Generic implementation of semi-analytical CI gradients for NDDO-type methods

Generic semi-analytical energy gradients are derived and implemented for NDDO-type methods, by using numerical integral and Fock matrix derivatives in the context of an otherwise analytical approach for configuration interaction (CI) and other non-variational treatments. The correctness, numerical precision, and performance of this hybrid approach are established through comparisons with fully numerical and fully analytical calculations. The semi-analytical evaluation of the CI gradient is generally much faster than the fully numerical computation, but somewhat slower than a fully analytical calculation, which however shows the same scaling behavior. It is the method of choice whenever a fully analytical CI gradient is not available due to the lack of analytical integral derivatives. The implementation is generic in the sense that it can easily be extended to any new NDDO-type Hamiltonian. The present development of a semi-analytical CI gradient will facilitate studies of electronically excited states with recently proposed NDDO methods that include orthogonalization corrections. Dedicated to Professor Karl Jug on the occasion of his 65th birthday     

Analytical energy gradient evaluation in relativistic and nonrelativistic density functional calculations

The expressions of analytical energy gradients in density functional theory and their implementation in programs are reported. The evaluation of analytical energy gradients can be carried out in the fully 4-component relativistic, approximate relativistic, and nonrelativistic density functional calculations under local density approximation or general gradient approximation with or without frozen core approximation using different basis sets in our programs. The translational invariance condition and the fact that the one-center terms do not contribute to the energy gradients are utilized to improve the calculation accuracy and to reduce the computational effort. The calculated results of energy gradients and optimized geometry as well as atomization energies of some molecules by the analytical gradient method are in very good agreement with results obtained by the numerical derivative method.     

Analytical calculation of the solvent-accessible surface area and its nuclear gradients by stereographic projection: A general approach for molecules,polymers, nanotubes,helices, and surfaces

In this article, we explore an alternative to the analytical Gauss–Bonnet approach for computing the solvent-accessible surface area (SASA) and its nuclear gradients. These two key quantities are required to evaluate the nonelectrostatic contribution to the solvation energy and its nuclear gradients in implicit solvation models. We extend a previously proposed analytical approach for finite systems based on the stereographic projection technique to infinite periodic systems such as polymers, nanotubes, helices, or surfaces and detail its implementation in the Crystal code. We provide the full derivation of the SASA nuclear gradients, and introduce an iterative perturbation scheme of the atomic coordinates to stabilize the gradients calculation for certain difficult symmetric systems. An excellent agreement of computed SASA with reference analytical values is found for finite systems, while the SASA size-extensivity is verified for infinite periodic systems. In addition, correctness of the analytical gradients is confirmed by the excellent agreement obtained with numerical gradients and by the translational invariance achieved, both for finite and infinite periodic systems. Overall therefore, the stereographic projection approach appears as a general, simple, and efficient technique to compute the key quantities required for the calculation of the nonelectrostatic contribution to the solvation energy and its nuclear gradients in implicit solvation models applicable to both finite and infinite periodic systems.     

Gradients of the Exchange-repulsion Energy in the General Effective Fragment Potential Method

Formulae for calculating the analytic gradients of the exchange-repulsion energy in the general effective fragment potential (EFP2) method are derived and implemented using a direct differentiation approach. The timings for the exchange repulsion gradient evaluations are approximately three times longer than the energy evaluations, orders of magnitude faster than a previous implementation. Since the direct differentiation approach is not approximate, the gradients can be used with confidence in molecular dynamics and Monte Carlo simulations with the EFP2 method.     

Analytic energy gradients of the optimized effective potential method

The analytic energy gradients of the optimized effective potential (OEP) method in density-functional theory are developed. Their implementation in the direct optimization approach of Yang and Wu [Phys. Rev. Lett. 89, 143002 (2002)] and Wu and Yang [J. Theor. Comput. Chem. 2, 627 (2003)] are carried out and the validity is confirmed by comparison with corresponding gradients calculated via numerical finite difference. These gradients are then used to perform geometry optimizations on a test set of molecules. It is found that exchange-only OEP (EXX) molecular geometries are very close to the Hartree-Fock results and that the difference between the B3LYP and OEP-B3LYP results is negligible. When the energy is expressed in terms of a functional of Kohn-Sham orbitals, or in terms of a Kohn-Sham potential, the OEP becomes the only way to perform density-functional calculations and the present development in the OEP method should play an important role in the applications of orbital or potential functionals.     

