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.     

Factorization of certain evolution operators using lie algebra: Formulation of the method

In this work, a new method to factorize certain evolution operators into an infinite product of simple evolution operators is presented. The method uses Lie operator algebra and the evolution operators are restricted to exponential form. The argument of these forms is a first-order linear partial differential operator. The method has broad applications, including the areas of sensitivity analysis, the solution of ordinary differential equations, and the solution of Liouville's equation. A sequence of -approximants is generated to represent the Lie operators. Under certain conditions, the convergence rate of the -approximant sequences is remarkably high. This work presents the general formulation of the scheme and some simple illustrative examples. Investigation of convergence properties is given in a companion paper.Supported by the Department of Energy.     

Error analysis of molecular dynamics and fractal time approximants from a combinatorial perspective

Trotter's theorem forms the theoretical basis of most modern molecular dynamics. In essence this theorem states that a time displacement operator (a Lie operator) constructed by exponentiating a sum of noncommuting operators can be approximated by a product of single operators provided the time interval is "very small." In theory "very small" implies infinitesimally small (at which point the approximate product becomes exact), while in practical analysis a finite time interval is divided into several small subintervals or steps. It follows, therefore, that the larger the number of steps the better the approximation to the exact time displacement operator. The question therefore arises: How many steps are sufficient? For bounded operators, standard theorems are available to provide the answer. In this paper we show that a very simple combinatorial formula can be derived which allows the computation of the global differences (as a function of the number of steps) between the Taylor coefficients of the exact time displacement operator and an approximate one constructed by using a finite number of steps. The formula holds for both bounded and nonbounded operators and shows, quantitatively, what is qualitatively expected-that the error decreases with increasing number of steps. Furthermore, the formula applies irrespective of the complexity of the system, boundary conditions, or the thermodynamic ensemble employed for averaging the initial conditions. The analysis yields explicit expressions for the Taylor coefficients which are then used to compute the errors. In the case of the algorithmically based practical numerical simulations in which fixed, albeit small, steps are repeatedly applied, the rise in the number of steps does not reduce the size of the steps but increases the total time of interest. The combinatorial formula shows that, here, the errors diverge. Furthermore, this work can be used to supplement other efforts such as the use of shadow Hamiltonians where the truncation of the series expansion of the latter will produce errors in the higher order propagator moments.     

Symmetrization of operator matrix elements

The method of Dupuis and King for generating matrix elements of a totally symmetric one-electron operator in terms of symmetry-distinct integrals only is generalized to the case of nontotally symmetric operators. For operators constructed from two-electron integrals, explicit reduction of integral processing to permutationally inequivalent symmetry-distinct integrals only is described, while for one-electron operators further reductions are derived using double coset decompositions. Finally, some computational consequences of this approach are briefly discussed.     

A probabilistic evolution approach trilogy, part 1: quantum expectation value evolutions, block triangularity and conicality, truncation approximants and their convergence

This is the first one of three companion papers focusing on the “probabilistic evolution approach (PEA)” which has been developed for the solution of the explicit ODE involving problems under certain consistent impositions. The main purpose here is the determination of the expectation value of a given operator in quantum mechanics by solving only ODEs, not directly using the wave function. To this end we first define a basis operator set over the Kronecker powers of an appropriately defined “system operator vector”. We assume that the target operator’s commutator with the system’s Hamiltonian can be expressed in terms of the above-mentioned basis operators. This assumption leads us to an infinite set of linear homogeneous ODEs over the expectation values of the basis operators. Its coefficient matrix is in block Hessenberg form when the target operator has no singularity, and beyond that, it may become block triangular when certain conditions over the system’s potential function are satisfied. The initial conditions are the basic determining agents giving the probabilistic nature to the solutions of the obtained infinite set of ODEs. They may or may not have fluctuations depending on the nature of the probability density. All these issues are investigated in a phenomenological and constructive theoretical manner in this paper. The remaining two papers are devoted to further details of PEA in quantum mechanics, and, the application of PEA to systems defined by Liouville equation.     

Quantal cumulant dynamics: general theory

The authors have derived coupled equations of motion of cumulants that consist of a symmetric-ordered product of the position and momentum fluctuation operators in one dimension. The key point is the utilization of a position shift operator acting on a potential operator, where the expectation value of the shift operator is evaluated using the cumulant expansion technique. In particular, the equations of motion of the second-order cumulant and the expectation values of the position and momentum operators are given. The resultant equations are expressed by those variables and a quantal potential that consists of an exponential function of the differential operators and the original potential. This procedure enables us to perform quantal (semiclassical) dynamics in one dimension. In contrast to a second-order quantized Hamilton dynamics by Prezhdo and Pereverzev which conserves the total energy only with an odd-order Taylor expansion of the potential [J. Chem. Phys. 116, 4450 (2002); 117, 2995 (2002)], the present quantal cumulant dynamics method exactly conserves the energy, even if a second-order approximation of the cumulants is adopted, because the present scheme does not truncate the given potential. The authors propose three schemes, (i) a truncation, (ii) a summation of derivatives, and (iii) a convolution method, for evaluating the quantal potentials for several types of potentials. The numerical results show that although the truncation method preserves the energy to some degree, the trajectory obtained gradually deviates from that of the summation scheme after 2000 steps. The phase space structure obtained by the truncation scheme is also different from that of the summation scheme in a strongly anharmonic region.     

