Mixed second and third energy derivatives from auxiliary density perturbation theory†

Abstract

The working equations for the calculation of mixed second- and third-order energy derivatives in the framework of auxiliary density functional theory are presented. The perturbations with respect to nuclear displacements and external homogeneous electric field components are calculated with auxiliary density perturbation theory. The presented energy derivative working equations were implemented in deMon2k and validated by vibrational spectra simulations within the double harmonic approximation. The effect of the auxiliary functions on the IR and Raman spectra simulation were analysed for the C₆₀ fullerene. As applications, vibrational spectra of icosahedral carbon fullerenes with up to 540 atoms are calculated without employing symmetry constraints.     

Structural and spectroscopic investigation of the N-methylformamide–water (NMF···3H2O) complex

In this work, theoretical studies on the structure, molecular properties, hydrogen bonding, and vibrational spectra of the N-methylformamide–water (NMF···3H₂O) complex will be presented. The molecular geometry was optimised by using Hartree–Fock (HF), second Møller–Plesset (MP2), and density functional theory methods with different basis sets. The harmonic vibrational frequencies are computed by using the B3LYP method with 6-311++G(d,p) as a basis set and then scaled with a suitable scale factor to yield good coherence with the observed values. The temperature dependence of various thermodynamic functions (heat capacity, entropy, and enthalpy changes) was also studied. A detailed analysis of the nature of the hydrogen bonding, using natural bond orbital (NBO) and topological atoms in molecules theory, has been reported.     

Entropy of flexible liquids from hierarchical force–torque covariance and coordination

ABSTRACT

New theory is presented to calculate the entropy of a liquid of flexible molecules from a molecular dynamics simulation. Entropy is expressed in two terms: a vibrational term, representing the average number of configurations and momentum states in an energy well, and a topographical term, representing the effective number of energy wells. The vibrational term is derived in a hierarchical manner from two force–torque covariance matrices, one at the molecular level and one at the united-atom level. The topographical term comprises conformations and orientations, which are derived from the dihedral distributions and coordination numbers, respectively. The method is tested on 14 liquids, ranging from argon to cyclohexane. For most molecules, our results lie within the experimental range, and are slightly higher than those by the 2PT method, the only other method currently capable of directly calculating entropy for such systems. As well as providing an efficient and practical way to calculate entropy, the theory serves to give a comprehensive characterisation and quantification of molecular structure.     

Frequency spectra of CFC metals and correlated spectral density between bulk atoms

The Green's function generating coefficients method is applied to the calculation of frequency spectra and correlated spectral density for bulk atoms of copper and platinum. The mean square and mean correlated displacements are then deduced. The method is certainly a powerful tool allowing precise calculation of all quantities related to vibrational properties of solids.     

Algorithm for solving the inverse vibronic problem on the basis of SVL fluorescence spectra

Computer methods whereby the inverse vibronic problem is solved on the basis of resonance fluorescence spectra with the use of modern quantum-mechanical methods for constructing structuraldynamic models of polyatomic molecules are discussed. An algorithm is proposed for solving the inverse vibronic problem according to resonance fluorescence spectra under laser excitation, and the corresponding calculation programs are constructed. The initial program data are acquired by means of an original software package which implements the scaling of quantum-mechanical force fields in two electronic states. The Duschinsky matrix and the initial matrix of shifts in normal coordinates caused by electron excitation are calculated in the Cartesian and natural vibrational coordinates. The program data are taken from quantum-molecular models based on calculations performed via ab initio modern quantum-mechanical methods and density functional theory. The algorithm is tested through the calculation of a model molecular system.     

Equations-of-motion approach to quantum mechanics: application to a model phase transition

We present a generalized equations-of-motion method that efficiently calculates energy spectra and matrix elements for algebraic models. The method is applied to a five-dimensional quartic oscillator that exhibits a quantum phase transition between vibrational and rotational phases. For certain parameters, 10 x 10 matrices give better results than obtained by diagonalizing 1000 x 1000 matrices.     