Structural and ITC Characterization of Peptide-Protein Binding: Thermodynamic Consequences of Cyclization Constraints,a Case Study on Vascular Endothelial Growth Factor Ligands

Macrocyclization constraints are widely used in the design of protein ligands to stabilize their bioactive conformation and increase their affinities. However, the resulting changes in binding entropy can be puzzling and uncorrelated to affinity gains. Here, the thermodynamic (Isothermal Titration Calorimetry) and structural (X-ray, NMR and CD) analysis of a complete series of lactam-bridged peptide ligands of the vascular endothelial growth factor, and their unconstrained analogs are reported. It is shown that differences in thermodynamics arise mainly from the folding energy of the peptide upon binding. The systematic reduction in conformational entropy penalty due to helix pre-organization can be counterbalanced by an unfavorable vibrational entropy change if the constraints are too rigid. The gain in configurational entropy partially escapes the enthalpy/entropy compensation and leads to an improvement in affinity. The precision of the analytical ITC method makes this study a possible benchmark for constrained peptides optimization.     

基本不变量神经网络解析梯度方法的研究

提升势能面的运行速度对于动力学模拟至关重要. 相对于计算简单、 但耗时更长的数值梯度计算, 直接求解势能面梯度的解析公式能够大幅提高势能面的运行效率. 本文发展了基本不变量神经网络解析梯度的生成方法. 计算解析梯度的代码可以通过程序自动生成. 对大量数据点进行测试后, 证明了该方法可以得到正确的势能面梯度输出结果. 通过测试不同势能面的调用时间, 发现采用解析梯度方法能够带来10倍以上的性能提升. 随着体系的增大, 这种性能提升也会越明显.     

Direct minimization of the energy functional in LCAO-MO density matrix formalism

The general multiconfiguration self-consistent-field method is presented along the density matrix formalism. The proposed optimization procedure for orbitals makes use of an orthogonal transformation in the space spanned by the fixed basis set. Acting on the unconstrained parameters of the transformation a direct minimization of the energy expression is performed using a gradient approach. A similar method may also be applied to the optimization of the expansion coefficients. The method works not only for the ground state of a given system, but also for any excited state, yielding an upper bound to the true energy of the considered state.     

High-performance chromatofocusing using linear and concave pH gradients formed with simple buffer mixtures. I. Effect of buffer composition on the gradient shape

Numerical calculations together with simplified analytical relations based on local equilibrium theory are used to determine the factors which govern the shape of the gradient formed during chromatofocusing when simple mixtures of buffering species are employed to produce linear or concave pH gradients. The numerical and analytical development is also used to determine the relation between the gradient shape and the buffering capacities of the adsorbed and liquid phases. Experiments which verify the theoretical methods are described where internally generated, retained pH gradients of various shapes are formed using high-performance chromatography columns. The resulting experimental and theoretical basis can be employed as means for the selection of the buffer composition for use in chromatofocusing.     

Density functional theory for molecular and periodic systems using density fitting and continuous fast multipole method: Analytical gradients

A full implementation of analytical energy gradients for molecular and periodic systems is reported in the TURBOMOLE program package within the framework of Kohn–Sham density functional theory using Gaussian‐type orbitals as basis functions. Its key component is a combination of density fitting (DF) approximation and continuous fast multipole method (CFMM) that allows for an efficient calculation of the Coulomb energy gradient. For exchange‐correlation part the hierarchical numerical integration scheme (Burow and Sierka, Journal of Chemical Theory and Computation 2011, 7, 3097) is extended to energy gradients. Computational efficiency and asymptotic O(N) scaling behavior of the implementation is demonstrated for various molecular and periodic model systems, with the largest unit cell of hematite containing 640 atoms and 19,072 basis functions. The overall computational effort of energy gradient is comparable to that of the Kohn–Sham matrix formation. © 2016 Wiley Periodicals, Inc.     

