Identification and thermodynamic treatment of several types of large-amplitude motions

We present a partially automated method for the thermodynamic treatment of large-amplitude motions. Starting from the molecular geometry and the Hessian matrix, we evaluate anharmonic partition functions for selected vibrational degrees of freedom. Supported anharmonic vibration types are internal rotation and inversion (oscillation in a double-well potential). By heuristic algorithms, we identify internal rotations in most cases automatically from the Hessian eigenvectors, and we also estimate the parameters of anharmonic partition functions (e.g., potential barrier, periodicity, and symmetry number) with thermodynamically sufficient precision. We demonstrate the validity of our schemes by comparison to pointwise calculated ab initio potential curves.     

Spinor optimization for a relativistic spin-dependent CASSCF program

Detailed formulae for the implementation of the multi-configuration SCF spinor optimization in a basis of Kramers pair 2-spinors – i.e. exploiting time-reversal symmetry – are presented. Full expressions for the spinor gradient and spinor Hessian elements are given in abstract form as well as within the usual CASSCF subspace division. As far as possible, the resulting terms are grouped to relativistic inactive and active Fock matrices, which have been introduced previously. Approximations for the Hessian are introduced so as to initialize it in an inverse Hessian update algorithm for a diagonal first approximation within the standard quasi-Newton-Raphson procedure. The effects of double group symmetry arising from spin dependence on Fock matrices and therefore gradient and Hessian are discussed and a group scheme for the implementation is proposed. Received: 14 January 1997 / Accepted: 3 February 1997     

Extremely localized molecular orbitals (ELMO): a non-orthogonal Hartree-Fock method

A new optimization method for extremely localized molecular orbitals (ELMO) is derived in a non-orthogonal formalism. The method is based on a quasi Newton-Raphson algorithm in which an approximate diagonal-blocked Hessian matrix is calculated through the Fock matrix. The Hessian matrix inverse is updated at each iteration by a variable metric updating procedure to account for the intrinsically small coupling between the orbitals. The updated orbitals are obtained with approximately n ² operations. No n ³ processes such as matrix diagonalization, matrix multiplication or orbital orthogonalization are employed. The use of localized orbitals allows for the creation of high-quality initial “guess” orbitals from optimized molecular orbitals of small systems and thus reduces the number of iterations to converge. The delocalization effects are included by a Jacobi correction (JC) which allows the accurate calculation of the total energy with a limited number of operations. This extension, referred to as ELMO(JC), is a variational method that reproduces the Hartree-Fock (HF) energy with an error of less than 2 kcal/mol for a reduced total cost compared to standard HF methods. The small number of variables, even for a very large system, and the limited number of operations potentially makes ELMO a method of choice to study large systems. Received: 30 December 1996 / Accepted: 5 June 1997     

Convergence optimization of restricted open-shell self-consistent field calculations

Different self-consistent field (SCF) iteration schemes for open-shell systems are discussed. After a brief summary of the well-known level shifting and damping procedure, we describe the quadratically convergent SCF (QCSCF) approach based on the gradient and the Hessian matrix in a space of orbital rotation parameters. An analytical expression for the latter is derived for the general many-shell case. Starting from the expression for the energy change obtained by the QCSCF method, we then present a simplified direct procedure avoiding matrix diagonalization but also the difficulties of the QCSCF method in handling the Hessian matrix. Numerical calculations on some open-shell systems involving transition-metal complexes show that this method leads to rapid and reliable convergence of the iteration process in cases where the usual SCF procedure of iterative diagonalization tends to diverge. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 62: 617–637, 1997     

Normal Coordinate Analysis for Polymer Systems: Capabilities and New Opportunities

Normal coordinate analysis is an important tool in studying the structure, dynamics, and physical properties of polymer systems. In this article the capabilities of normal coordinate analysis (NCA) are explored in some detail. The use of the eigenvalues and eigenvectors from NCA is catalogued for a wide variety of purposes: for assigning or interpreting polymer spectra, for structural determination, for constructing force fields, for computing heat capacity and other thermodynamic properties, and for computing other physical properties. Examples are given for crystals, melts, and amorphous systems. Also described are methods for characterizing the normal mode vectors that are especially useful for larger systems, in which a large amount of data must be analyzed or where visualization or animation fails. Finally, a recently developed method for eliminating negative eigenvalues in systems with tens of thousands of atoms, trajectory averaging, is presented. Also described are several advances in numerical linear algebra for speeding up the diagonalization phase and for computing physical properties without requiring full diagonalization of the Hessian matrix.     

