Correlated wave-function theory for many-center many-electron problems

The correlated electronic wave-function theory developed by S. Obara and K. Hirao [Bull. Chem. Soc. Jpn. 66 , 3300 (1993)], as applied to two-electron molecular systems, is generalized to many-center many-electron problems. The exact formulas for effective Hamiltonian operators are given. The rules for the calculation of matrix elements with three-electron operators over Slater determinants are formulated. From the energy-minimum principle, the system of master equations is derived for variational coefficients of a trial wave function for the molecules with closed electronic shells. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 69: 639–648, 1998     

The rotation–vibration coupling equations for polyatomic molecules in internal coordinates

An exact vibration–rotation kinetic energy operator for polyatomic molecules has been obtained on the basis of Sutcliffe's method, in terms of curvilinear internal coordinates and rotational angular moment operators. This operator is derived from the kinetic energy operator in Cartesian coordinates by the successive transformations using the chain rule. This kinetic energy operator can be used not only for the system of any triatomic and tetraatomic molecules and common polyatomic molecules in chemistry, but also for the investigation of the collision problems between two molecules after some modifications. Finally, using this Hamiltonian, the rotation–vibration coupling equations of polyatomic molecules have been derived and discussed. © 1992 John Wiley & Sons, Inc.     

General criteria for assessing the accuracy of approximate wave functions and their densities

Master equations for propagators in quantum open systems and their spectral resolutions are derived. The Zwanzig partitioning scheme along the superoperator algebra are used to derive equations of motion for partitioned operators in a Liouville space. The reservoir influence on the dynamical evolution of operators is shown to lead explicitly to dissipative effects arising from memory terms in the evolution equations of such operators. It is also shown that spectral representations may be written in a self-consistent analytic way by means of the self-energy fields for transition energies of the system by taking into account the lack of the complete knowledge about the reservoir. A kinematic fluid interpretation of the resultant equations is given and an explicit form of the “collision” superoperator is obtained. Finally, a simple example to illustrate the determination of self-energy fields for the system–reservoir interaction corrections is given. © 1995 John Wiley & Sons, Inc.     

Unitary transformations with the unitary operators depending on projection operators

Unitary transformations of operators and variables are considered for those cases when a Hermitian operator in the expression U = exp (i S ) explicitly depends on the projection operators. Some simple examples of such transformations are given and the equations obtained which define projectors (among special sets of them), thus bringing forth better estimations for upper and lower bounds. The unitary transformations are also applied to the reduced resolvent operator and equations are presented for the operators U that minimize the energy functional.     

Dynamics,hyperfine structure and symmetry of nonrigid molecules

The transformation properties of the Born—Oppenheimer problem and of two probability densities for electronic singlet states of semirigid molecules are reviewed, based on the isometric group concept. A definition of sets of equivalent nuclei of semirigid molecules is put forward, which will be used to derive general transformation formulae for vector operators associated with nuclear spin operators of equivalent sets. Furthermore transformation formulae for general tensor operators and for phenomenological field gradient and spin—rotation coupling tensors of semirigid molecules will be presented and illustrated by examples. The phenomenological relations will be verified by considering the transformation properties of some of the relevant terms of the molecular hamiltonian.     

Selection rules and line-strength factors for multiphoton transitions in gas phase molecular spectroscopy

Multiphoton transitions are discussed in terms applicable to experimental spectroscopy. A simple tensor operator for the equations governing multiphoton transitions in molecules is obtained. Selection rules for these processes are derived by symmetry considerations and procedures for calculation of rotational line-strength factors are given. For the general transition, the rovibronic n-photon selection rules can be determined by the symmetry product of a n-photon operator with the rovibronic wavefunctions. The line-strength factors are found by considering the integral over rotations of the nth rank tensor operators. The line-strength evaluations do not depend on exact knowledge of the vibronic overlap, to within a constant factor.     

Self-consistent embedded clusters: Building block equations for localized orthogonal orbitals

In order to be able to study a large electronic system following a building block approach, in which smaller tractable subsystems are handled at a time rather than the system as a whole, equations are proposed in this paper whose solutions are variational orthogonal orbitals localized on the subsystems. The equations for a given subsystem correspond to a molecular cluster embedded in the field created by the rest of the system, and are coupled to the corresponding equations for all subsystems under consideration, so that they must be solved self-consistently. While the localized nature of the solutions makes the equations appropriate for use in conjunction with local basis sets in practical implementations without significant loss of precision due to truncation errors, their orthogonality properties allow for the use of the advantages of the theory of separability of McWeeny in order to calculate total energies and (generalized product) wave functions. Since the building block equations proposed involve inter-subsystem interactions very cumbersome to calculate, an approximation is proposed in order to make their application to actual problems feasible: the representation of the cumbersome interaction operators by ab initio model potentials which are obtained directly from them, without resorting to any parametrization procedure based on a reference. This ab initio model potential approximation has been found to provide considerable computational savings without significant loss of accuracy in frozen-core calculations on molecules and frozen-lattice calculations on imperfect crystals.     

