Analytical Energy Gradients Research Articles

Density functional crystal orbital study on the normal vibrations and phonon dispersion curves of all-trans polyethylene

Optimized structural parameters and frequencies of the infrared- and Raman-active vibrations are obtained for all-trans polyethylene by using the analytical energy gradient scheme in the density functional crystal orbital formalism. The Slater–Vosko–Wilk–Nusair (SVWN), the Becke–Lee–Yang–Parr (BLYP), and the Becke3–Lee–Yang–Parr (B3LYP) functionals are used with the 3-21G and 6-31G* basis sets. The frequencies calculated with the 6-31G* basis set are found to be in better agreement with the observed frequencies than those calculated with the 3-21G basis set regardless of the exchange-correlation functionals used. The root mean square errors between the calculated and observed frequencies are 21, 20, and 15 cm−1 for the SVWN/6-31G*, the BLYP/6-31G*, and the B3LYP/6-31G* calculations, respectively. Optical branches of the phonon dispersion curves are calculated at the SVWN/6-31G* level by adopting a C7H14 unit as a reference unit cell. The calculated phonon dispersion curves are in reasonable agreement with the curves experimentally determined and with the curves obtained with an empirical force field except for the skeletal stretching branches. Inelastic neutron scattering (INS) spectrum is also calculated by using the force field derived at the SVWN/6-31G* level. The overall intensity profile of the observed INS spectrum is well reproduced by the present calculations in which the effects of the Debye–Waller factors and the phonon wings are taken into account.

Analytic energy gradients for multiconfigurational self-consistent field second-order quasidegenerate perturbation theory (MC-QDPT)

An analytic energy gradient method for second-order quasidegenerate perturbation theory with multiconfigurational self-consistent field reference functions (MC-QDPT) is derived along the lines of the response function formalism (RFF). According to the RFF, the gradients are calculated without solving coupled perturbed equations. Instead, it is necessary to solve seven sets of linear equations in order to determine Lagrangian multipliers, corresponding to four sets of parameter constraining conditions and three sets of additional parameter defining conditions in the Lagrangian. Just one of these linear equations is a large scale linear equation; the others are reducible to just partial differentiations or simple equations solvable by straightforward subroutines.

Analytical energy gradients for local second-order Mo/ller–Plesset perturbation theory

Based on the orbital invariant formulation of Mo/ller–Plesset (MP) perturbation theory, analytical energy gradients have been formulated and implemented for local second order MP (LMP2) calculations. The geometry-dependent truncation terms of the LMP2 energy have to be taken into account. This leads to a set of coupled-perturbed localization (CPL) equations which must be solved together with the coupled-perturbed Hartree–Fock (CPHF) equations. In analogy to the conventional non-local theory, the repeated solution of these equations for each degree of freedom can be avoided by using the z-vector method of Handy and Schaefer. Explicit equations are presented for the Pipek–Mezey localization. Test calculations on smaller organic molecules demonstrate that the local approximations introduce only minor changes of computed equilibrium structures.

Density functional crystal orbital study on the normal vibrations of polyacetylene and polymethineimine

Optimized molecular structures and k=0 (Brillouin zone center) vibrational frequencies are obtained for trans- and cis-transoid polyacetylene and for anti- and syn-transoid polymethineimine by the density functional crystal orbital method with the Becke3–Lee–Yang–Parr functional. The analytical energy gradient scheme is implemented in the density functional crystal orbital method, and the force constants of the infinite polymers are evaluated by numerical differentiation of the analytical energy gradients. For the trans- and cis-transoid isomers of polyacetylene, the vibrational frequencies calculated and then uniformly scaled by a single scale factor are in reasonable agreement with the observed frequencies. For polymethineimine, it is found that the calculated frequencies of the anti-transoid isomer completely disagree with the observed frequencies. In contrast, the calculated frequencies of syn-transoid polymethineimine are in agreement with the observed, although there are only three observed frequencies available. The total energy of syn-transoid polymethineimine is found to be lower than that of the anti-transoid isomer by 15.6 kJ mol−1.

Analytical energy gradient of the ground, excited, ionized and electron-attached states calculated by the SAC/SAC-CI method

The method of calculating the analytical energy gradients for the ground, excited, ionized and electron-attached states calculated by the SAC (symmetry adapted cluster)/SAC-CI (configuration interaction) method is formulated and implemented. It is adapted with the configuration selection procedure used in the SAC/SAC-CI method. The performance of the method is investigated for the Li 2 and CO molecules.