液态RbCl的态密度

给出了在分子动力学模拟基础上Fumi-Tosi势离子液体的正则模式分析方法,用Fumi-Tosi势(包括长程势)代替Lennard-Jones势,并且用等效Coulomb势处理长程Coulomb作用.讨论了Hessian矩阵元的计算方法和Hessian矩阵特征值的计算方法.计算实践表明，取用余误差函数形式的等效库仑势,可以合理地得到Hessian矩阵和态密度.液态RbCl中构型平均态密度的数值结果表明，液态RbCl的态密度表现出与Lennard-Jones液体的态密度相仿的特点.     

Extracting effective normal modes from equilibrium dynamics at finite temperature

A general method for obtaining effective normal modes of a molecular system from molecular dynamics simulations is presented. The method is based on a localization criterion for the Fourier transformed velocity time-correlation functions of the effective modes. For a given choice of the localization function used, the method becomes equivalent to the principal mode analysis (PMA) based on covariance matrix diagonalization. On the other hand, a proper choice of the localization function leads to a novel method with a strong analogy with the usual normal mode analysis of equilibrium structures, where the Hessian system at the minimum energy structure is replaced by the thermal averaged Hessian, although the Hessian itself is never actually calculated. This method does not introduce any extra numerical cost during the simulation and bears the same simplicity as PMA itself. It can thus be readily applied to ab initio molecular dynamics simulations. Three such examples are provided here. First we recover effective normal modes of an isolated formaldehyde molecule computed at 20 K in very good agreement with the results of a normal mode analysis performed at its equilibrium structure. We then illustrate the applicability of the method for liquid phase studies. The effective normal modes of a water molecule in liquid water and of a uracil molecule in aqueous solution can be extracted from ab initio molecular dynamics simulations of these two systems at 300 K.     

Elements of irreducible tensorial matrices generated by finite groups with applications to ligand field Hamiltonians

Using symmetry to determine Hamiltonian matrix elements for quantum systems with finite group symmetry is a special case of obtaining group-generated irreducible tensorial matrices. A group-generated irreducible tensorial matrix transforms irreducibly under the group and is a linear combination of group transformations on a reference matrix. The reference matrix elements may be appropriate integrals or parameters. The methods of normalized irreducible tensorial matrices (NITM) are employed to express elements of the generated matrix in terms of those of the reference matrix without performing the actual transformations. Only NTTM components of the reference matrix with the same transformation properties as the group-generated matrix will contribute to its elements. The elements of invariant symmetry-generated matrices are proportional to simple averages of certain elements of the reference matrix. This relation is substantially more efficient than previous techniques for evaluating matrix elements of octahedral and tetragonal d-type ligand-field Hamiltonians.     

Analytical hessian fitting schemes for efficient determination of force‐constant parameters in molecular mechanics

Building upon our recently developed partial Hessian fitting (PHF) method (Wang et al., J. Comput. Chem. 2016 , 37, 2349), we formulated and implemented two other rapid force‐field parameterization schemes called full Hessian fitting (FHF) and internal Hessian fitting (IHF), and comparisons were made among these three parameterization schemes to assess their performance. FHF minimizes deviation between the Hessian matrices in Cartesian coordinates computed by quantum mechanics (QM) and molecular mechanics (MM), to determine the best possible MM force‐constant parameters. While PHF requires step‐by‐step fittings of 3 × 3 partial Hessian matrices, FHF compares the lower triangular part of the QM and MM Hessian matrices, which allows simultaneous determination of all force‐constant parameters. In addition to this simple FHF scheme, IHF was developed such that it considers the Hessian matrices in redundant internal coordinates, where all possible internal coordinates that arise from the user‐defined interatomic connectivity are utilized. The results show that IHF performs best overall, followed by PHF and then FHF. Python‐based programing codes were developed to automate various tedious steps involved in the parameterization processes. © 2017 Wiley Periodicals, Inc.     

