wave functions was proposed (UHF-LPNO-CCSD) [A. Hansen et al., J. Chem. Phys. 135, 214102 (2011)]. After a few years of experience with this method, a few shortcomings of UHF-LPNO-CCSD were noticed that required a redesign of the method, which is the subject of this paper. To this end, we employ the high-spin open-shell variant of the N-electron valence perturbation theory formalism to define the initial guess wave function, and consequently also the open-shell PNOs. The new PNO ansatz properly converges to the closed-shell limit since all truncations and approximations have been made in strict analogy to the closed-shell case. Furthermore, given the fact that the formalism uses a single set of orbitals, only a single PNO integral transformation is necessary, which offers large computational savings. We show that, with the default PNO truncation parameters, approximately 99.9% of the total CCSD correlation energy is recovered for open-shell species, which is comparable to the performance of the method for closed-shells. UHF-DLPNO-CCSD shows a linear scaling behavior for closed-shell systems, while linear to quadratic scaling is obtained for open-shell systems. The largest systems we have considered contain more than 500 atoms and feature more than 10 000 basis functions with a triple-ζ quality basis set.

Intrinsic Resolution of Molecular Electronic Wave Functions and Energies in Terms of Quasi-atoms and Their Interactions.

A general intrinsic energy resolution has been formulated for strongly correlated wave functions in the full molecular valence space and its subspaces. The information regarding the quasi-atomic organization of the molecular electronic structure is extracted from the molecular wave function without introducing any additional postulated model state wave functions. To this end, the molecular wave function is expressed in terms of quasi-atomic molecular orbitals, which maximize the overlap between subspaces of the molecular orbital space and the free-atom orbital spaces. As a result, the molecular wave function becomes the superposition of a wave function representing the juxtaposed nonbonded quasi-atoms and a wave function describing the interatomic electron migrations that create bonds through electron sharing. The juxtaposed nonbonded quasi-atoms are shown to consist of entangled quasi-atomic states from different atoms. The binding energy is resolved as a sum of contributions that are due to quasi-atom formation, quasiclassical electrostatic interactions, and interatomic interferences caused by electron sharing. The contributions are further resolved according to orbital interactions. The various transformations that generate the analysis are determined by criteria that are independent of the working orbital basis used for calculating the molecular wave function. The theoretical formulation of the resolution is quantitatively validated by an application to the C2 molecule.

A massively parallel tensor contraction framework for coupled-cluster computations

Precise calculation of molecular electronic wavefunctions by methods such as coupled-cluster requires the computation of tensor contractions, the cost of which has polynomial computational scaling with respect to the system and basis set sizes. Each contraction may be executed via matrix multiplication on a properly ordered and structured tensor. However, data transpositions are often needed to reorder the tensors for each contraction. Writing and optimizing distributed-memory kernels for each transposition and contraction is tedious since the number of contractions scales combinatorially with the number of tensor indices. We present a distributed-memory numerical library (Cyclops Tensor Framework (CTF)) that automatically manages tensor blocking and redistribution to perform any user-specified contractions. CTF serves as the distributed-memory contraction engine in Aquarius, a new program designed for high-accuracy and massively-parallel quantum chemical computations. Aquarius implements a range of coupled-cluster and related methods such as CCSD and CCSDT by writing the equations on top of a C++ templated domain-specific language. This DSL calls CTF directly to manage the data and perform the contractions. Our CCSD and CCSDT implementations achieve high parallel scalability on the BlueGene/Q and Cray XC30 supercomputer architectures showing that accurate electronic structure calculations can be effectively carried out on top of general distributed-memory tensor primitives.

