Low‐memory iterative density fitting

A new low‐memory modification of the density fitting approximation based on a combination of a continuous fast multipole method (CFMM) and a preconditioned conjugate gradient solver is presented. Iterative conjugate gradient solver uses preconditioners formed from blocks of the Coulomb metric matrix that decrease the number of iterations needed for convergence by up to one order of magnitude. The matrix‐vector products needed within the iterative algorithm are calculated using CFMM, which evaluates them with the linear scaling memory requirements only. Compared with the standard density fitting implementation, up to 15‐fold reduction of the memory requirements is achieved for the most efficient preconditioner at a cost of only 25% increase in computational time. The potential of the method is demonstrated by performing density functional theory calculations for zeolite fragment with 2592 atoms and 121,248 auxiliary basis functions on a single 12‐core CPU workstation. © 2015 Wiley Periodicals, Inc.     

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     

Gradient extremals and valley floor bifurcations on potential energy surfaces

Gradient extremals are curves in configuration space denned by the condition that the gradient of the potential energy is an eigenvector of the Hessian matrix. Solutions of a corresponding equation go along a valley floor or along a crest of a ridge, if the norm of the gradient is a minimum, and along a cirque or a cliff or a flank of one of the two if the gradient norm is a maximum. Properties of gradient extremals are discussed for simple 2D model surfaces including the problem of valley bifurcations.     

Complete basis set, Gaussian, and hybrid density functional theory evaluation of the proton affinities of water and ammonia

An ab inito computation of reorganization energy for the electron transfer (ET) reactions between metal–benzene and metal ion–benzene complexes is presented. The geometry optimization of the metal–benzene complexes was performed. The metal atoms (or metal ions)– benzene molecule separation distances computed using an ab initio method were found to agree with earlier reported results. Values of reorganization energies using George-Griffith Marcus (GGM) method (the contribution from only diagonal elements of force constant matrix) and Hessian matrix method (including the contribution from both diagonal and off-diagonal elements) were computed. Results of reorganization energy show that the GGM method gives much lower values compared to those obtained using the Hessian method, suggesting that the coupling interactions between different vibrational modes are important to the inner-sphere reorganization energy for the ET reactions in gaseous phase.     

Modified one-electron hamiltonian method in the MC SCF theory

For orbital optimization within the MC SCF theory a modification of the OEH method is proposed with the direction of descent determined according to the Fletcher–Reeves gradient method. The combined method developed on this basis ensures the convergence of the iterative process when the Hessian singularities occur. The convergence properties of the method proposed are studied by performing the ab initio water molecule calculations using two types of multiconfigurational wave functions.     

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.     

On the quadratic reaction path evaluated in a reduced potential energy surface model and the problem to locate transition states*

Knowledge of the location of saddle points is crucial to the study the chemical reactivity. Using a path following method defined in a reduced potential energy surface, and starting at either the reactant or product region, we propose an algorithm that locates the corresponding saddle point. The reduced potential energy surface is defined by the set of molecular geometry parameters, namely bond distances, bond angles, and dihedral angles that undergo the largest change for the reaction under consideration; the rest of the coordinates are forced to have a null gradient. Consequently, the proposed method can be seen as a new formulation of the distinguished coordinate method. The method is based on a quadratic model; consequently, it only requires the calculation of the energy and the gradient. The Hessian matrix is normally updated except in the first step and the steps where the resulting updated Hessian matrix is not adequate. Some examples are presented and analyzed. © 2001 John Wiley & Sons, Inc. J Comput Chem 22: 387–406, 2001     

Technique for incorporating the density functional Hessian into the geometry optimization of biomolecules, solvated molecules, and large floppy molecules

Traditional geometry optimization methods require the gradient of the potential surface, together with a Hessian which is often approximated. Approximation of the Hessian causes difficulties for large, floppy molecules, increasing the number of steps required to reach the minimum. In this article, the costly evaluation of the exact Hessian is avoided by expanding the density functional to second order in both the nuclear and electronic variables, and then searching for the minimum of the quadratic functional. The quadratic search involves the simultaneous determination of both the geometry step and the associated change in the electron density matrix. Trial calculations on Taxol indicate that the cost of the quadratic search is comparable to the cost of the density functional energy plus gradient. While this procedure circumvents the bottleneck coupled-perturbed step in the evaluation of the full Hessian, the second derivatives of the electron-repulsion integrals are still required for atomic-orbital-based calculations, and they are presently more expensive than the energy plus gradient. Hence, we anticipate that the quadratic optimizer will initially find application in fields in which existing optimizers breakdown or are inefficient, particularly biochemistry and solvation chemistry.     