Novel method for geometry optimization of molecular clusters: application to benzene clusters

A heuristic and unbiased method for searching optimal geometries of clusters of nonspherical molecules was constructed from the algorithm recently proposed for Lennard-Jones atomic clusters. In the method, global minima are searched by using three operators, interior, surface, and orientation operators. The first operator gives a perturbation on a cluster configuration by moving molecules near the center of mass of a cluster, and the second one modifies a cluster configuration by moving molecules to the most stable positions on the surface of a cluster. The moved molecules are selected by employing a contribution of the molecules to the potential energy of a cluster. The third operator randomly changes the orientations of all molecules. The proposed method was applied to benzene clusters. It was possible to find new global minima for (C6H6)11, (C6H6)14, and (C6H6)15. Global minima for (C6H6)16 to (C6H6)30 are first reported in this article.     

The loge theory as a starting point for variational calculations. I. General formalism

It is shown that any expectation value of any observable associated with a molecule is the sum of loge contributions and of loge pair contributions. This result provides a rigorous theoretical basis for the study of additive properties of molecules. It is demonstrated that molecular wave functions (exact or approximate) can be expressed as a sum of functions corresponding to the various electronic events. Furthermore any of these event functions can be expressed in terms of correlated loge functions. This expression suggests many kinds of variational procedures of calculating wave functions (known methods and new ones). The case in which noncorrelated completely localized loge functions are used is discussed. If continuous functions are used the variational equation reduces to a sum of independent variational equations, each one corresponding to a particular electronic event. This is not so when discontinuous functions are used or when a delocalized function is added to replace the correlation interloge function. The noncorrelated completely localized loge model is analyzed in more detail. It is seen that local spin operators can be introduced and that each event density operator is the product of the loge density operators. Therefore that model is an independent loge model. The corresponding generalized self-consistent field equations are derived. This treatment helps us to understand how a localized state of a molecule can produce an ion containing a delocalized region, a phenomenon which is sometimes at the origin of some misunderstanding in photoelectron spectroscopy. Finally it is seen how virtual loge functions can be introduced to describe excited states.     

The use of partitioning technique in the solution of operator equations for ionizations and excitations

An inhomogeneous operator equation is associated with the equations of motion for ionization- and excitation operators. The partitioning technique is applied to solve the equation. Formal expressions for the ionization- and excitation operators are obtained and examples of approximations to these are given.     

Nonadiabatic electron wavepacket dynamics of molecules in an intense optical field: an ab initio electronic state study

A theory of quantum electron wavepacket dynamics that nonadiabatically couples with classical nuclear motions in intense optical fields is studied. The formalism is intended to track the laser-driven electron wavepackets in terms of the linear combination of configuration-state functions generated with ab initio molecular orbitals. Beginning with the total quantum Hamiltonian for electrons and nuclei in the vector potential of classical electromagnetic field, we reduce the Hamiltonian into a mixed quantum-classical representation by replacing the quantum nuclear momentum operators with the classical counterparts. This framework gives equations of motion for electron wavepackets in an intense laser field through the time dependent variational principle. On the other hand, a generalization of the Newtonian equations provides a matrix form of forces acting on the nuclei for nonadiabatic dynamics. A mean-field approximation to the force matrix reduces this higher order formalism to the semiclassical Ehrenfest theory in intense optical fields. To bring these theories into a practical quantum chemical package for general molecules, we have implemented the relevant ab initio algorithms in it. Some numerical results in the level of the semiclassical Ehrenfest-type theory with explicit use of the nuclear kinematic (derivative) coupling and the velocity form for the optical interaction are presented.     

Combined Open Shell Hartree–Fock Theory of Atomic–Molecular and Nuclear Systems

In this study, the combined Hartree–Fock (HF) and Hartree–Fock–Roothaan equations are derived for multideterminantal single configuration states with any number of open shells of atoms, molecules and nuclei. It is shown that the postulated orbital-dependent energy and Fock operators are invariant to the unitary transformation of orbitals. This new methodology is based entirely on the spin-restricted HF theory. As an application of combined open shell theory of atomic–molecular and nuclear systems presented in this paper, we have solved Hartree–Fock–Roothaan equations for the ground state of electronic configuration C(1s ²2s ²2p ²) using Slater type orbitals as a basis.     