Quantum statistical mechanics with Gaussians: equilibrium properties of van der Waals clusters

The variational Gaussian wave-packet method for computation of equilibrium density matrices of quantum many-body systems is further developed. The density matrix is expressed in terms of Gaussian resolution, in which each Gaussian is propagated independently in imaginary time beta=(k(B)T)(-1) starting at the classical limit beta=0. For an N-particle system a Gaussian exp[(r-q)(T)G(r-q)+gamma] is represented by its center qinR(3N), the width matrix GinR(3Nx3N), and the scale gammainR, all treated as dynamical variables. Evaluation of observables is done by Monte Carlo sampling of the initial Gaussian positions. As demonstrated previously at not-very-low temperatures the method is surprisingly accurate for a range of model systems including the case of double-well potential. Ideally, a single Gaussian propagation requires numerical effort comparable to the propagation of a single classical trajectory for a system with 9(N(2)+N)/2 degrees of freedom. Furthermore, an approximation based on a direct product of single-particle Gaussians, rather than a fully coupled Gaussian, reduces the number of dynamical variables to 9N. The success of the methodology depends on whether various Gaussian integrals needed for calculation of, e.g., the potential matrix elements or pair correlation functions could be evaluated efficiently. We present techniques to accomplish these goals and apply the method to compute the heat capacity and radial pair correlation function of Ne(13) Lennard-Jones cluster. Our results agree very well with the available path-integral Monte Carlo calculations.     

Another way to implement the Powell formula for updating Hessian matrices related to transition structures

A way to update the Hessian matrix according to the Powell formula is given. With this formula one does not need to store the full Hessian matrix at any iteration. A method to find transition structures, which is a combination of the quasiNewton–Raphson augmented Hessian algorithm with the proposed Powell update scheme, is also given. The diagonalization of the augmented Hessian matrix is carried out by Lanczoslike methods. In this way, during all the optimization process, one avoids to store full matrices.     

A survey of optimization procedures for stable structures and transition states

We examine a variety of methods for obtaining the stable geometry of molecules and the transition states of simple systems and summarize some of our findings. We find the most efficient methods for optimizing structure to be those based on calculated gradients and estimated second derivative (Hessian) matrices, the later obtained either from the Broyden–Fletcher–Goldfarb–Shanno (BFGS ) quasi-Newton update method or from approximations to the coupled perturbed Hartree–Fock method. For uncovering transition states we find particularly useful a variety of the augmented Hessian theory used to uncover regions of the potential energy hypersurface with one and only one negative eigenvalue of the Hessian matrix characterizing the catchment region of the transition state. Once this region is found we minimize the norm of the gradient vector to catch the nearest extreme point of the surface. Examples of these procedures are given.     

Rotational symmetry of the molecular potential energy in the Cartesian coordinates

We consider the molecular Born-Oppenheimer potential energy as a function of atomic Cartesian coordinates and discuss the non-stationary Hessian properties arising due to rotational symmetry. A connection with the extended Hessian theory is included. New applications of Cartesian representation for examining and correcting raw numerical Hessian data and a simple formulation of harmonic vibrational analysis of partially optimized systems are proposed. Exemplary calculations for the porphyrin molecule with an internal proton transfer are reported. We also develop the normal transformation method to incorporate the rotational symmetry into the approximate analytical potentials, which are parametrized in the Cartesian coordinates. The transformation converts the coordinates from the space fixed frame to the frame which translates and rotates with the molecule and is determined by the Eckart conditions. New simple analytical formulas for the first and second derivatives of the transformed potential are derived. This fast method can be used to calculate the potential and its derivatives in the simulations of chemical reaction dynamics in the space fixed Cartesian frame without the need to constrain the molecular rotation or to define the local non-redundant internal coordinates.     

Use of symmetric rank-one Hessian update in molecular geometry optimization

The use of the symmetric rank-one Hessian update and the Broyden–Fletcher–Goldfarb–Shano (BFGS) update formula are considered in an ab initio molecular geometry optimization algorithm. It is noted that the symmetric rank-one Hessian update has an advantage when compared with the BFGS update formula and this advantage must be more evident in the optimization of molecular geometry, because the total energy surface is a near-quadratic function with a small nonlinearity close to a minimum point. The results obtained in geometry optimization of a test group of molecules support this proposal and show that the use of the symmetric rank-one Hessian update formula permits reduction of the number of energy and gradient evaluations needed to locate a minimum on the energy surface. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 1877–1886, 1998     

High symmetries in quantum chemistry

A method of simplifying the solution of secular equations occurring in electronic structure, normal mode and nuclear spin state calculations for molecules possessing symmetry is illustrated by applying it to the π-electrons of the recently discovered C₆₀ cluster. In contrast to the usual procedure, the method of reduction to characters does not require actual matrix realizations of the irreducible representations of the symmetry group but only the information contained in the character and multiplication tables. In the case of C₆₀ it leads to explicit algebraic solutions for all the π-orbital energies.     