Optimization of equilibrium geometries and transition structures

A modified conjugate gradient algorithm for geometry optimization is outlined for use with ab initio MO methods. Since the computation time for analytical energy gradients is approximately the same as for the energy, the optimization algorithm evaluates and utilizes the gradients each time the energy is computed. The second derivative matrix, rather than its inverse, is updated employing the gradients. At each step, a one-dimensional minimization using a quartic polynomial is carried out, followed by an n-dimensional search using the second derivative matrix. By suitably controlling the number of negative eigenvalues of the second derivative matrix, the algorithm can also be used to locate transition structures. Representative timing data for optimizations of equilibrium geometries and transition structures are reported for ab initio SCF –MO calculations.     

An efficient algorithm for energy gradients and orbital optimization in valence bond theory

An efficient algorithm for energy gradients in valence bond theory with nonorthogonal orbitals is presented. A general Hartree-Fock-like expression for the Hamiltonian matrix element between valence bond (VB) determinants is derived by introducing a transition density matrix. Analytical expressions for the energy gradients with respect to the orbital coefficients are obtained explicitly, whose scaling for computational cost is m(4), where m is the number of basis functions, and is thus approximately the same as in HF method. Compared with other existing approaches, the present algorithm has lower scaling, and thus is much more efficient. Furthermore, the expression for the energy gradient with respect to the nuclear coordinates is also presented, and it provides an effective algorithm for the geometry optimization and the evaluation of various molecular properties in VB theory. Test applications show that our new algorithm runs faster than other methods.     

Energy gradients with respect to atomic positions and cell parameters for the Kohn-Sham density-functional theory at the Gamma point

The application of theoretical methods based on density-functional theory is known to provide atomic and cell parameters in very good agreement with experimental values. Recently, construction of the exact Hartree-Fock exchange gradients with respect to atomic positions and cell parameters within the Gamma-point approximation has been introduced. In this article, the formalism is extended to the evaluation of analytical Gamma-point density-functional atomic and cell gradients. The infinite Coulomb summation is solved with an effective periodic summation of multipole tensors. While the evaluation of Coulomb and exchange-correlation gradients with respect to atomic positions are similar to those in the gas phase limit, the gradients with respect to cell parameters needs to be treated with some care. The derivative of the periodic multipole interaction tensor needs to be carefully handled in both direct and reciprocal space and the exchange-correlation energy derivative leads to a surface term that has its origin in derivatives of the integration limits that depend on the cell. As an illustration, the analytical gradients have been used in conjunction with the QUICCA algorithm to optimize one-dimensional and three-dimensional periodic systems at the density-functional theory and hybrid Hartree-Fock/density-functional theory levels. We also report the full relaxation of forsterite supercells at the B3LYP level of theory.     

Communication: extended multi-state complete active space second-order perturbation theory: energy and nuclear gradients

The extended multireference quasi-degenerate perturbation theory, proposed by Granovsky [J. Chem. Phys. 134, 214113 (2011)], is combined with internally contracted multi-state complete active space second-order perturbation theory (XMS-CASPT2). The first-order wavefunction is expanded in terms of the union of internally contracted basis functions generated from all the reference functions, which guarantees invariance of the theory with respect to unitary rotations of the reference functions. The method yields improved potentials in the vicinity of avoided crossings and conical intersections. The theory for computing nuclear energy gradients for MS-CASPT2 and XMS-CASPT2 is also presented and the first implementation of these gradient methods is reported. A number of illustrative applications of the new methods are presented.     

A method for constrained energy minimization of macromolecules

Two algorithms for the local energy minimization of the structure of macromolecules in the presence of constraints are proposed. They are a combination of the method of steepest descents and the method of conjugate gradients with the procedure SHAKE, by which distance constraints can be satisfied. The two algorithms are tested by applying them to a small protein, the bovine pancreatic trypsin inhibitor (BPTI), and compared with the penalty function method for conserving constraints. The efficiency of the proposed methods depends on the level of interdependence of the constraints. For bond-length constraints, the use of SHAKE is superior to the penalty function method. However, when bond-angle constraints are included, SHAKE is more efficient only if the curvature of the penalty function is considerably greater than that of the potential function being minimized. The results indicate that with bond-length constraints the minimization behavior is similar to that without constraints. However, the simultaneous application of bond-length and bond-angle constraints appears to confine the molecule to a very limited part of configuration space, very different from the part covered by an unconstrained minimization. This conclusion calls into question energy minimizations of protein systems in which only the dihedral angles are allowed to vary.     