On the convergence of zero-point vibrational corrections to nuclear shieldings and shielding anisotropies towards the complete basis set limit in water

ABSTRACT

The method and basis set dependence of zero-point vibrational corrections (ZPVCs) to nuclear magnetic resonance shielding constants and anisotropies has been investigated using water as a test system. A systematic comparison has been made using the Hartree–Fock, second-order Møller–Plesset perturbation theory (MP2), coupled cluster singles and doubles (CCSD), coupled cluster singles and doubles with perturbative triples corrections (CCSD(T)) and Kohn–Sham density functional theory with the B3LYP exchange-correlation functional methods in combination with the second-order vibrational perturbation theory (VPT2) approach for the vibrational corrections. As basis sets, the correlation consistent basis sets cc-pVXZ, aug-cc-pVXZ, cc-pCVXZ and aug-cc-pCVXZ with X = D, T, Q, 5, 6 and the polarisation consistent basis sets aug-pc-n and aug-pcS-n with n = 1, 2, 3, 4 were employed. Our results show that basis set convergence of the vibrational corrections is not monotonic and that very large basis sets are needed before a reasonable extrapolation to the basis set limit can be performed. Furthermore, our results suggest that coupled cluster methods and a decent basis set are required before the error of the electronic structure approach is lower than the inherent error of the VPT2 approximation.     

Determination of vibrational energy levels and transition dipole moments of CO2 molecules by density functional theory

An efficient method is presented to calculate the intra-molecular potential energies and electrical dipole moments of CO₂ molecules at the electronic ground state by solving the Kohn-Sham (KS) equation for a total of 101 992 nuclear configurations. The projector-augmented wave (PAW) exchange-correlation potential functionals and plane wave (PW) basis functions were used in solving the KS equation. The calculated intra-molecular potential function was then included in the pure vibrational Schrödinger equation to determine the vibrational energy eigen values and eigen functions. The vibrational wave functions combined with the calculated dipole moment function were used to determine the transition dipole moments. The calculated results were compared with the experimental data.     

Matrix isolation studies and potential function for perfluorocyclobutane

The infrared and Raman spectra of perfluorocyclobutane isolated in argon matrices at 1:100 and 1:200 mole ratios have been measured between 200 and 2000 cm^?1. Although the ring-puckering fundamental (ν₁₆) was not observed directly, an assignment for the 2 ← 1 (30 cm^?1) transition of ν₁₆ has been deduced from sum and difference bands resolved in the infrared spectrum. Potential functions based upon valence force models are considered in detail and correlated with those of similar ring systems. By using the frequency of the 2 ← 1 transition for ν₁₆ and a vibrational reduced mass of 1501 amu, an approximate model potential function calculation yields a slightly puckered equilibrium conformation with a barrier on the order of 124 cm^?1. The vibrational assignment for perfluorocyclobutane is discussed in terms of the new matrix isolation spectra.     

The impact of vibrational wave function on low energy electron vibrational scattering from nitrogen molecule

The vibrational wave function of the target theoretically plays an important role in the calculation of vibrational excitation cross sections. By a careful study of the differential cross sections resulting from different vibrational wave functions we find that cross sections are susceptible to vibrational wave functions. Minor changes in the vibration wave function may cause a significant change in the cross section. Even more surprising is that by selecting a few numbers of potential models(which determine the vibrational wave functions) we can often calculate the differential scattering cross section in much closer agreement with experiment in the framework of body-frame vibrational close-coupling theory, which suggest that an accurate potential energy may play a more important role in scattering than we thought before.     

Low-rank canonical-tensor decomposition of potential energy surfaces: application to grid-based diagrammatic vibrational Green's function theory

ABSTRACT

A new method is proposed for a fast evaluation of high-dimensional integrals of potential energy surfaces (PES) that arise in many areas of quantum dynamics. It decomposes a PES into a canonical low-rank tensor format, reducing its integral into a relatively short sum of products of low-dimensional integrals. The decomposition is achieved by the alternating least squares (ALS) algorithm, requiring only a small number of single-point energy evaluations. Therefore, it eradicates a force-constant evaluation as the hotspot of many quantum dynamics simulations and also possibly lifts the curse of dimensionality. This general method is applied to the anharmonic vibrational zero-point and transition energy calculations of molecules using the second-order diagrammatic vibrational many-body Green's function (XVH2) theory with a harmonic-approximation reference. In this application, high dimensional PES and Green's functions are both subjected to a low-rank decomposition. Evaluating the molecular integrals over a low-rank PES and Green's functions as sums of low-dimensional integrals using the Gauss–Hermite quadrature, this canonical-tensor-decomposition-based XVH2 (CT-XVH2) achieves an accuracy of 0.1 cm^?1 or higher and nearly an order of magnitude speedup as compared with the original algorithm using force constants for water and formaldehyde.     

