Analytical Energy Derivatives Research Articles

Towards the blackbox computation of magnetic exchange coupling parameters in polynuclear transition-metal complexes: Theory, implementation, and application

We present a method for calculating magnetic coupling parameters from a single spin-configuration via analytic derivatives of the electronic energy with respect to the local spin direction. This method does not introduce new approximations beyond those found in the Heisenberg-Dirac Hamiltonian and a standard Kohn-Sham Density Functional Theory calculation, and in the limit of an ideal Heisenberg system it reproduces the coupling as determined from spin-projected energy-differences. Our method employs a generalized perturbative approach to constrained density functional theory, where exact expressions for the energy to second order in the constraints are obtained by analytic derivatives from coupled-perturbed theory. When the relative angle between magnetization vectors of metal atoms enters as a constraint, this allows us to calculate all the magnetic exchange couplings of a system from derivatives with respect to local spin directions from the high-spin configuration. Because of the favorable computational scaling of our method with respect to the number of spin-centers, as compared to the broken-symmetry energy-differences approach, this opens the possibility for the blackbox exploration of magnetic properties in large polynuclear transition-metal complexes. In this work we outline the motivation, theory, and implementation of this method, and present results for several model systems and transition-metal complexes with a variety of density functional approximations and Hartree-Fock.

Analytical second derivatives of excited-state energy within the time-dependent density functional theory coupled with a conductor-like polarizable continuum model

This work extends our previous works [J. Liu and W. Z. Liang, J. Chem. Phys. 135, 014113 (2011); J. Liu and W. Z. Liang, J. Chem. Phys. 135, 184111 (2011)] on analytical excited-state Hessian within the framework of time-dependent density functional theory (TDDFT) to couple with a conductor-like polarizable continuum model (CPCM). The formalism, implementation, and application of analytical first and second energy derivatives of TDDFT/CPCM excited state with respect to the nuclear and electric perturbations are presented. Their performances are demonstrated by the calculations of excitation energies, excited-state geometries, and harmonic vibrational frequencies for a number of benchmark systems. The calculated results are in good agreement with the corresponding experimental data or other theoretical calculations, indicating the reliability of the current computer implementation of the developed algorithms. Then we made some preliminary applications to calculate the resonant Raman spectrum of 4-hydroxybenzylidene-2,3-dimethyl-imidazolinone in ethanol solution and the infrared spectra of ground and excited states of 9-fluorenone in methanol solution.

Analytic calculation of second-order electric response properties with the normalized elimination of the small component (NESC) method

Analytic second derivatives of the relativistic energy for the calculation of electric response properties are derived utilizing the normalized elimination of the small component (NESC) method. Explicit formulas are given for electric static dipole polarizabilities and infrared intensities by starting at the NESC representation of electric dipole moments. The analytic derivatives are implemented in an existing NESC program and applied to calculate dipole moments, polarizabilities, and the infrared spectra of gold- and mercury-containing molecules as well as some actinide molecules. Comparison with experiment reveals the accuracy of NESC second order electric response properties.

Symmetry-adapted cluster and symmetry-adapted cluster-configuration interaction method in the polarizable continuum model: Theory of the solvent effect on the electronic excitation of molecules in solution

In this paper we present the theory and implementation of the symmetry-adapted cluster (SAC) and symmetry-adapted cluster-configuration interaction (SAC-CI) method, including the solvent effect, using the polarizable continuum model (PCM). The PCM and SAC/SAC-CI were consistently combined in terms of the energy functional formalism. The excitation energies were calculated by means of the state-specific approach, the advantage of which over the linear-response approach has been shown. The single-point energy calculation and its analytical energy derivatives are presented and implemented, where the free-energy and its derivatives are evaluated because of the presence of solute-solvent interactions. We have applied this method to s-trans-acrolein and metylenecyclopropene of their electronic excitation in solution. The molecular geometries in the ground and excited states were optimized in vacuum and in solution, and both the vertical and adiabatic excitations were studied. The PCM-SAC/SAC-CI reproduced the known trend of the solvent effect on the vertical excitation energies but the shift values were underestimated. The excited state geometry in planar and nonplanar conformations was investigated. The importance of using state-specific methods was shown for the solvent effect on the optimized geometry in the excited state. The mechanism of the solvent effect is discussed in terms of the Mulliken charges and electronic dipole moment.