Analytic energy gradients with frozen molecular orbitals in coupled-cluster and many-body perturbation theory methods: Systematic study of the magnitude and trends of the effects of frozen molecular orbitals

Analytic coupled-cluster (CC) and many-body perturbation theory (MBPT) energy gradient methods with restricted Hartree–Fock (RHF), unrestricted Hartree–Fock (UHF), restricted open-shell Hartree–Fock (ROHF), and quasi-RHF(QRHF) reference functions are extended to permit dopping core and excited orbitals. By using the canonical property of the semicanonical ROHF orbitals and the RHF orbitals from which the QRHF reference function is constructed, it is shown that a general procedure can be established not only for RHF and UHF, but also for ROHF and QRHF reference functions. The basic theory and implementation are reported. To provide a systematic study of the trends and magnitudes of the effects of dropped molecular orbitals (MOs) on the structures, harmonic frequencies, and ir intensities, we study HCN, C2H2, CO2, HO2, and C2H4 at increasing levels of correlation and basis sets. The effects of the dropped MOs with the largest basis sets are about 0.003 Å and 0.1° in structures and about 1% on harmonic frequencies and ir intensities. The magnitude and the direction of the drop-MO effect tend to be almost constant from MBPT(2) to CCSD(T) methods. The two isomers of S3 are studied by the drop-MO-method, yielding very accurate results.

Simple recipe for implementing computation of first-order relativistic corrections to electron correlation energies in framework of direct perturbation theory

The computation of the relativistic correction to the first order in 1/c2, where c is the velocity of light, is implemented at the levels of coupled cluster and many-body perturbation theory. The relativistic correction is obtained by applying direct perturbation theory through the first order, and it is shown that its implementation is straightforward if analytical energy gradients of the methods under consideration are available. Preliminary results were obtained by a numerical procedure and are reported for some closed-shell atoms (He, Be, Ne, and Ar) and molecules (CuH and SiH4). © 1997 by John Wiley & Sons, Inc.

Density functional based structure optimization for molecules containing heavy elements: analytical energy gradients for the Douglas-Kroll-Hess scalar relativistic approach to the LCGTO-DF method

The self-consistent scalar-relativistic linear combination of Gaussian-type orbitals density functional (LCGTO-DF) method has been extended to calculate analytical energy gradients. The method is based on the use of a unitary second order Douglas-Kroll-Hess (DKH) transformation for decoupling large and small components of the full four-component Dirac-Kohn-Sham equation. The approximate DKH transformation most common in molecular calculations has been implemented; this variant employs nuclear potential based projectors and it leaves the electron-electron interaction untransformed. Examples are provided for the geometry optimization of a series of heavy metal systems which feature a variety of metal-ligand bonds, like Au2, AuCl, AuH, Mo(CO)6 and W(CO)6 as well as the d10 complexes [Pd(PH3)2O2] and [Pt(PH3)2O2]. The calculated results, obtained with several gradient-corrected exchange-correlation potentials, compare very well with experimental data and they are of similar or even better accuracy than those of other high quality relativistic calculations reported so far.

An effective fragment method for modeling solvent effects in quantum mechanical calculations

An effective fragment model is developed to treat solvent effects on chemical properties and reactions. The solvent, which might consist of discrete water molecules, protein, or other material, is treated explicitly using a model potential that incorporates electrostatics, polarization, and exchange repulsion effects. The solute, which one can most generally envision as including some number of solvent molecules as well, is treated in a fully ab initio manner, using an appropriate level of electronic structure theory. In addition to the fragment model itself, formulae are presented that permit the determination of analytic energy gradients and, therefore, numerically determined energy second derivatives (hessians) for the complete system. Initial tests of the model for the water dimer and water-formamide are in good agreement with fully ab initio calculations.

Analytical energy gradient for the reference interaction site model multiconfigurational self-consistent-field method: Application to 1,2-difluoroethylene in aqueous solution

We have reformulated the reference interaction site model self-consistent-field (RISM-SCF) method to calculate the molecular properties of solute in solutions. The key feature is to introduce the Helmholtz free energy given as a sum of the solute electronic energy estimated by ab initio molecular orbital (MO) methods and the hypernetted-chain (HNC) excess chemical potential of solvation for the extended reference interaction site model (XRISM) method. The variational conditions for the multiconfigurational self-consistent-field (MCSCF) wave functions were derived. It was also shown that, in the case of the Hartree–Fock (HF) method, the present expression gives the same Fock matrix defined previously. Moreover the expression for first derivatives of the free energy with respect to the solute nuclear coordinates were obtained. Sample calculations for the chemical equilibrium of cis- and trans-isomers of 1,2-difluoroethylene are presented. With the use of analytical energy gradients for the MCSCF and HF methods, the solute molecular geometries were determined in an aqueous solution.

