Grassmann coherent states representation of the path integral: Evaluation of the generating function for spin systems

An interacting spin system is investigated within the scenario of the Feynman path integral representation of quantum mechanics. Short‐time propagator algorithms and a discrete time formalism are used in combination with a basis set involving Grassmann variables coherent states to get a many‐body analytic propagator. The generating function thus obtained leads, after an adequate tracing over Grassmann variables in the imaginary time domain, to the partition function. A spin 1/2 Hamiltonian involving the whole set of interactions is considered. Fermion operators satisfying the standard anticommutation relations are constructed from the raising and lowering spin operators via the Jordan–Wigner transformation. The partition function obtained is more general than the partition function of the traditional Ising model involving only first‐neighbor interactions. Computations were performed assuming that the coupling as a function of the distance can be reasonably well represented by an Airy function. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002     

Hydrodynamic interactions between spheres in a viscous fluid with a flat free surface or hard wall

Hydrodynamic interactions between spheres immersed in a low-Reynolds-number fluid flow close to a flat free surface or hard wall are investigated. The spheres may have different or equal radii, and may be separated from the boundary or at contact with the free surface. A simple and useful expression is derived for the propagator (Green operator) connecting centers of two spheres. In the derivation, the method of images and the displacement theorems are used. Symmetry of the displacement operators is explicitly shown. The significance of these results in efficient Stokesian and Brownian dynamics simulations is outlined. An example of an application is shown.     

A semiclassical initial-value representation for quantum propagator and boltzmann operator

Starting from the position-momentum integral representation, we apply the correction operator method to the derivation of a uniform semiclassical approximation for the quantum propagator and then extend it to approximate the Boltzmann operator. In this approach, the involved classical dynamics is determined by the method itself instead of given beforehand. For the approximate Boltzmann operator, the corresponding classical dynamics is governed by a complex Hamiltonian, which can be described as a pair of real Hamiltonian systems. It is demonstrated that the semiclassical Boltzmann operator is exact for linear systems. A quantum propagator in the complex time is thus proposed and preliminary numerical results show that it is a reasonable approximation for calculating thermal correlation functions of general systems. © 2018 Wiley Periodicals, Inc.     

Asymmetric rotor-like probes to polarized fluorescence study of the macroscopically oriented uniaxial media: Model parameters recognition

In this paper the possibility of the numerical data modelling in the case of angle- and time-resolved fluorescence spectroscopy is investigated. The asymmetric fluorescence probes are assumed to undergo the restricted rotational diffusion in a hosting medium. This process is described quantitatively by the diffusion tensor and the aligning potential. The evolution of the system is expressed in terms of the Smoluchowski equation with an appropriate time-developing operator. A matrix representation of this operator is calculated, then symmetrized and diagonalized. The resulting propagator is used to generate the synthetic noisy data set that imitates results of experimental measurements. The data set serves as a groundwork to the χ² optimization, performed by the genetic algorithm followed by the gradient search, in order to recover model parameters, which are diagonal elements of the diffusion tensor, aligning potential expansion coefficients and directions of the electronic dipole moments. This whole procedure properly identifies model parameters, showing that the outlined formalism should be taken in the account in the case of analysing real experimental data.     

The extended Koopmans' theorem Fock operator and the generalized overlap amplitude one-electron operator

The wave function of a system may be expanded in terms of eigenfunctions of the N −1 electron Hamiltonian times one-particle functions known as generalized overlap amplitudes (GOAS). The one-electron operator whose eigenfunctions are the GOAS is presented, without using an energy-dependent term as in the one-particle Green function or propagator approach. It is shown that this operator and the extended Koopmans' theorem (EKT) one-electron operator are of similar form, but perform complementary roles. The GOA operator begins with one-electron densities and total energies of N −1 electron states to generate the two-matrix and total energy of an N-electron state. The EKT operator begins with the two-matrix of an N-electron state to generate one-electron densities and ionization potentials (or approximations thereto) for N −1 electron states. However, whereas the EKT orbitals must be linearly independent, no such restriction applies to the GOAS. © 1996 John Wiley & Sons, Inc.     

Self-consistent approximation to the polarization propagator

A theorem is presented that characterizes approximate states and truncated operator manifolds associated with self-consistent approximate propagators. This theorem establishes a natural relationship between Hermiticity, stationarity, nonredundance and completeness of operator manifolds, model time evolution, and the vacuum condition. For the case of the polarization propagator we describe algorithms by which we can construct states and manifolds that satisfy this theorem and the vacuum condition.     