Accurate ab initio energy gradients in chemical compound space

Analytical potential energy derivatives, based on the Hellmann-Feynman theorem, are presented for any pair of isoelectronic compounds. Since energies are not necessarily monotonic functions between compounds, these derivatives can fail to predict the right trends of the effect of alchemical mutation. However, quantitative estimates without additional self-consistency calculations can be made when the Hellmann-Feynman derivative is multiplied with a linearization coefficient that is obtained from a reference pair of compounds. These results suggest that accurate predictions can be made regarding any molecule's energetic properties as long as energies and gradients of three other molecules have been provided. The linearization coefficent can be interpreted as a quantitative measure of chemical similarity. Presented numerical evidence includes predictions of electronic eigenvalues of saturated and aromatic molecular hydrocarbons.

An efficient algorithm for the density-functional theory treatment of dispersion interactions

The quasi-self-consistent-field dispersion-corrected density-functional theory formalism (QSCF-DC-DFT) is developed and presented as an efficient and reliable scheme for the DFT treatment of van der Waals dispersion complexes, including full geometry optimizations and frequency calculations with analytical energy derivatives in a routine way. For this purpose, the long-range-corrected Perdew-Burke-Ernzerhof exchange functional and the one-parameter progressive correlation functional of Hirao and co-workers are combined with the Andersson-Langreth-Lundqvist (ALL) long-range correlation functional. The time-consuming self-consistent incorporation of the ALL term in the DFT iterations needed for the calculation of forces and force constants is avoided by an a posteriori evaluation of the ALL term and its gradient based on an effective partitioning of the coordinate space into global and intramonomer coordinates. QSCF-DC-DFT is substantially faster than SCF-DC-DFT would be. QSCF-DC-DFT is used to explore the potential energy surface (PES) of the benzene dimer. The results for the binding energies and intermolecular distances agree well with coupled-cluster calculations at the complete basis-set limit. We identify 16 stationary points on the PES, which underlines the usefulness of analytical energy gradients for the investigation of the PES. Furthermore, the inclusion of analytically calculated zero point energies reveals that large-amplitude vibrations connect the eight most stable benzene dimer forms and make it difficult to identify a dominating complex form. The tilted T structure and the parallel-displaced sandwich form have the same D(0) value of 2.40 kcal/mol, which agrees perfectly with the experimental value of 2.40+/-0.40 kcal/mol.

Analytic derivatives for the Cholesky representation of the two-electron integrals

We propose a formalism for calculating analytic derivatives of the electronic energy with respect to nuclear coordinates using Cholesky decomposition of the two-electron integrals. The formalism is derived by exploiting the equivalence of Cholesky decomposition and density fitting when a suitable auxiliary basis set is used for expanding atomic orbital product densities in the latter. An implementation of gradients at the nonhybrid density functional theory level is presented, and sample calculations demonstrate that the errors in equilibrium geometries due to the Cholesky representation of the integrals can be controlled by adjusting the decomposition threshold.

New parallel algorithm for MP2 energy gradient calculations

A new parallel algorithm has been developed for calculating the analytic energy derivatives of full accuracy second order Møller-Plesset perturbation theory (MP2). Its main projected application is the optimization of geometries of large molecules, in which noncovalent interactions play a significant role. The algorithm is based on the two-step MP2 energy calculation algorithm developed recently and implemented into the quantum chemistry program, GAMESS. Timings are presented for test calculations on taxol (C47H51NO14) with the 6-31G and 6-31G(d) basis sets (660 and 1032 basis functions, 328 correlated electrons) and luciferin (C11H8N2O3S2) with aug-cc-pVDZ and aug-cc-pVTZ (530 and 1198 basis functions, 92 correlated electrons). The taxol 6-31G(d) calculations are also performed with up to 80 CPU cores. The results demonstrate the high parallel efficiency of the program.