Analytic energy gradients for the Gaussian very fast multipole method (GvFMM)

The first use of the Gaussian very fast multipole method (GvFMM) for calculating the integral derivatives that arise in the Coulomb terms of density-functional theory (DFT) energy gradients is reported. Tests of the GvFMM gradient algorithm indicate that its accuracy, speed, and near-linear scaling behavior are similar to the GvFMM molecular energy algorithm. Specifically, 10 −8 hartree per Bohr accuracy in the energy gradient has been achieved, and the ratio of the computational cost for the GvFMM energy gradient to the GvFMM energy has been found to be lower than the ratio for previous state-of-the-art method.

The implementation of analytical energy gradients based on a quasi‐relativistic density functional method: The application to metal carbonyls

AbstractThe capabilities of the quasi‐relativistic scheme due to J. G. Snijders at al. [Mol. Phys, 38, 1969 (1979) Ibid., J. Phys. Chem. 93, 3050 (1989)] has been extended by deriving expressions for the energy gradients with respect to the total energy EQR and implementing them into the ADF program system [B. to Velde and E. J. Baerends, Int. J. Quantum Chem. 33, 87 (1988)]. This implementation enables automated geometry optimization at the relativistic level. The new scheme has been applied together with a self‐consistent nonlocal density functional scheme, NL–SCF + QR, to the calculation of MCO bond lengths and the first bond dissociation energy (FBDE) in the binary transition metal carbonyls M(CO)5 (M = Fe, Ru, Os) and M(CO)6 (M = Cr, Mo, W). The calculated MCO bond lengths are in good agreement with available experimental data with an error typically smaller than 0.01 Å. The calculated FBDES are 45.7 (Fe), 33.0 (Ru), 34.7 (Os), 46.2 (Cr), 39.7 (Mo), and 43.7 (W) kcal/mol, respectively. These values compare well with the available experimental estimates of 42 (Fe), 28 (Ru), 31 (Os), 37 (Cr), 41 (Mo), and 46 (W), respectively. The relativistic effects are found to contract MCO bonds by between 0.07 and 0.16 A and strengthen the FBDEs by 5‐11 kcal/mol for third‐row compounds. The relativistic stabilization of the FBDES among the 5d elements makes, in general, the MCO bond of the 4d element weakest within a triad. © 1995 John Wiley & Sons, Inc.

Grid-free density-functional technique with analytical energy gradients.

We have implemented a completely analytical linear combination of atomic orbitals-X\ensuremath{\alpha} local density-functional method with exact energy gradients. The superiority of the analytical method over the corresponding grid-based technique in terms of speed, smooth energy surfaces, and accurate gradients is demonstrated in calculations for the umbrella inversion mode of ammonia. The equilibrium geometry and inversion barrier height are also shown to compare well to experimental data.

Analytical energy gradients and geometry optimization in the divide-and-conquer method for large molecules

Based on the divide-and-conquer method in the density-functional theory, an efficient approach is developed to compute analytically the energy gradients with respect to the nuclear coordinates. Tests performed show that both energy gradients and optimized molecular geometry converge to the corresponding results of the Kohn–Sham method when the nearest neighbor contributions are increased.

On the orbital contribution to analytical derivatives of perturbation theory energies

A new approach for determining the orbital contribution to analytical derivatives of perturbation theory energies is presented. The usefulness of the new approach is demonstrated by application to closed-shell second-order Moller-Plesset perturbation theory (MP2). This new method is related to that used previously (Lee, T. J., and Rendell, A. P., 1991, J. chem. Phys., 94, 6229) for analytical energy gradients of the closed-shell singles and doubles coupled-cluster method augmented with a perturbational estimate of the effects of connected triple excitations, CCSD(T), but it is more appealing in that potential singularities are rigorously eliminated. The computational savings for MP2 are modest (two n 5 steps, where n is the number of molecular orbitals), but the savings for fourth-order perturbation theory (MP4) or CCSD(T) energy gradients (two n 7 steps) are significant. Hence the new approach is less expensive than previous applications of analytical gradient theory to perturbation theory electron corre...

Ab initio self-consistent-field studies of the structure, energetics and bonding of small gallium arsenide clusters

Ab initio all-electron self-consistent-field calculations are peformed on smallGa m As n(m+n≤5) clusters using large basis set. The structure, energetics and bonding are analyzed for all clusters, including the positive and negative charge states. In total 18 species are utilized to give a coherent unified picture. Geometry optimization for all species are performed using analytic energy gradient. For crucial or controversial cases post-self-consistent-field calculations at the second-order Moller-Plesset level are performed. In special, theD 3h geometry ofGa 2 As 3 as suggested experimentally from ESR experiment is obtained only after inclusion of electron correlation effects. We suggest an explanation to the experimentally observed mass spectrum of clusters with a given number of atoms. The observed alternating behavior for the ionization potentials and electron affinities is discussed and it is suggested that electron correlation corrections are important for a quantitative theoretical description.

