Parallel coupled perturbed CASSCF equations and analytic CASSCF second derivatives

A parallel algorithm for solving the coupled-perturbed MCSCF (CPMCSCF) equations and analytic nuclear second derivatives of CASSCF wave functions is presented. A parallel scheme for evaluating derivative integrals and their subsequent use in constructing other derivative quantities is described. The task of solving the CPMCSCF equations is approached using a parallelization scheme that partitions the electronic hessian matrix over all processors as opposed to simple partitioning of the 3 N solution vectors among the processors. The scalability of the current algorithm, up to 128 processors, is demonstrated. Using three test cases, results indicate that the parallelization of derivative integral evaluation through a simple scheme is highly effective regardless of the size of the basis set employed in the CASSCF energy calculation. Parallelization of the construction of the MCSCF electronic hessian during solution of the CPMCSCF equations varies quantitatively depending on the nature of the hessian itself, but is highly scalable in all cases.     

Higher analytic derivatives. I. A new implementation for the third derivative of the SCF energy

This is the first of a series of papers on the ab initio calculation of the second, third, and fourth derivatives of the energy with respect to nuclear coordinates. The knowledge of these derivatives yields anharmonic spectroscopic constants. Here, we present efficient formulae for the analytic evaluation of these derivatives for closed-shell SCF wave functions. We discuss our implementation of the third derivative formula, in particular the integral and vectorization procedures. Applications are reported for H₂S, CHOF, and HCCF.     

Molecules-in-molecules calculations with fixed resonance integrals

A simple equation for the evaluation of resonance integrals from overlap integrals and ionization potentials of the molecular fragments is suggested for molecules-in-molecules π electron calculations. The singlet π → π* transition energies of some benzene derivatives containing donor substituents were calculated. The best results were obtained if in the expression of the resonance integral the first experimental ionization potential of the methyl derivative of the donor groups is used.     

Computation of one and two electron spin-orbit integrals

Methods for the computation of one- and two-electron spin-orbit integrals over Gaussian-type basis functions are presented. We show that existing nuclear-attraction and electron-repulsion integral codes can be readily adapted for the efficient evaluation of spin-orbit integrals; in particular, one can take advantage of recent advances in the computation of derivative integrals. Recurrence relations for the nuclear attraction integrals are also developed.     

Analytical GIAO and hybrid-basis integral derivatives: application to geometry optimization of molecules in strong magnetic fields

Analytical integral evaluation is a central task of modern quantum chemistry. Here we present a general method for evaluating differentiated integrals over standard Gaussian and mixed Gaussian/plane-wave hybrid orbitals. The main idea is to have a representation of basis sets that is flexible enough to enable differentiated integrals to be reinterpreted as standard integrals over modified basis functions. As an illustration of the method, we report a very simple implementation of Hartree-Fock level geometrical derivatives in finite magnetic fields for gauge-origin independent atomic orbitals, within the London program. As a quantum-chemical application, we optimize the structure of helium clusters and some well-known covalently bound molecules (water, ammonia and benzene) subject to strong magnetic fields.     

Ab-initio calculations on large molecules and solids by desirable computational procedures

Desirable computational procedures developed here recently for ab-initio calculations on large molecules are outlined. Effective core model potentials (MODPOT) permit calculations of valence electrons only explicitly, yet accurately; a charge-conserving integral prescreening evaluation to decide whether a block of integrals will be larger than a preset threshold and thus be calculated explicitly is effective for spatially extended systems; an efficient MERGE technique to save and reuse common invariant skeletal integrals is useful for geometry variations and for adding basis fcuntions, substituent groups and molecules; and an effective configuration interaction (CI) Hamiltonian into which are folded the effects of the occupied molecular orbitals from which no excitations are allowed is useful for molecular decompositions and intermolecular reactions. These techniques have been extended for CI calculations on breaking a chemical bond in a molecule in a crystal or solid; atom-class/atomic-class potential functions and dispersion calculations have been added. In a new program, POLY-CRYST, all the integral strategies for large molecules are meshed.     