Connection between parameters of the Murrell–Sorbie and Fayyazuddin potentials

A connection between the Murrell–Sorbie and the Fayyazuddin potential energy functions were established by obtaining conversion matrices that convert the former's parameters into the latter and vice versa. This was attained by rearranging the Fayyazuddin potential into another, but equivalent, form comparable to the Murrell–Sorbie function with the aid of series expansion. A list of 71 sets of Fayyazuddin diatomic parameters were generated by applying one of the conversion matrices on the Huxley–Murrell data [J. Chem. Soc. Faraday Trans. II 79, 323 (1983)]. Verification of the developed conversion matrices was performed by plotting the Murrell–Sorbie and the Fayyazuddin functions using the Huxley–Murrell data and the converted data, respectively. Potential energy plots of typical (OSi) and extreme (FO, BeS and HH) diatomic parameters exhibit very good agreement between the two potential functions considered, thereby confirming the conversion matrices’ validity. Slight discrepancies of the plotted diatomic energy curves further suggest the comparative suitability of the Fayyazuddin potential for curve-fitting harder, rather than softer, bonds.     

TiO基态 (X 3 Δr) 的势能函数与光谱常数

应用群论及原子分子反应静力学方法推导了TiO分子基态(X³Δ_r)的离解极限.采用不同的计算方法,包括密度泛函B3LYP,B3P86,BP86,B3PW91和MP2,MP4方法,结合不同基组计算了TiO分子基态的平衡核间距、能量和振动频率.研究表明,使用B3LYP方法,对O原子使用6-311+G基组,Ti原子使用6-311+ +G^**基组时计算得到的平衡几何结构、分子离解能和谐振频率与实验值符合得最好.使用优选出的方法和基组对T 关键词： TiO 势能函数 光谱常数 密度泛函理论     

Lattice dynamics of a Rb2KScF6 crystal in unstable cubic and tetragonal phases and in a stable monoclinic phase

The results of a nonempirical calculation of the static and dynamic properties of a Rb₂KScF₆ crystal with elpasolite structure in cubic, tetragonal, and monoclinic phases are presented. The calculation is performed on the basis of a microscopic model of an ionic crystal that takes account of the deformability and polarizability of the ions. The deformability parameters of the ions are determined from the condition that the total energy of the crystal is minimum. The computational results for the equilibrium lattice parameters are in satisfactory agreement with experimental data. Unstable vibrational modes are found in the vibrational spectrum of the lattice in the cubic and tetragonal phases. These modes occupy the phase space throughout the entire Brillouin zone. The characteristic vectors of the most unstable mode at the center of the Brillouin zone of the cubic phase are related to the displacements of the fluorine ions and correspond to rotation of ScF₆ octahedra. Condensation of this mode leads to a tetragonal distortion of the structure. In the tetragonal phase the most unstable mode belongs to the boundary point of the Brillouin zone and condensation of this mode leads to monoclinic distortion with doubling of the unit-cell volume. In the monoclinic phase unstable modes are absent in the vibrational spectrum of the lattice. Fiz. Tverd. Tela (St. Petersburg) 41, 1297–1305 (July 1999)     

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提出了一种以多目标参数优化确定热斑的温度和密度径向分布的方法。用热斑中心位置和燃料界面位置对应的电子温度、靶剩余烧蚀层峰值密度、剩余烧蚀层质量与初始烧蚀层质量的比值以及燃料与烧蚀层分界面位置构成五维变量空间参数空间的每个点描述一组温度和密度分布。将实验测量的X射线图像的归一化强度分布与理论模型计算的强度分布的方差作为两个目标函数;另一个目标函数是实验测量的热斑的平均温度与理论模型计算的平均温度的方差。通过多目标遗传算法搜索五维参数空间获得最优参数,从而获得最优的温度和密度分布。对KB显微镜配合X射线胶片测量的实验结果进行了计算,给出了热斑的最佳温度和密度的径向分布。由测试算例发现,温度分布对目标函数的依赖较强。     

Ab initio Compton maps of small molecules

A closed form expression for the evaluation of directional Compton profiles (DCP) from the position space one-electron reduced density matrix expanded in Cartesian Gaussian basis functions is presented. The Compton map, i.e. a two-dimensional contour plot of the DCP function J(q) in a selected plane, is introduce as a convenient method for describing the DCP Examples of theoretical Compton maps are given for the hydrogen molecule ion (both ground and excited states), water, ammonia, ethylene and ¹A₁ methylene molecules in their ground states. Compton maps generated with Hartree-Fock, configuration interaction and density functional theory methods are compared.     