Nonadiabatic coupling of the 1 1A″ and 2 1A″ states of ozone

We report the results of ab initio calculations of the nonadiabatic Born coupling terms for the two lowest-lying singlet A″ excited states of ozone ( 1 A 2 and 1 B 1 in C 2v symmetry) whose electronic potential energy surfaces exhibit a crossing at a bond angle near 120°. Since photoexcitation from the ground state to these A″ states is believed to be responsible for the Chappuis band of ozone, determination of the nonadiabatic coupling terms in the molecular Hamiltonian is therefore likely to be of importance in understanding the predissociative features of the spectrum. The nuclear derivative coupling matrix elements are computed between ab initio electronic wavefunctions obtained from multi-reference configuration interaction (MRD-CI) calculations in a basis of configuration state functions constructed from MCSCF orbitals. The first-derivative nonadiabatic coupling terms are calculated analytically in a manner similar to the calculation of analytic energy gradients, thereby avoiding the cumbersome numerical differentiation of CI and SCF coefficients. The calculated analytical derivative coupling matrix elements are used to determine a suitable transformation to a quasidiabatic basis.

Analytic energy gradients for the two-determinant coupled cluster method with application to singlet excited states of butadiene and ozone

Recently, the coupled cluster method has been generalized to a two-determinant reference (TD-CC method) applicable to open-shell singlet states. In this paper, we report the formulas and an efficient algorithm to calculate analytic energy derivatives for this new method. The implementation has been carried out in the ACES II many-body ab initio program system. The new method was first applied to the 1Bu state of butadiene and several excited states of ozone. We report geometries and one-electron properties for those states.

Ge(100) 2 x 1 and c(4 x 2) surface reconstructions studied by ab initio total-energy molecular-force calculations.

The 2\ifmmode\times\else\texttimes\fi{}1 and c(4\ifmmode\times\else\texttimes\fi{}2) surface reconstructions on Ge(100) are investigated by the ab initio, all-electron, molecular-cluster method, which solves the local-density-functional equations and provides analytical energy gradients. We use finite-size clusters (up to 71 atoms including 39 Ge atoms) to model the Ge(100) surfaces. Atomic-force calculations are extensively used to obtain the minimum-energy geometry for the different structures investigated. We determine and compare the binding energy and geometry up to the fourth layer of the symmetric (2\ifmmode\times\else\texttimes\fi{}1), buckled (2\ifmmode\times\else\texttimes\fi{}1), as well as the higher-order c(4\ifmmode\times\else\texttimes\fi{}2) reconstruction. Important energetic and structural differences are found compared to the corresponding Si(100)2\ifmmode\times\else\texttimes\fi{}1 surface. The asymmetric dimer model is found to be 0.34 eV/dimer lower than the symmetric one with the up-dimer atom being 0.19 \AA{} above the plane of the unreconstructed surface and a dimer tilt of 15\ifmmode^\circ\else\textdegree\fi{}. The buckled 2\ifmmode\times\else\texttimes\fi{}1 and c(4\ifmmode\times\else\texttimes\fi{}2) reconstructions are found to be close in energy, which suggests that both could be present on the surface at room temperature. These results are in excellent agreement with scanning-tunneling-microscopy experiments and previous theoretical studies using a slab geometry. This energy is well below the energy of the symmetric dimer reconstruction indicating that dimer flipping recently suggested for the Si(100)2\ifmmode\times\else\texttimes\fi{}1 surface is unlikely to occur in the case of the Ge(100) surface. In significant contrast to the Si(100) surface, we found that the Ge-Ge dimer is weaker with bond lengths that are slightly above the bulk value of 2.44 \AA{}, at 2.48 and 2.50 \AA{} for the asymmetric 2\ifmmode\times\else\texttimes\fi{}1 and c(4\ifmmode\times\else\texttimes\fi{}2) reconstructions, respectively. It suggests that the Ge(100) surface might show some different behavior towards adsorption.

Analytic energy gradients for the equation-of-motion coupled-cluster method: Implementation and application to the HCN/HNC system

A recently developed theory for calculating analytic first derivatives of the energy for excited states treated by the equation-of-motion coupled-cluster (EOM-CC) method has been implemented and applied to study potential energy surfaces in the HCN/HNC system. The EOM-CC singles and doubles (EOM-CCSD) approximation is used to predict equilibrium structures, energies, dipole moments, harmonic vibrational frequencies and infrared intensities of HCN and HNC isomers in both the à 1A″ and B̃ 1A′ excited states. In addition, the activated complex for HCN→HNC isomerization in these states is investigated theoretically for the first time.