Finding transition states when second-order Jahn–Teller instability occurs

When second-order Jahn–Teller couplings become strong along “streambeds” on potential energy surfaces, instability reflected in negative curvature along a symmetry-lowering distortion coordinate can take place. The point where such negative curvature sets in is usually not a transition state because the gradient of the potential is usually large there. In this paper, it is demonstrated how to use the local energy, local gradient, local Hessian, and knowledge of how quickly the curvature for the symmetry-breaking mode evolves along the streambed (i.e., the derivative of this curvature) to predict how far to move in the symmetry-breaking mode in search of the desired transition state. It is shown that the Hessian matrix evaluated at the symmetry-broken geometry suggested by this analysis has only one negative eigenvalue. Because this analysis is based on a local approximation to the potential, its predictions are, of course, approximate. As such, they only “suggest” the proper direction and magnitude that one should “step” to move toward a transition state. © 1993 John Wiley & Sons, Inc.     

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.     

Computation of the amide I band of polypeptides and proteins using a partial Hessian approach

A partial Hessian approximation for the computation of the amide I band of polypeptides and proteins is introduced. This approximation exploits the nature of the amide I band, which is largely localized on the carbonyl groups of the backbone amide residues. For a set of model peptides, harmonic frequencies computed from the Hessian comprising only derivatives of the energy with respect to the displacement of the carbon, oxygen, and nitrogen atoms of the backbone amide groups introduce mean absolute errors of 15 and 10 cm(-1) from the full Hessian values at the Hartree-Fock/STO-3G and density functional theory EDF16-31G(*) levels of theory, respectively. Limiting the partial Hessian to include only derivatives with respect to the displacement of the backbone carbon and oxygen atoms yields corresponding errors of 24 and 22 cm(-1). Both approximations reproduce the full Hessian band profiles well with only a small shift to lower wave number. Computationally, the partial Hessian approximation is used in the solution of the coupled perturbed Hartree-Fock/Kohn-Sham equations and the evaluation of the second derivatives of the electron repulsion integrals. The resulting computational savings are substantial and grow with the size of the polypeptide. At the HF/STO-3G level, the partial Hessian calculation for a polypeptide comprising five tryptophan residues takes approximately 10%-15% of the time for the full Hessian calculation. Using the partial Hessian method, the amide I bands of the constituent secondary structure elements of the protein agitoxin 2 (PDB code 1AGT) are calculated, and the amide I band of the full protein estimated.     

Vibrational modes in partially optimized molecular systems

In this paper the authors develop a method to accurately calculate localized vibrational modes for partially optimized molecular structures or for structures containing link atoms. The method avoids artificially introduced imaginary frequencies and keeps track of the invariance under global translations and rotations. Only a subblock of the Hessian matrix has to be constructed and diagonalized, leading to a serious reduction of the computational time for the frequency analysis. The mobile block Hessian approach (MBH) proposed in this work can be regarded as an extension of the partial Hessian vibrational analysis approach proposed by Head [Int. J. Quantum Chem. 65, 827 (1997)]. Instead of giving the nonoptimized region of the system an infinite mass, it is allowed to move as a rigid body with respect to the optimized region of the system. The MBH approach is then extended to the case where several parts of the molecule can move as independent multiple rigid blocks in combination with single atoms. The merits of both models are extensively tested on ethanol and di-n-octyl-ether.     

On the determination of the minimum on the crossing seam of two potential energy surfaces

An improved gradient-based algorithm is presented for the determination of the minimum energy point on the crossing seam hypersurface between two arbitrary potential energy hypersurfaces. The Hessian matrix is updated employing the gradient information. The method is demonstrated in a study of some representative cases including charge-transfer states of a typical molecular-device molecule (a rigid spiro π – σ – π molecular cation) with, as well as without, an external electric field.     

C library for topological study of the electronic charge density

The topological study of the electronic charge density is useful to obtain information about the kinds of bonds (ionic or covalent) and the atom charges on a molecule or crystal. For this study, it is necessary to calculate, at every space point, the electronic density and its electronic density derivatives values up to second order. In this work, a grid‐based method for these calculations is described. The library, implemented for three dimensions, is based on a multidimensional Lagrange interpolation in a regular grid; by differentiating the resulting polynomial, the gradient vector, the Hessian matrix and the Laplacian formulas were obtained for every space point. More complex functions such as the Newton–Raphson method (to find the critical points, where the gradient is null) and the Cash–Karp Runge–Kutta method (used to make the gradient paths) were programmed. As in some crystals, the unit cell has angles different from 90°, the described library includes linear transformations to correct the gradient and Hessian when the grid is distorted (inclined). Functions were also developed to handle grid containing files (grd from DMol® program, CUBE from Gaussian® program and CHGCAR from VASP® program). Each one of these files contains the data for a molecular or crystal electronic property (such as charge density, spin density, electrostatic potential, and others) in a three‐dimensional (3D) grid. The library can be adapted to make the topological study in any regular 3D grid by modifying the code of these functions. © 2012 Wiley Periodicals, Inc.     