Spin-Independent alternant systems

Spin-independent alternant systems are analyzed using the alternancy symmetry adapted (ASA ) approach. This approach is based on an explicit construction of ASA operators that either commute (altemant) or anticommute (antialternant) with the particle-hole symmetry operator K?. The method yields an explicit construction and identification of all spin-independent Hamiltonians that describe altemant systems (neutral and/or ionic) as well as an explicit identification of all linear properties characteristic to these systems. It also establishes the connection between the spin-eigenstates and the particle-hole symmetry.     

Validation of raw data measurements in the comet assay

General guidance recently proposed for the comet assay concluded that “the method should be adjusted scientifically at each laboratory to obtain valid and reproducible results”. However, the comet widely used metrics, Tail DNA and Tail moment, are actually based on a ratio of fluorescence signals, a relative and semi-quantitative measurement, and are quite difficult to validate according to classical criteria. As the validation of analytical methods increasingly becomes an absolute requirement in many fields, this paper investigates a scheme to study the variability of raw data measurements for computer-assisted comet measurement, including the between-operators reproducibility. In the overall analysis process, we show that the image acquisition step gives the highest variability, notably for the Tail length parameter that negatively influences the Olive tail moment. However, when the operator interacts with the system to correct obviously mistaken measurements, the reproducibility is sensibly improved. For the metrics Tail DNA and Olive tail moment, the total variability in measurements for a panel of comets quantified by different operators in real conditions is about 4%. The proposed validation scheme allows to assess the measurement process and to verify if there are any major difference between trained operators, an essential requirement for long-term investigations.     

Addition theorem and matrix elements of radiative multipole operators

The addition theorem for radiative multipole operators, i.e., electric-dipole, electric-quadropole, or magnetic-dipole, etc., is derived through a translational transformation. The addition theorem of μth component of the angular momentum operator, L _μ (r), is also derived as a simple expression that represents a general translation of the angular momentum operator along an arbitrary orientation of a displacement vector and when this displacement is along the Z-axis. The addition theorem of the multipole operators is then used to analytically evaluate the matrix elements of the electric and magnetic multipole operators over the basis functions, the spherical Laguerre Gaussian-type function (LGTF), . The explicit and simple formulas obtained for the matrix elements of these operators are in terms of vector-coupling coefficients and LGTFs of the internuclear coordinates. The matrix element of the magnetic multipole operator is shown to be a linear combination of the matrix element of the electric multipole operator     

量子混沌在谱涨落统计特征上的表象

通过对Ｈｅｎｏｎ－Ｈｅｉｌｅｓ模型、Ｂａｒｂａｎｉｓ模型和刚环转球模型这两类三种守恒体系之对应量子体系的研究，具体地剖析了本征值谱涨落统计特征与体系动力学行为的联系，说明了量子混沌现象的两种表现形式。同时，通过将本质上属于经典耗散系范畴的Ｌｏｔｋａ－Ｖｏｌｔｅｒｒａ模型之演化方程哈密顿化，发现其量子对应系本征值谱的涨落统计特征超出了Ｐｏｉｓｓｏｎ－ＧＯＥ（Ｗｉｇｎｅｒ）或ＧＵＥ框架。揭示出经典耗散     

No fluctuation approximation in any desired precision for univariate function matrix representations

The operator involving problems are mostly handled by using the matrix representations of the operators over a finite set of appropriately chosen basis functions in a Hilbert space as long as the related problem permits. The algebraic operator which multiplies its operand by a function is the focus of this work. We deal with the univariate case for simplicity. We show that a rapidly converging scheme can be constructed by defining an appropriate fluctuation operator which projects, in fact, to the complement of the space spanned by appropriately chosen finite number of basis functions. What we obtain here can be used in efficient numerical integration also.     

Addition Theorem and Matrix Elements of Radiative Multipole Operators

The addition theorem for radiative multipole operators, i.e., electric-dipole, electric-quadropole or magnetic-dipole, etc., is derived through a translational transformation. The addition theorem of the μth component of the angular momentum operator, L_μ (r), is also derived as a simple expression that represents a general translation of the angular momentum operator along an arbitrary orientation of a displacement vector and when this displacement is along the Z-axis. The addition theorem of the multipole operators is then used to analytically evaluate the matrix elements of the electric and magnetic multipole operators over the basis functions, the spherical Laguerre Gaussian-type function (LGTF), . The explicit and simple formulas obtained for the matrix elements of these operators are in terms of vector-coupling coefficients and LGTFs of the internuclear coordinates. The matrix element of the magnetic multipole operator is shown to be a linear combination of the matrix element of the electric multipole operator.     

A Hierarchical-Equation-of-Motion Based Semiclassical Approach to Quantum Dissipation?