A self-contained and portable density functional theory library for use in Ab Initio quantum chemistry programs

A major unresolved problem of density functional theory is the yet unknown exchange-correlation functional, which leads to a proliferation of its less or more successful approximations. A practical implementation of these numerous functionals can present a substantial challenge particularly if the higher order functional derivatives are required. We present a systematic method of functional implementation. The method allows a clean handling of a large number of functionals in a mutually independent way. We developed an extensive set of automatic test routines to facilitate functional and derivative testing with respect to the implementation correctness and numerical stability. An integral part of the presented solution is a program for automatic code generation from analytical formulas that uses only freely available tools. Code for evaluation of functionals and their first, second, third, and fourth derivatives can be generated, which accelerates the development, implementation, and testing of new functionals.     

A unified scheme for the calculation of differentiated and undifferentiated molecular integrals over solid-harmonic Gaussians

Utilizing the fact that solid-harmonic combinations of Cartesian and Hermite Gaussian atomic orbitals are identical, a new scheme for the evaluation of molecular integrals over solid-harmonic atomic orbitals is presented, where the integration is carried out over Hermite rather than Cartesian atomic orbitals. Since Hermite Gaussians are defined as derivatives of spherical Gaussians, the corresponding molecular integrals become the derivatives of integrals over spherical Gaussians, whose transformation to the solid-harmonic basis is performed in the same manner as for integrals over Cartesian Gaussians, using the same expansion coefficients. The presented solid-harmonic Hermite scheme simplifies the evaluation of derivative molecular integrals, since differentiation by nuclear coordinates merely increments the Hermite quantum numbers, thereby providing a unified scheme for undifferentiated and differentiated four-center molecular integrals. For two- and three-center two-electron integrals, the solid-harmonic Hermite scheme is particularly efficient, significantly reducing the cost relative to the Cartesian scheme.     

Higher order alchemical derivatives from coupled perturbed self-consistent field theory

We present an analytical approach to treat higher order derivatives of Hartree-Fock (HF) and Kohn-Sham (KS) density functional theory energy in the Born-Oppenheimer approximation with respect to the nuclear charge distribution (so-called alchemical derivatives). Modified coupled perturbed self-consistent field theory is used to calculate molecular systems response to the applied perturbation. Working equations for the second and the third derivatives of HF/KS energy are derived. Similarly, analytical forms of the first and second derivatives of orbital energies are reported. The second derivative of Kohn-Sham energy and up to the third derivative of Hartree-Fock energy with respect to the nuclear charge distribution were calculated. Some issues of practical calculations, in particular the dependence of the basis set and Becke weighting functions on the perturbation, are considered. For selected series of isoelectronic molecules values of available alchemical derivatives were computed and Taylor series expansion was used to predict energies of the "surrounding" molecules. Predicted values of energies are in unexpectedly good agreement with the ones computed using HF/KS methods. Presented method allows one to predict orbital energies with the error less than 1% or even smaller for valence orbitals.     

Analytic second derivatives for the spin-free exact two-component theory

The formulation and implementation of the spin-free (SF) exact two-component (X2c) theory at the one-electron level (SFX2c-1e) is extended in the present work to the analytic evaluation of second derivatives of the energy. In the X2c-1e scheme, the four-component one-electron Dirac Hamiltonian is block diagonalized in its matrix representation and the resulting "electrons-only" two-component Hamiltonian is then used together with untransformed two-electron interactions. The derivatives of the two-component Hamiltonian can thus be obtained by means of simple manipulations of the parent four-component Hamiltonian integrals and derivative integrals. The SF version of X2c-1e can furthermore exploit available nonrelativistic quantum-chemical codes in a straightforward manner. As a first application of analytic SFX2c-1e second derivatives, we report a systematic study of the equilibrium geometry and vibrational frequencies for the bent ground state of the copper hydroxide (CuOH) molecule. Scalar-relativistic, electron-correlation, and basis-set effects on these properties are carefully assessed.     