Quantum wavepacket ab initio molecular dynamics: generalizations using an extended Lagrangian treatment of diabatic states coupled through multireference electronic structure

We present a generalization to our previously developed quantum wavepacket ab initio molecular dynamics (QWAIMD) method by using multiple diabatic electronic reduced single particle density matrices, propagated within an extended Lagrangian paradigm. The Slater determinantal wavefunctions associated with the density matrices utilized may be orthogonal or nonorthogonal with respect to each other. This generalization directly results from an analysis of the variance in electronic structure with quantum nuclear degrees of freedom. The diabatic electronic states are treated here as classical parametric variables and propagated simultaneously along with the quantum wavepacket and classical nuclei. Each electronic density matrix is constrained to be N-representable. Consequently two sets of new methods are derived: extended Lagrangian-QWAIMD (xLag-QWAIMD) and diabatic extended Lagrangian-QWAIMD (DxLag-QWAIMD). In both cases, the instantaneous potential energy surface for the quantum nuclear degrees of freedom is constructed from the diabatic states using an on-the-fly nonorthogonal multireference formalism. By introducing generalized grid-based electronic basis functions, we eliminate the basis set dependence on the quantum nucleus. Subsequent reuse of the two-electron integrals during the on-the-fly potential energy surface computation stage yields a substantial reduction in computational costs. Specifically, both xLag-QWAIMD and DxLag-QWAIMD turn out to be about two orders of magnitude faster than our previously developed time-dependent deterministic sampling implementation of QWAIMD. Energy conservation properties, accuracy of the associated potential surfaces, and vibrational properties are analyzed for a family of hydrogen bonded systems.     

Coarse-graining of protein structures for the normal mode studies

The coarse-grained structural model such as Gaussian network has played a vital role in the normal mode studies for understanding protein dynamics related to biological functions. However, for the large proteins, the Gaussian network model is computationally unfavorable for diagonalization of Hessian (stiffness) matrix for the normal mode studies. In this article, we provide the coarse-graining method, referred to as "dynamic model condensation," which enables the further coarse-graining of protein structures consisting of small number of residues. It is shown that the coarser-grained structures reconstructed by dynamic model condensation exhibit the dynamic characteristics, such as low-frequency normal modes, qualitatively comparable to original structures. This sheds light on that dynamic model condensation and may enable one to study the large protein dynamics for gaining insight into biological functions of proteins.     

Harmonic modes as variables to approximately account for receptor flexibility in ligand–receptor docking simulations: Application to DNA minor groove ligand complex

An approach to approximately account for receptor flexibility in ligand–receptor docking simulations is described and applied to a DNA/Hoechst 33258 analogue complex. Harmonic modes corresponding to eigenvectors with small eigenvalues of the Hessian matrix of the potential energy function were used as independent variables to describe receptor flexibility. For the DNA minor groove ligand case most of the conformational difference between an energy minimized free DNA and ligand-bound structure could be assigned to 5–40 harmonic receptor modes with small eigenvalues. During docking, deformations of the DNA receptor structure in the subset of harmonic modes were limited using a simple penalty function that avoided the summation over all intrareceptor atom pairs. Significant improvement of the sterical fit between ligand and receptor was found upon relaxation of the DNA in the subset of harmonic modes after docking of the ligand at the position found in the known crystal structure. In addition, the harmonic mode relaxation resulted in DNA structures that were more similar to the energy minimized ligand-bound form. Although harmonic mode relaxation also leads to improved sterical fit for other ligand placements, the placement as observed in the crystal structure could still be identified as the site with the most favorable sterical interactions. Because relaxation in the harmonic modes is orders of magnitude faster than conventional energy minimization using all atom coordinates as independent variables, the approach might be useful as a preselection tool to recognize ligand binding sites accessible only upon small conformational changes of the receptor. The harmonic mode relaxed structures can only be considered as approximate structures because deformation of the receptor in the harmonic modes can lead to small perturbations of the stereochemical geometry of the molecule. Energy minimization of preselected ligand–DNA docking candidates in all atom coordinates is required to reduce these deviations. ©1999 John Wiley & Sons, Inc. J Comput Chem 20: 287–300, 1999     

Efficient method for computation of representation matrices of the unitary group generators

The representation matrices of the unitary group generators E_k1 are equivalent to the representation matrices of cyclic permutations (k, k ± l, …, l ? 1, l). A method is presented for simultaneous computation of matrices corresponding to all different generators at a cost of less than one multiplication per nonzero element. The number of operations necessary for calculation of individual matrices for single generators or for products of two generators is at most proportional to the number of matrix elements of the final matrix. This approach eliminates the need to store the representation matrices in CI calculations.