Independent particle theory with electron correlation

We formulate an effective independent particle model where the effective Hamiltonian is composed of the Fock operator and a correlation potential. Within the model the kinetic energy and the exchange energy can be expressed exactly leaving the correlation energy functional as the remaining unknown. Our efforts concentrate on finding a correlation potential such that exact ionization potentials and electron affinities can be reproduced as orbital energies. The equation-of-motion coupled-cluster approach enables us to define an effective Hamiltonian from which a correlation potential can be extracted. We also make the connection to electron propagator theory. The disadvantage of the latter is the inherit energy dependence of the potential resulting in a different Hamiltonian for each orbital. Alternatively, the Fock space coupled-cluster approach employs an effective Hamiltonian which is energy independent and universal for all orbitals. A correlation potential is extracted which yields the exact ionization potentials and electron affinities and a set of associated molecular orbitals. We also describe the close relationship to Brueckner theory.     

Assessment of transition operator reference states in electron propagator calculations

The transition operator method combined with second-order, self-energy corrections to the electron propagator (TOEP2) may be used to calculate valence and core-electron binding energies. This method is tested on a set of molecules to assess its predictive quality. For valence ionization energies, well known methods that include third-order terms achieve somewhat higher accuracy, but only with much higher demands for memory and arithmetic operations. Therefore, we propose the use of the TOEP2 method for the calculation of valence electron binding energies in large molecules where third-order methods are infeasible. For core-electron binding energies, TOEP2 results exhibit superior accuracy and efficiency and are relatively insensitive to the fractional occupation numbers that are assigned to the transition orbital.     

The antisymmetrized geminal power approximation to the excitation propagator

A consistent propagator approximation, denoted as the excitation propagator, is introduced. This propagator describes excitations between N-particle states and its approximation has properties required of consistent random phase approximation schemes. Several properties of this propagator are explored when based on a generalized antisymmetrized geminal power wavefunction. How singularities in the metric occur and how to remove them is discussed in detail. The excitation propagator is also contrasted with the principal (polarization) propagator.     

The use of localized molecular orbitals and the polarization propagator to identify transmission mechanisms in nuclear spin-spin couplings

An extension of the IPPP (inner projections of the polarization propagator) method to theoretically analyze transmission mechanisms of indirect nuclear spin-spin couplings is presented. The localization technique used is modified so that all the canonical molecular orbitals of a compound may be localized to represent chemical bonds, lone pairs, and the corresponding antibonding molecular orbitals. These localized molecular orbitals, together with the polarization propagator, are used to obtain an intuitive picture of how a coupling is generated as a sum of terms, each one consisting of two particle-hole single excitations. This picture can be used to identify underlying transmission mechanisms and quantitatively evaluate their importance toward the total coupling. The paramagnetic spin-orbit and the spin-dipole interactions are studied in detail.     

Comparison of perturbative and multiconfigurational electron propagator methods

Ionization energies below 20 eV of 10 molecules calculated with electron propagator techniques employing Hartree-Fock orbitals and multiconfigurational self-consistent field orbitals are compared. Diagonal and nondiagonal self-energy approximations are used in the perturbative formalism. Three diagonal methods based on second- and third-order self-energy terms, all known as the outer valence Green's function, are discussed. A procedure for selecting the most reliable of these three versions for a given calculation is tested. Results with a polarized, triple ζ basis produce root mean square errors with respect to experiment of approximately 0.3 eV. Use of the selection procedure has a slight influence on the quality of the results. A related, nondiagonal method, known as ADC (3), performs infinite-order summations on several types of self-energy contributions, is complete through third-order, and produces similar accuracy. These results are compared to ionization energies calculated with the multiconfigurational spin-tensor electron propagator method. Complete active space wave functions or close approximations constitute the reference states. Simple field operators and transfer operators pertaining to the active space define the operator manifold. With the same basis sets, these methods produce ionization energies with accuracy that is comparable to that of the perturbative techniques. © 1996 John Wiley & Sons, Inc.     

On-the-fly, electric-field-driven, coupled electron-nuclear dynamics

An on-the-fly, electric field driven, coupled electron-nuclear dynamics approach is developed and applied to model the photodissociation of water in the A((1)B1) excited state. In this method, a quantum propagator evolves the photon-induced electronic dynamics in the ultrafast time scale, and a quasi-classical surface hopping approach describes the nuclear dynamics in the slower time scale. In addition, strong system-field interactions are explicitly included in the electronic propagator. This theoretical development enables us to study rapid photon-induced bond dissociation dynamics and demonstrates the partial breakdown of electronic coherence as well as electronic population trapping in the excited state when the molecular vibrations detune the system with respect to the applied field. The method offers a practical way to use on-the-fly dynamics for modeling light-molecule interactions that lead to interesting photochemical events.     