Intrinsic minimal atomic basis representation of molecular electronic wavefunctions

The problem of finding an effective minimal atomic basis that spans the exact occupied wavefunctions of a mean-field theory at a given molecular geometry, which has a number of special properties, is studied and a new general procedure is developed that (1) solves for a raw minimal set of strongly atom-centered functions—products of spherical harmonics and molecule-optimized radial parts—that approximately span the occupied molecular wavefunctions and minimize the sum of their energies, (2) uses projection operators to get a new set of deformed atom-centered functions that exactly span the occupied space and fall into core and valence subsets, (3) applies a new zero-bond-dipole orthogonalization scheme to the core-orthogonalized valence subset so that for each two-center product of these functions the projection of its dipole moment along the line going through the two centers is zero. The resulting effective minimal atomic basis is intrinsic to the molecular problem and does not need a free-atoms input. Some interesting features of the zero-bond-dipole orthogonalization are showing up in the atomic population analysis of a diverse set of molecules. The new procedure may be useful for the interpretation of electronic structure, for the construction of model Hamiltonians in terms of transferable molecular integrals, and for the definition of active valence space in the treatment of electron correlation. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011

First order simultaneous optimization of molecular geometry and electronic wave function

We present a very efficient simultaneous optimization method for both molecular geometry and electronic wave function. We introduce a simultaneous least squares scheme to minimize errors in a full local quadratic space of both geometric and wave function vectors. Several self-consistent-field iterations are taken until a uniform search direction is achieved in every geometry optimization step. Not only does this new simultaneous optimization method significantly reduce the number of self-consistent-field iterations to reach convergence, it is able to overcome shallow potential wells to find a better stationary point. This new method exhibits up to approximately 60% savings in computational cost compared to conventional geometry optimization methods.

Molecular orbital tomography using short laser pulses

Recently, a method to image molecular electronic wave functions using high harmonic generation (HHG) was introduced by Itatani \textit{et al.\} [Nature {\textbf{432}}, 876 (2004)]. We show that, while the tomographic reconstruction of general orbitals with arbitrary symmetry cannot be performed with long laser pulses, this becomes possible when extremely short pulses are used. An alternative reconstruction equation based on momentum matrix elements, rather than on dipole matrix elements, is proposed. We present simulations of the procedure for 2D model systems based on numerical solutions of the time-dependent Schr\{o}dinger equation, and present results from further post-processing of the reconstructed orbitals.

Molecular Recollision Interferometry in High Harmonic Generation

We use extreme-ultraviolet interferometry to measure the phase of high-order harmonic generation from transiently aligned CO(2) molecules. We unambiguously observe a reversal in phase of the high-order harmonic emission for higher harmonic orders with a sufficient degree of alignment. This results from molecular-scale quantum interferences between the molecular electronic wave function and the recolliding electron as it recombines with the molecule, and is consistent with a two-center model. Furthermore, using the combined harmonic intensity and phase information, we extract accurate information on the dispersion relation of the returning electron wave packet as a function of harmonic order. This analysis shows evidence of the effect of the molecular potential on the recolliding electron wave.

Reformulation in mathematical programming: An application to quantum chemistry

This paper concerns the application of reformulation techniques in mathematical programming to a specific problem arising in quantum chemistry, namely the solution of Hartree–Fock systems of equations, which describe atomic and molecular electronic wave functions based on the minimization of a functional of the energy. Their traditional solution method does not provide a guarantee of global optimality and its output depends on a provided initial starting point. We formulate this problem as a multi-extremal nonconvex polynomial programming problem, and solve it with a spatial Branch-and-Bound algorithm for global optimization. The lower bounds at each node are provided by reformulating the problem in such a way that its convex relaxation is tight. The validity of the proposed approach was established by successfully computing the ground-state of the helium and beryllium atoms.

Intrinsic local constituents of molecular electronic wave functions. II. Electronic structure analyses in terms of intrinsic oriented quasi-atomic molecular orbitals for the molecules FOOH, H2BH2BH2, H2CO and the isomerization HNO → NOH

The electronic bonding structures of the molecules FOOH, H2CO, H2BH2BH2 and the isomerization HNO → NOH are analyzed in terms of the intrinsic, oriented quasi-atomic molecular orbitals that are extracted from the optimized full-valence multi-configuration self-consistent-field wavefunctions by the unbiased, basis-set-independent procedure described in the preceding paper. In all cases, the method brings to light the essentials of the bonding interactions. Detailed insights are furnished regarding the hyper-conjugation between lone pairs and nearby antibonding orbitals in FOOH and H2CO, regarding the three-center bonding in diborane, and regarding the transition state structure in HNO. The versatility of the use of the quasi-atomic orbitals is exemplified.