We present a semiclassical (SC) approach for quantum dissipative dynamics, constructed on basis of the hierarchical-equation-of-motion (HEOM) formalism. The dynamical components considered in the developed SC-HEOM are wavepackets' phase-space moments of not only the primary reduced system density operator but also the auxiliary density operators (ADOs) of HEOM. It is a highly numerically efficient method, meanwhile taking into account the high-order effects of system-bath couplings. The SC-HEOM methodology is exemplified in this work on the hierarchical quantum master equation[J. Chem. Phys. 131 , 214111 (2009)] and numerically demonstrated on linear spectra of anharmonic oscillators.     

Somehow emancipating Probabilistic Evolution Theory (PREVTH) from singularities via getting single monomial PREVTH

This paper deals with the combination of the evolver dynamics of a hydrogen-like quantum system with the conical PREVTH (Probabilistic Evolution Theory) and excessively with the single monomial PREVTH. The new original applications are degree escalations in each monomial of a set of Poisson Bracket equations with multinomial right hand side and arbitrary, optimisable, parameters insertion to the resulting single monomial via an approach based on commutativity relations amongst the system basis operators. What we have shown that the additions based on commutativities with the Constancy Adding Space Extension (CASE) operator which is in fact proportional to identity operator, does not contribute to the suppression of the norm square of the single monomial coefficient matrix. This is just an observation for a specific family of systems but may be signaling a more general reality. If so it needs rather a rigorous proof.     

Theoretical description of dissipative vibrational dynamics using the density matrix in the state representation

Ultrafast dissipative dynamics of vibrational degrees of freedom in molecular systems in the condensed phase are studied here. Assuming that the total system is separable into a relevant part and a reservoir, the dynamics of the relevant part can be described by means of a reduced statistical density operator. For a weak or intermediate coupling between the relevant part and the reservoir, it is possible to derive a second-order master equation for this operator. Using a representation of the reduced statistical operator in an appropriate molecular basis set, vibrational dynamics in a variety of potential energy surfaces can be studied. In the numerical calculations, we focus on the dissipative dynamics under the influence of external laser fields. In the first example, vibrational wave-packet dynamics and time-resolved pump-probe spectroscopy of molecular systems with nonadiabatically coupled excited-state potential energy surfaces is presented. In the second part, we show how an intense laser field modifies the wave-packet motion onto two radiatively coupled potential energy surfaces. Finally, the controlled preparation of definite vibrational states in a triatomic molecule with infrared laser pulses is considered taking relaxation and dephasing processes into account. © 1996 John Wiley & Sons, Inc.     

On the Construction of n-Electron States with Symplectic Symmetry

Under quasispin scheme, a complete group theoretical classification of fermion states with symplectlc symmetry is proposed. Furthermore, the first and second order irreducible tensor operators are investigated in detail to approach the fermion states with explicit forms.     

Towards a “Chemical” Hamiltonian

An analysis of the LCAO Hamiltonian is performed in terms of a “mixed” formulation of the second quantization for nonorthogonal orbitals, compressing the different interactions to one- and two-center terms as far as possible by performing appropriate projections. For this purpose an operator of atomic charge is also introduced, the expectation values of which are the Mulliken gross atomic populations on the individual atoms. The LCAO Hamiltonian is decomposed into terms having different physical meaning and significance: (i) sum of effective atomic Hamiltonians; (ii) the electrostatic interactions in the point-charge approximation; (iii) the electrostatic effects connected with the deviation of the actual charge distribution from the pointlike one; (iv) two-center overlap effects; (v) finite basis (“counterpoise”) correction terms related to the individual atoms; and (vi) similar finite basis correction terms with respect to the two-center interactions. Only terms of types (i) to (iv), containing no three- or four-center integrals, are considered as having physical significance. Based on the analysis of the Hamiltonian, an energy partitioning scheme is developed, and explicit expressions are given for one- and two-center (and basis extension) components of the SCF energy. The approach is also applied to the problem of intermolecular interactions, and an explicit formula is given permitting calculation of the “counterpoise” part of the supermolecule energy by properly taking into account that it depends not only on the extension of the basis, but also on the occupation of the additional orbitals in the intervening molecule—a factor completely overlooked in the usual scheme of calculations.     

组态相互作用中的特征波函数

提出了完全活性空间组态相互作用方法中对称化算符、本征基、特征矩阵、特征波函数等概念和将多电子波函救按空间与自旋对称分类的统一方法。按此法对称化算符并作用在特征波函数上,再按等价权空间求和即得全组态波函数。因对称性使组态函数中物理因子与几何因子完全分离,可消除了变分求解中多余的变量。特征基化学概念明确,直接对应给定的价键结构。     

Spectral analysis of the Green's function

The energy eigenvalue spectrum for a conservative dynamical system is contained implicitly in its Green's function. It becomes explicit in the Fourier transform of either the Green's function or its trace. The trace exists only when the spectrum is entirely discrete. Applications are made to the free particle, the linear harmonic oscillator, and the hydrogen atom. In the latter two cases determination of the Green's function can be considerably simplified by similarity transformations on the Hamiltonian operator.