Analytical energy gradients for local second-order Møller-Plesset perturbation theory using density fitting approximations.

An efficient method to compute analytical energy derivatives for local second-order Møller-Plesset perturbation energy is presented. Density fitting approximations are employed for all 4-index integrals and their derivatives. Using local fitting approximations, quadratic scaling with molecular size and cubic scaling with basis set size for a given molecule is achieved. The density fitting approximations have a negligible effect on the accuracy of optimized equilibrium structures or computed energy differences. The method can be applied to much larger molecules and basis sets than any previous second-order Møller-Plesset gradient program. The efficiency and accuracy of the method is demonstrated for a number of organic molecules as well as for molecular clusters. Examples of geometry optimizations for molecules with 100 atoms and over 2000 basis functions without symmetry are presented.

Analytic second derivatives for general coupled-cluster and configuration-interaction models.

Analytic second derivatives of energy for general coupled-cluster (CC) and configuration-interaction (CI) methods have been implemented using string-based many-body algorithms. Wave functions truncated at an arbitrary excitation level are considered. The presented method is applied to the calculation of CC and CI harmonic frequencies and nuclear magnetic resonance chemical shifts up to the full CI level for some selected systems. The present benchmarks underline the importance of higher excitations in high-accuracy calculations.

Molecular gradients for the second-order generalized Van Vleck variant of multireference perturbation theory

Recently, a revised second-order generalized Van Vleck perturbation theory (GVVPT2) for the description of molecular electronic structure has been reported [J. Chem. Phys. 117, 4133 (2002)] that is both state selective and of the “perturb-then-diagonalize” type of multireference perturbation theory (MRPT). Herein, formulas for analytic derivatives of the GVVPT2 energy with respect to nuclear perturbations are presented, as are illustrative calculations on model problems. Specifically, it is shown that the modification of the energy denominator, which addresses the so-called intruder-state problem of MRPT, is analytically differentiable with respect to nuclear perturbation and only requires use of matrices available, or directly obtainable, from the underlying multiconfigurational self-consistent field calculation. The developed formalism takes full advantage of the theoretical and computational characteristics of the GVVPT2 energy. In particular, the calculations are performed directly in a spin-adapted basis and utilize the recently introduced concept of macroconfigurations. Moreover, the full flexibility of the energy calculations with respect to arbitrariness of reference—i.e., no restriction to complete active space self-consistent field—is retained. Test calculations on N2 and O3 comparing the analytic derivatives with the results of finite-difference calculations corroborate the formulas and implementation.

Analytic energy derivatives for regular approximations of relativistic effects applicable to methods with and without correlation corrections

Analytic expressions are derived for the evaluation of derivatives of the total molecular energy with respect to external parameters (nuclear coordinates, external electric fields, etc.) within the relativistic regular approximation. The presented formalism employs the spectral resolution of the identity avoiding, however, the explicit use of an auxiliary basis set in the calculation of the matrix elements of the regular relativistic Hamiltonian. The final formulas for the total energy and energy derivatives are presented in matrix form suitable for implementation into standard quantum chemical packages. Results of benchmark calculations for gold containing diatomic molecules and for xenone hexafluoride performed at the Hartree–Fock and various correlation corrected levels of theory are presented and discussed.

New challenges in quantum chemistry: Quests for accurate calculations for large molecular systems.