Intrinsic local constituents of molecular electronic wave functions. I. Exact representation of the density matrix in terms of chemically deformed and oriented atomic minimal basis set orbitals

A coherent, intrinsic, basis-set-independent analysis is developed for the invariants of the first-order density matrix of an accurate molecular electronic wavefunction. From the hierarchical ordering of the natural orbitals, the zeroth-order orbital space is deduced, which generates the zeroth-order wavefunction, typically an MCSCF function in the full valence space. It is shown that intrinsically embedded in such wavefunctions are elements that are local in bond regions and elements that are local in atomic regions. Basis-set-independent methods are given that extract and exhibit the intrinsic bond orbitals and the intrinsic minimal-basis quasi-atomic orbitals in terms of which the wavefunction can be exactly constructed. The quasi-atomic orbitals are furthermore oriented by a basis-set independent method (viz. maximization of the sum of the fourth powers of all off-diagonal density matrix elements) so as to exhibit clearly the chemical interactions. The unbiased nature of the method allows for the adaptation of the localized and directed orbitals to changing geometries.

Intensities of the photoelectron spectra of weakly bound anions: A complex of an atomic anion with a diatomic molecule

A closed formula for the state-resolved photodetachment probabilities of an electron from a weakly bound complex of an atomic anion with a diatomic molecule residing in a nondegenerate electronic state is obtained. Due to the reduction of the dipole moment matrix elements in terms of molecular electronic wave functions to the matrix elements in terms of atomic wave functions, the formula allows one to calculate the relative intensities of the photoelectron spectrum taking into account rovibrational and spin-orbit interactions and using practically convenient representations of the wave functions of the bound initial state of an anionic complex and of the bound or unbound final state of a neutral system. The methods of estimation of the atomic dipole moment matrix elements, the error due to neglect of the photoelectron rescattering, and possible modifications and generalizations of the formula are discussed.

Characterization of the molecular iodine electronic wave functions and potential energy curves through hyperfine interactions in the B0+_u(3Pi_u) state

We present a high-resolution analysis of the six electronic states that share the same dissociation limit with the second excited electronic state B in molecular iodine. These six states are coupled to the B state via hyperfine interactions. The four hyperfine parameters, C_B, delta_B, d_B, and eqQ_B, are calculated with the available potential energy curves and wave functions constructed from the separated-atom basis set. We obtain a maximum separation of the respective contributions from all six electronic states and compare each individual contribution with high-precision spectroscopic data, providing an independent verification of the relevant electronic structure.

Using an interval branch‐and‐bound algorithm in the Hartree–Fock method

AbstractThe Hartree–Fock (HF) method is widely used to obtain atomic and molecular electronic wave functions, based on the minimization of a functional of the energy. We propose to use a deterministic global optimization algorithm, based on a branch‐and‐bound method, that applies techniques of interval arithmetic. This algorithm is applied directly to the minimization of the energy expression derived from the HF method. The proposed approach was successfully applied to the ground state of He and Be atoms. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005

Ab initio molecular quantum defect theory: I. Method of artificial well potentials

A new method is presented for the calculation of highly excited bound and continuum molecular electronic wavefunctions. It combines the methods of ab initio quantum chemistry with multichannel quantum defect theory. The approach is based on artificial well potentials (AWPs) designed such that square-integrable Cartesian Gaussian basis sets can be used for the calculation of continuum states at arbitrary energy. The method will be illustrated in the following paper by an application to the NO molecule.

Extensive, high-resolution measurement of hyperfine interactions: precise investigations of molecular potentials and wave functions

We report high-resolution and high-precision measurement of hyperfine interactions of the first excited electronic state (B) of I 2 over an extensive range of vibrational and rotational quantum numbers. Systematic variations in the hyperfine parameters, including both rovibrational level dependence and internuclear separation dependence, are confirmed by calculations based on ab initio molecular potential energy curves and electronic wave functions derived from a separated-atom basis set. We have accurately determined the state-dependent quantitative changes of hyperfine interactions caused by perturbations from other electronic states and identified the respective perturbing states.