Nonclassical Field States in Quantum Optics and Particle Physics

The primary aim of the present paper is to draw the attention of particle physicists to new developments in studying squeezed and correlated states of the electromagnetic field, and those working on the latest developments to new findings about multiplicity distributions and other specific effects in quantum chromodynamics. New types of nonclassical states used in quantum optics such as squeezed states, correlated states, and even and odd coherent states (Schrödinger cat states) for one-mode and multimode interactions are reviewed. Their distribution functions are analyzed according to the method first used for multiplicity distributions in high-energy particle interactions. The phenomenon of oscillations of particle distribution functions of squeezed fields is described and related to the phenomenon of oscillations of cumulant moments of some distributions for squeezed and correlated field states. Possible extension of the method to fields different from the electromagnetic field (gluons, pions, etc.) is conjectured, and some predictions of specific effects in nucleus-nucleus interactions at high energies are presented.     

A generalized internal axis method for high barrier tunneling problems,as applied to the water dimer

When more than one large-amplitude vibrational motion is present in a molecule, it is often not possible to define a global internal-axis-method (IAM) coordinate system and set of basis functions. In the present work, a method is presented for extending the IAM treatment to tunneling problems in such cases, using as an illustration a model for the water dimer with three large-amplitude vibrational coordinates. The method involves the construction of two different sets of local IAM-like coordinate systems. The first of these contains n coordinate systems, one for the small neighborhood surrounding each of the n equilibrium frameworks. The second contains on the order of $\text{n}{}^2\text{2}$ coordinate systems, one for each feasible tunneling path between each pair of frameworks. Basis functions written in the second set of local IAM-like coordinates are used to determine the complex phase factors associated in this method with tunneling matrix elements of the phenomenological rotational Hamiltonian in the high barrier limit. These phase factors govern the way in which the various real tunneling frequencies in the molecule constructively and/or destructively interfere in the Hamiltonian matrix elements and final energy expressions. Various mathematical approximations are involved in using the local IAM-like basis sets to obtain matrix elements; the full extent of the adverse effects of these approximations will not be known until an attempt to fit experimental data is carried out.     

Calculation of the energy density function by the method of steepest descent

We present a unified theory of diatomic molecules which reconciles bound state spectroscopy and atomic scattering theory. The total wave-function is expanded in a complete set of atomic channel states which is entirely equivalent to an expansion in Hund's case (e) electronic-rotational states. An analysis of the coupled radial, that is vibrational, functions places strong constraints on the asymptotic properties of the molecular wave-functions. These are presented in terms of the reactance K and scattering S matrices of atomic scattering theory which offers a uniform treatment for open channels (inelastic scattering and continuum spectroscopy), closed channels (bound state spectroscopy) and mixtures of both (predissociation). The normalization of the total wavefunction is derived and related to the asymptotic boundary conditions both for continuum and bound states.     

Efficient implementation of the analytic second derivatives of Hartree–Fock and hybrid DFT energies: a detailed analysis of different approximations

In this paper, various implementations of the analytic Hartree–Fock and hybrid density functional energy second derivatives are studied. An approximation-free four-centre implementation is presented, and its accuracy is rigorously analysed in terms of self-consistent field (SCF), coupled-perturbed SCF (CP-SCF) convergence and prescreening criteria. The CP-SCF residual norm convergence threshold turns out to be the most important of these. Final choices of convergence thresholds are made such that an accuracy of the vibrational frequencies of better than 5 cm^?1 compared to the numerical noise-free results is obtained, even for the highly sensitive low frequencies (<100–200 cm^?1). The effects of the choice of numerical grid for density functional exchange–correlation integrations are studied and various weight derivative schemes are analysed in detail. In the second step of the work, approximations are introduced in order to speed up the computation without compromising its accuracy. To this end, the accuracy and efficiency of the resolution of identity approximation for the Coulomb terms and the semi-numerical chain of spheres approximation to the exchange terms are carefully analysed. It is shown that the largest performance improvements are realised if either Hartree–Fock exchange is absent (pure density functionals) and otherwise, if the exchange terms in the CP-SCF step of the calculation are approximated by the COSX method in conjunction with a small integration grid. Default values for all the involved truncation parameters are suggested. For vancomycine (176 atoms and 3593 basis functions), the RIJCOSX Hessian calculation with the B3LYP functional and the def2-TZVP basis set takes ～3 days using 16 Intel® Xeon® 2.60GHz processors with the COSX algorithm having a net parallelisation scaling of 11.9 which is at least ～20 times faster than the calculation without the RIJCOSX approximation.