The W algorithm and the D transformation for the numerical evaluation of three‐center nuclear attraction integrals

Three‐center nuclear attraction integrals over exponential‐type functions are required for ab initio molecular structure calculations and density functional theory (DFT). These integrals occur in many millions of terms, even for small molecules, and they require rapid and accurate numerical evaluation. The use of a basis set of B functions to represent atomic orbitals, combined with the Fourier transform method, led to the development of analytic expressions for these molecular integrals. Unfortunately, the numerical evaluation of the analytic expressions obtained turned out to be extremely difficult due to the presence of two‐dimensional integral representations, involving spherical Bessel integral functions. % The present work concerns the development of an extremely accurate and rapid algorithm for the numerical evaluation of these spherical Bessel integrals. This algorithm, which is based on the nonlinear D transformation and the W algorithm of Sidi, can be computed recursively, allowing the control of the degree of accuracy. Numerical analysis tests were performed to further improve the efficiency of our algorithm. The numerical results section demonstrates the efficiency of this new algorithm for the numerical evaluation of three‐center nuclear attraction integrals. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007     

Analytical second derivatives of molecular electronic energy integrals obtained by spherical gaussian orbital

In this research, the complete general formulas for the analytical second derivative of the molecular integrals for spherical gaussian orbitals of electronic energy are presented. Formulas were given for the second derivative for orbital exponent, orbital and nuclear cartesian coordinates and coefficients of contracted gaussians. In order to save computational time, the formulas for the second derivative are written in terms of the original integrals. Although the formulas were presented in general for any type of application, the Floating Spherical Gaussian Orbital (FSGO) method is applied to some molecules such as LiH, H₂O and CH₂ (singlet) to check the formulas. The results were compared with the results of the finite difference method. Besides the accuracy of the analytical derivative, the saving in computational time is significant.     

Overlap integrals of B functions

Two different methods for the evaluation of overlap integrals of B functions with different scaling parameters are analyzed critically. The first method consists of an infinite series expansion in terms of overlap integrals with equal scaling parameters [14]. The second method consists of an integral representation for the overlap integral which has to be evaluated numerically. Bhattacharya and Dhabal [13] recommend the use of Gauss-Legendre quadrature for this purpose. However, we show that Gauss-Jacobi quadrature gives better results, in particular for larger quantum number. We also show that the convergence of the infinite series can be improved if suitable convergence accelerators are applied. Since an internal error analysis can be done quite easily in the case of an infinite series even if it is accelerated, whereas it is very costly in the case of Gauss quadratures, the infinite series is probably more efficient than the integral representation. Overlap integrals of all commonly occurring exponentially declining basis functions such as Slater-type functions, can be expressed by finite sums of overlap integrals of B functions, because these basis functions can be represented by linear combinations of B functions.Dedicated to Professor J. Koutecký on the occasion of his 65th birthday     

On the possibility of improving the integral evaluation section in ab initio Hartree–Fock calculations on large molecules

A modified integral package for evaluation of two-electron integrals over Gaussian basis functions is described. Modifications are implemented in the MOLECULE program system and are especially suited for the study of large molecules and molecular complexes.     

Elastic forward scattering of high-energy electrons and molecular electron density

The elastic forward scattering of high-energy electrons from molecules has been studied in the second Born approximation. An integral transformation has been adopted to evaluate the second Born integrals analytically without explicit use of molecular wave functions. In the high-energy limit, the differential cross section for the forward scattering is expressed in terms of electric dipole and quadrupole moments, the second moment of charge distribution with respect to the molecular center, and transition dipole moments. All these quantities are shown to be computable from molecular electron densities in the ground state.     

Compact formulae for three-center nuclear attraction integrals over exponential type functions

In any ab initio molecular orbital calculations, the major task involves the computation of the so-called molecular multi-center integrals. Multi-center integral calculations is a very challenging mathematical problem in nature. Quantum mechanics only determines which integrals we evaluate, but the techniques employed for their evaluations are entirely mathematical. The three-center nuclear attraction integrals occur in a very large number even for small molecules and are among of the most difficult molecular integrals to compute efficiently. In the present contribution, we report analytical expressions for the three-center nuclear attraction integrals over exponential type functions. We describe how to compute the formula to obtain an efficient evaluation in double precision arithmetic. This requires the rational minimax approximants that minimize the maximum error on the interval of evaluation.