Some solutions in the graph theory of alternant benzenoids

A new formulation of the eigenvector equations for certain polymeric molecules is given. The electronic bands for the infinite molecules are deduced. For the finite molecules, the use of ghost molecules allows an exact specialization of the infinite to the finite eigenvector equations. The complete solution of the equations depends on the solution of a simple transcendental equation. Approximate solutions of this equation are given that bracket the accurate solutions so that, for molecules with many repeated moieties, the approximations become good. The general theory is illustrated by applications to the chrysene series and to the polyphenyls. © 1993 John Wiley & Sons, Inc.     

The evaluation of matrix elements in the analysis of anharmonic molecular vibrations: Functional taylor series expansions

This article presents methods for computing Cartesian Gaussian matrix elements using a Taylor series for general potential energy operators that admit well-behaved radial derivatives. These operators arise in the analyses of anharmonic vibrations in molecules. Application to the evaluation of matrix elements for hydrogen associated two wells illustrates the method. © 1995 John Wiley & Sons, Inc.     

Some analytical properties of the dilated SCF Equations

The self-consistent field (SCF ) equations for many-electron systems, suitable within the complex-coordinate method, are derived. The formulation is based on a general bivariational theorem for non-Hermitian operators, with an emphasis on the analytic structure invoked by the complex dilation of the total Hamiltonian. The dilation structure of the resulting SCF equations is stressed and the concomitant analytical properties are discussed. The solutions are classified with respect to these properties, and interpreted in terms of a general form of the symmetry dilemma. The role of the dilated SCF equations for resonance calculations is discussed.     

Graph neural network based coarse-grained mapping prediction

The selection of coarse-grained (CG) mapping operators is a critical step for CG molecular dynamics (MD) simulation. It is still an open question about what is optimal for this choice and there is a need for theory. The current state-of-the art method is mapping operators manually selected by experts. In this work, we demonstrate an automated approach by viewing this problem as supervised learning where we seek to reproduce the mapping operators produced by experts. We present a graph neural network based CG mapping predictor called Deep Supervised Graph Partitioning Model (DSGPM) that treats mapping operators as a graph segmentation problem. DSGPM is trained on a novel dataset, Human-annotated Mappings (HAM), consisting of 1180 molecules with expert annotated mapping operators. HAM can be used to facilitate further research in this area. Our model uses a novel metric learning objective to produce high-quality atomic features that are used in spectral clustering. The results show that the DSGPM outperforms state-of-the-art methods in the field of graph segmentation. Finally, we find that predicted CG mapping operators indeed result in good CG MD models when used in simulation.

We propose a scalable graph neural network-based method for automating coarse-grained mapping prediction for molecules.     

Study of simulation method of time evolution of atomic and molecular systems by quantum electrodynamics

We discuss a method to follow step‐by‐step time evolution of atomic and molecular systems based on quantum electrodynamics. Our strategy includes expanding the electron field operator by localized wavepackets to define creation and annihilation operators and following the time evolution using the equations of motion of the field operator in the Heisenberg picture. We first derive a time evolution equation for the excitation operator, the product of two creation or annihilation operators, which is necessary for constructing operators of physical quantities such as the electronic charge density operator. We, then, describe our approximation methods to obtain time differential equations of the electronic density matrix, which is defined as the expectation value of the excitation operator. By solving the equations numerically, we show “electron‐positron oscillations,” the fluctuations originated from virtual electron‐positron pair creations and annihilations, appear in the charge density of a hydrogen atom and molecule. We also show that the period of the electron‐positron oscillations becomes shorter by including the self‐energy process, in which the electron emits a photon and then absorbs it again, and it can be interpreted as the increase in the electron mass due to the self‐energy. © 2014 Wiley Periodicals, Inc.     

Internal symmetry groups of non-rigid molecules

Hougen has established, for quasi-rigid molecules, the relationship between permutationinversions acting on the molecular Hamiltonians written in Cartesian co-ordinates and permutation-rotations (perrotations) of symmetry acting on nuclear equilibrium configurations. We extend these relations to the case of non-rigid molecules. For this, we introduce kinetic perrotations which act on nuclear equilibrium configurations in the same way as do Altmann's isodynamic operators. We show that isodynamic operators do not always form a group. Moreover, their action cannot be extended to the electrons. They cannot be used for the classification of molecular wave functions. This classification is achieved by using the group of Longuet-Higgins and the group of the corresponding feasible perrotations.