Wavepacket propagation using time-sliced semiclassical initial value methods

A new semiclassical initial value representation (SC-IVR) propagator and a SC-IVR propagator originally introduced by Kay [J. Chem. Phys. 100, 4432 (1994)], are investigated for use in the split-operator method for solving the time-dependent Schrodinger equation. It is shown that the SC-IVR propagators can be derived from a procedure involving modified Filinov filtering of the Van Vleck expression for the semiclassical propagator. The two SC-IVR propagators have been selected for investigation because they avoid the need to perform a coherent state basis set expansion that is necessary in other time-slicing propagation schemes. An efficient scheme for solving the propagators is introduced and can be considered to be a semiclassical form of the effective propagators of Makri [Chem. Phys. Lett. 159, 489 (1989)]. Results from applications to a one-dimensional, two-dimensional, and three-dimensional Hamiltonian for a double-well potential are presented.     

Stereochemical dependence of NMR geminal spin-spin coupling constants

In this work it was sought to explore the versatility of geminal spin-spin coupling constants, (2)J(XY) SSCCs, as probes for stereochemical studies. A set of compounds, where their experimental (2)J(XY) SSCCs through the X-C-Y molecular fragment are predicted to be sensitive to hyperconjugative interactions involving either bonding or antibonding orbitals containing the C carbon atom ('coupling pathway'), were analyzed. SSCC calculations were performed for some selected examples using the second order polarization propagator approximation (SOPPA) method or within the DFT-B3LYP framework. Hyperconjugative interactions were calculated within the Natural Bond Orbital (NBO) approach. Results are condensed in two qualitative rules: Rule I(M)-hyperconjugative interactions transferring charge into the coupling pathway yield a positive increase to the Fermi contact (FC), contribution to (2)K(XY) reduced spin-spin coupling constants (RSSCC), and Rule II(M)-hyperconjugative interactions transferring charge from the coupling pathway yield a negative increase to the FC contribution to (2)K(XY) RSSCC.     

The shift operator matrix method applied to the two-dimensional nearest and next nearest neighbor problem

The shift operator matrix (SOM) method is discussed. We show that in the thermodynamic limit, the largest eigenvalue of the SOM determines the grand canonical partition function for situations when simple, nearest-neighbor and next-nearest neighbor interacting particles are distributed on anM ×N lattice space. In addition, we present a method for calculating the appropriate shift operator matrices.     

A semiclasical justification for the use of non-spreading wavepackets in dynamics calculations

A justification is given for the use of non-spreading or frozen gaussian packets in dynamics calculations. In this work an initial wavefunction or quantum density operator is expanded in a complete set of grussian wavepackets. It is demonstrated that the time evolution of this wavepacket expansion for the quantum wavefunction or density is correctly given within the approximations employed by the classical propagation of the avarage position and momentum of each gaussian packet, holding the shape of these individual gaussians fixed. The semiclassical approximation is employed for the quantum propagator and the stationary phase approximation for certain integrals is utilized in this derivation. This analysis demonstrates that the divergence of the classical trajectories associated with the individual gaussian packets accounts for the changes in shape of the quantum wavefunction or density, as has been suggested on intuitive grounds by Heller. The method should be exact for quadratic potentials and this is verified by explicitly applying it for the harmonic oscillator example.     

Acidity predictions in an electron propagator approach

An alternative way to calculate vertical ionization potentials (VIP) and vertical electron affinity (VEA) is the application of Koopman's theorem, using the electron propagator theory. In the present work, the results of the application of this theorem using the electron propagator formalism have been compared with the experiment in order to validate different basis set. Using the basis set with the best performance, the acidity tendencies in some substituted acetic acid molecules have been analyzed by correlating the proton affinity (PA) with molecular electronegativity (χ) and hardness (η); these last indexes were obtained from the calculated VIP and VEA considering the finite difference approximation. The above correlations were compared with equivalent correlations using the energy of the frontier Hartree–Fock orbitals and the corresponding Kohn–Sham orbitals, which were calculated with the B3LYP‐DFT procedure. The results indicate that the electron propagator theory could be an interesting alternative to evaluate reactivity indexes, since this theory gives reliable values of VIP and VEA. It was also found that (i) the VIP values are very close to experiment, with only a 0.38% of error; (ii) acceptable results are inferred for VEA; (iii) a triple zeta quality function works quite well in these calculations, and particularly the 6‐311G(d,p) basis set is the best, as it had been reported; and (iv) using the depronation energy (DPE), good results were obtained in the correlations δDPE‐VEA and δDPE‐χ. The results tested that P3 approximation in the electron propagator approach can be a new and interesting alternative in predicting VIP, VEA, and some reactivity indexes, such as χ and η, at least for the compounds studied. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2006     

Electronic structure analysis and electron detachment energies of polynitrogen pentagonal aromatic anions

Various decouplings of the electron propagator have been employed to provide theoretical comparison to experimental electron detachment energies for the pyrrolide, imidazolide, and pyrazolide anions. Predictions for isoelectronic anions in which CH groups are replaced by N atoms also are reported. The ab initio electron propagator results agree closely with experimental values, and the associated Dyson orbitals provide a detailed catalog of bonding changes as the number and positions of N atoms vary within the set of pentagonal aromatic anions.