     

Geometrical energy derivative evaluation with MRCI wave functions

The theory of MCSCF and CI energy derivatives with respect to geometrical variations is briefly reviewed with special attention given to the MCSCF and MRCI energy gradients. A computational procedure is proposed for MRCI energy gradients that does not require the solution to any “coupled-perturbed MCSCF ” equations, it does not require any expensive direct-CI matrix-vector products involving derivative integrals, and it does not require any derivative integrals to be transformed from the AO basis to the MO basis. An additional feature is that it does not require any changes to existing MCSCF gradient evaluation programs in order to compute MRCI gradients. The only difference in the two cases is the exact nature of the data passed to the gradient evaluation program from the previous steps in the computational procedure. The additional effort required to compute the entire MRCI energy gradient vector is approximately that required for one additional iteration of the MRCI diagonalization procedure and for one additional MCSCF iteration. For large scale MRCI wave functions, the MRCI energy gradient evaluation should only require about 10% of the effort of computing the wave function itself. This computational procedure removes a major computational botleneck of potential energy surface evaluation.     

Derivative of electron repulsion integral using accompanying coordinate expansion and transferred recurrence relation method for long contraction and high angular momentum

In this study, an early‐working algorithm is designed to evaluate derivatives of electron repulsion integrals (DERIs) for heavy‐element systems. The algorithm is constructed to extend the accompanying coordinate expansion and transferred recurrence relation (ACE‐TRR) method, which was developed for rapid evaluation of electron repulsion integrals (ERIs) in our previous article (M. Hayami, J. Seino, and H. Nakai, J. Chem. Phys. 2015, 142, 204110). The algorithm was formulated using the Gaussian derivative rule to decompose a DERI of two ERIs with the same sets of exponents, different sets of contraction coefficients, and different angular momenta. The algorithms designed for segmented and general contraction basis sets are presented as well. Numerical assessments of the central processing unit time of gradients for molecules were conducted to demonstrate the high efficiency of the ACE‐TRR method for systems containing heavy elements. These heavy elements may include a metal complex and metal clusters, whose basis sets contain functions with long contractions and high angular momenta.     

Gradients for two-component quasirelativistic methods. Application to dihalogenides of element 116

The authors report the implementation of geometry gradients for quasirelativistic two-component Hartree-Fock and density functional methods using either the zero-order regular approximation Hamiltonian or spin-dependent effective core potentials. The computational effort of the resulting program is comparable to that of corresponding nonrelativistic calculations, as it is dominated by the evaluation of derivative two-electron integrals, which is the same for both types of calculations. Besides the implementation of derivatives of matrix elements of the one-particle Hamiltonian with respect to nuclear displacements, the calculation of the derivative exchange-correlation energy for the open shell case involves complicated expressions because of the noncollinear approach chosen to define the spin density. A pilot application to dihalogenides of element 116 shows how spin-orbit coupling strongly affects the chemistry of the superheavy p-block elements. While these molecules are bent at a scalar-relativistic level, spin-orbit coupling is so strong that only the 7p3/2 atomic orbitals of element 116 are involved in bonding, which favors linear molecular geometries for dihalogenides with heavy terminal halogen atoms.     

Molecular properties from perturbation theory: A Unified treatment of energy derivatives

The definition of a molecular property as a derivative of the electronic energy with respect to one or more applied perturbations is reviewed. The explicit enumeration of terms entering the derivative formulas is performed by considering in turn the various parameter spaces on which the energy and wave function depend. After deriving general expressions for first, second, and third derivatives for different types of perturbation, the parameter spaces involved in MCSCF and CI cases are identified and used to obtain expressions for the first and second derivatives. An example of an MCSCF third derivative is also given. In addition, the various equation systems defining the perturbed wave functions in each order are derived. Some attention is given to the efficient computer implementation of derivative calculations, and the present work is compared with that of other authors.