The experimental design for computing the harmonic and anharmonic force constant matrices of polyatomic molecules

In the present work we propose a numerical approach to estimate the harmonic and anharmonic force constant matrices, supposing we are able to compute analytically the first order derivative vector of the potential energy surface with respect to the internal coordinates. We use a polynomial least square fit to interpolate this gradient in the stationary point region. The structure of the regression matrix shows that the harmonic force constant matrix may be obtained even for large molecules; the evaluation of the anharmonic contributions request slightly more labor but is possible for 5 to 7 atoms. The present work is applicable even at the CI level and the number of computations remains small. We use the experimental planification to select the geometries to be computed in order to improve the estimation of the regression coefficients i.e. this means to lower their variance.chercheur qualifié du Fonds National Belge de la Recherche Scientifique.     

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     

Partial hessian fitting for determining force constant parameters in molecular mechanics

We present a new protocol for deriving force constant parameters that are used in molecular mechanics (MM) force fields to describe the bond‐stretching, angle‐bending, and dihedral terms. A 3 × 3 partial matrix is chosen from the MM Hessian matrix in Cartesian coordinates according to a simple rule and made as close as possible to the corresponding partial Hessian matrix computed using quantum mechanics (QM). This partial Hessian fitting (PHF) is done analytically and thus rapidly in a least‐squares sense, yielding force constant parameters as the output. We herein apply this approach to derive force constant parameters for the AMBER‐type energy expression. Test calculations on several different molecules show good performance of the PHF parameter sets in terms of how well they can reproduce QM‐calculated frequencies. When soft bonds are involved in the target molecule as in the case of secondary building units of metal‐organic frameworks, the MM‐optimized geometry sometimes deviates significantly from the QM‐optimized one. We show that this problem is rectified effectively by use of a simple procedure called Katachi that modifies the equilibrium bond distances and angles in bond‐stretching and angle‐bending terms. © 2016 Wiley Periodicals, Inc.     

Analytical Hessian of electronic excited states in time-dependent density functional theory with Tamm-Dancoff approximation

We present the analytical expression and computer implementation for the second-order energy derivatives of the electronic excited state with respect to the nuclear coordinates in the time-dependent density functional theory (TDDFT) with Gaussian atomic orbital basis sets. Here, the Tamm-Dancoff approximation to the full TDDFT is adopted, and therefore the formulation process of TDDFT excited-state Hessian is similar to that of configuration interaction singles (CIS) Hessian. However, due to the replacement of the Hartree-Fock exchange integrals in CIS with the exchange-correlation kernels in TDDFT, many quantitative changes in the derived equations are arisen. The replacement also causes additional technical difficulties associated with the calculation of a large number of multiple-order functional derivatives with respect to the density variables and the nuclear coordinates. Numerical tests on a set of test molecules are performed. The simulated excited-state vibrational frequencies by the analytical Hessian approach are compared with those computed by CIS and the finite-difference method. It is found that the analytical Hessian method is superior to the finite-difference method in terms of the computational accuracy and efficiency. The numerical differentiation can be difficult due to root flipping for excited states that are close in energy. TDDFT yields more exact excited-state vibrational frequencies than CIS, which usually overestimates the values.     

Franck-Condon factors within damped displacement harmonic oscillators: Solvent-enhanced absorption and fluorescence spectra

Damped harmonic oscillators that take into account local-mode nuclear vibrations interacting with solvent molecules are developed into Franck-Condon factors within displaced harmonic oscillator approximation. This is practically done by scaling an unperturbed Hessian matrix that represents local modes of force constants for molecules in a gaseous phase, and then by diagonalizing the perturbed Hessian matrix it results in direct modification of Huang–Rhys factors which represent normal modes of solute molecule perturbed by solvent environment. For highly symmetric polycyclic aromatic hydrocarbon molecules in which hydrogen atom vibrations in a solution can be scaled equally, one-set scaling parameters constructed into damped Franck-Condon factors can reproduce solvent-enhanced absorption and fluorescence spectra in solution. However, for low symmetry molecules with atoms other than hydrogen and carbon atoms, multi-set scaling parameters constructed into damped Franck-Condon factors can also reproduce solvent-enhanced absorption and fluorescence spectra in solution. Examples for high symmetry perylene in benzene solution with one-set scaling parameters and for low symmetry carbazole in n-hexane solution with multi-set scaling parameters are given, in both cases, the present damped Franck-Condon simulation can reproduce solvent-enhanced absorption and fluorescence spectra in solution.     

Pattern of separatrices and intrinsic reaction coordinates for degenerate thermal rearrangements

A structurally stable model of the standard adiabatic gradient field of the potential energy surface for certain pericyclic reactions is derived.These reactions are not subjected to the principles of orbital isomerism or to the Woodward-Hoffmann rules.Use is made of a principle established by Ariel Fernández and Oktay Sinanolu which precludes direct meta-IRC connections between transition states.It is shown that Jahn-Teller isomers of the singlet biradicals involved in the process are not interconvertible since the biradical configuration is not a transition state but a critical point with Hessian matrix with two negative eigenvalues.The topological features of the PES obtained by combinatorial methods are in full agreement with earlier results obtained from MINDO calculations.