Energy gradients in combined fragment molecular orbital and polarizable continuum model (FMO/PCM) calculation

The analytic energy gradients for the combined fragment molecular orbital and polarizable continuum model (FMO/PCM) method are derived and implemented. Applications of FMO/PCM geometry optimization to polyalanine show that the structures obtained with the FMO/PCM method are very close to those obtained with the corresponding full ab initio PCM methods. FMO/PCM (RHF/6‐31G* level) is used to optimize the solution structure of the 304‐atom Trp‐cage miniprotein and the result is in agreement with NMR experiments. The key factors determining the relative stability of the α‐helix, β‐turn and the extended form in solution are elucidated for polyalanine. © 2009 Wiley Periodicals, Inc. J Comput Chem, 2010     

Molecular tailoring approach for geometry optimization of large molecules: energy evaluation and parallelization strategies

A linear-scaling scheme for estimating the electronic energy, gradients, and Hessian of a large molecule at ab initio level of theory based on fragment set cardinality is presented. With this proposition, a general, cardinality-guided molecular tailoring approach (CG-MTA) for ab initio geometry optimization of large molecules is implemented. The method employs energy gradients extracted from fragment wave functions, enabling computations otherwise impractical on PC hardware. Further, the method is readily amenable to large scale coarse-grain parallelization with minimal communication among nodes, resulting in a near-linear speedup. CG-MTA is applied for density-functional-theory-based geometry optimization of a variety of molecules including alpha-tocopherol, taxol, gamma-cyclodextrin, and two conformations of polyglycine. In the tests performed, energy and gradient estimates obtained from CG-MTA during optimization runs show an excellent agreement with those obtained from actual computation. Accuracy of the Hessian obtained employing CG-MTA provides good hope for the application of Hessian-based geometry optimization to large molecules.     

An algorithm for geometry optimization without analytical gradients

A numerical algorithm for locating both minima and transition states designed for use in the ab initio program package GAUSSIAN 82 is presented. It is based on the RFO method of Simons and coworkers and is effectively the numerical version of an analytical algorithm (OPT = EF) previously published in this journal. The algorithm is designed to make maximum use of external second derivative information obtained from prior optimizations at lower levels of theory. It can be used with any wave function for which an energy can be calculated and is about two to three times faster than the default DFP algorithm (OPT = FP) supplied with GAUSSIAN 82.     

A method for the calculation of normal-mode vibrational frequencies using symmetry coordinates. Application to the calculation of secondary deuterium isotope effects on carbocations

A method is introduced for the calculation of normal-mode vibrational frequencies of polyatomic molecules based on numerical differencing of analytical gradients in symmetry coordinates. This procedure requires a number of gradient evaluations equal to the largest number of symmetry coordinates belonging to any single irreducible representation of the molecular point group (plus a single gradient evaluation at the equilibrium configuration), which is fewer than the 3N-6 (N atoms) gradient evaluations needed for schemes based on Cartesian or internal coordinates. While the proposed method will not generally be as efficient as procedures which involve the direct calculation of energy second derivatives analytically (as are now available for single-determinant wavefunctions) it appears to be equally accurate, and it should be the method of choice for frequency calculations involving multideterminant wavefunctions for which analytical second-derivative algorithms have yet to be developed. The method is illustrated by the calculation of equilibrium secondary deuterium-isotope effects on a number of reactions involving simple carbocations.     

Faster gradients for semiempirical methods

A new algorithm for the numerical evaluation of gradients in semiempirical methods is described. The method is approximately twice as fast as the schemes currently employed and produces gradients of comparable accuracy. This method has been tested by comparing the results obtained by the new method with those of the previous numerical scheme, and also with those calculated analytically. The results of using the new gradients in geometry optimizations are also presented. ©1999 John Wiley & Sons, Inc. J Comput Chem 20: 629–635, 1999