As quantum chemistry plays a more and more central role in many complicated chemical problems, it has become necessary to obtain accurate results for large molecular systems. Conventional quantum chemistry methods are either too expensive to apply to large systems or too approximate for the results to be reliable, and they fail to satisfy this requirement. A variety of different approaches is being developed with the aim of achieving this goal: local correlation methods; divide-and-conquer methods; linear-scaling density functional methods based on the fast multipole and other approximations; effective potential methods; and hybrid methods. ONIOM (our N-layered integrated molecular orbital plus molecular mechanics method), developed by the authors, is a hybrid method in which a large molecular system is divided into onion-skin-like layers, and different quantum chemistry/molecular mechanics methods are used for different parts of the system; the results are combined to extrapolatively estimate the results of high-level calculation for the real system. Several applications of ONIOM will be discussed.

Analytical derivatives, π bonding and d–s mixing in the ligand field molecular mechanics method

Expressions are presented for the analytical energy derivatives of the ligand field stabilisation energy (LFSE) within the angular overlap model formalism. In addition to the σ-bonding contributions already incorporated in the ligand field molecular mechanics (LFMM) code, explicit contributions from π-bonding and d–s mixing have been added. Illustrative LFMM parameters are developed for planar [CuCl4]2− and tetragonal octahedral [CuCl6]4− to demonstrate that it is now possible to reproduce to high accuracy both the structures and the associated d–d transition energies.

Analytic Energy Derivatives for the Direct Iterative Approach to the Generalized Bloch Equation

A general formalism for the analytic energy derivatives in the context of the recently developed state-selective version of the direct iterative approach to the generalized Bloch equation is presented. An explicit formalism is developed for both the gradients and the Hessian by exploiting the so-called Z-vector method. A procedure for the development of the corresponding algorithm for higher than the second-order properties is also briefly outlined.

Fast Evaluation of Geometries and Properties of Excited Molecules in Solution: A Tamm-Dancoff Model with Application to 4-Dimethylaminobenzonitrile

We present a method to evaluate ab initio energy, wave function, and gradient of a solvated molecule in an electronically excited state. In particular, this paper extends the Polarizable Continuum Model (PCM) to a specific level of theory for the interpretation of the electronic spectra: configuration interaction (CI) among all singly substituted determinants using a Hartree−Fock reference state. This method, in very wide use, allows investigations of both structures and properties of electronically excited molecules through the evaluation of analytical energy derivatives with respect to various parameters. The most relevant formal aspects on the extension of the theory to solvated systems are discussed, and numerical applications to the study of geometries and one-electron properties of the low-lying excited states of 4-dimethylamino benzonitrile (DMABN) in acetonitrile solution are presented.

Optimization of quantum Monte Carlo wave functions using analytical energy derivatives

An algorithm is proposed to optimize quantum Monte Carlo (QMC) wave functions based on Newton’s method and analytical computation of the first and second derivatives of the variational energy. This direct application of the variational principle yields significantly lower energy than variance minimization methods when applied to the same trial wave function. Quadratic convergence to the local minimum of the variational parameters is achieved. A general theorem is presented, which substantially simplifies the analytic expressions of derivatives in the case of wave function optimization. To demonstrate the method, the ground-state energies of the first-row elements are calculated.

A grid-free DFT implementation of non-local functionals and analytical energy derivatives

A grid-free DFT implementation of non-local functionals and analytical energy derivatives

A grid-free DFT implementation of non-local functionals and analytical energy derivatives

We have previously developed a grid-free approach to implement local density functionals based entirely on matrix representations of operators known from conventional electronic structure methods (Y.C. Zheng and J. Almlöf, Chem. Phys. Lett., 214 (1993) 397). In this article we present a scheme to implement non-local gradient corrected functionals and analytical energy derivatives of density functional theory. The results obtained show that our grid-free DFT approach is indeed a viable alternative to these grid-based methods.

Generalized product functions: energy analytic derivatives. The variational theory of subsystems as a guide for solvation models

The analytic energy derivatives for the Hartree-Fock generalized product functions are presented. The results of the present theoretical model are analyzed and compared with the discrete-continuous cavity model for the treatment of solvation problems.