Nicholas C. Handy

Reaction path Hamiltonian for polyatomic molecules

The reaction path on the potential energy surface of a polyatomic molecule is the steepest descent path (if mass‐weighted Cartesian coordinates are used) connecting saddle points and minima. For an N‐atom system in 3d space it is shown how the 3N‐6 internal coordinates can be chosen to be the reaction coordinate s, the arc length along the reaction path, plus (3N‐7) normal coordinates that describe vibrations orthogonal to the reaction path. The classical (and quantum) Hamiltonian is derived in terms of these coordinates and their conjugate momenta for the general case of an N atom system with a given nonzero value of the total angular momentum. One of the important facts that makes this analysis feasible (and therefore interesting) is that all the quantities necessary to construct this Hamiltonian, and thus permit dynamical studies, are obtainable from a relatively modest number of ab initio quantum chemistry calculations of the potential energy surface. As a simple example, it is shown how the effects o...

DEVELOPMENT AND ASSESSMENT OF NEW EXCHANGE-CORRELATION FUNCTIONALS

We recently presented a new method for developing generalized gradient approximation (GGA) exchange-correlation energy functionals, using a least-squares procedure involving numerical exchange-correlation potentials and experimental energetics and nuclear gradients. In this paper we use the same method to develop a new GGA functional, denoted HCTH, based on an expansion recently suggested by Becke [J. Chem. Phys. 107, 8554 (1997)]. For our extensive training set, the new functional yields improved energetics compared to both the BLYP and B3LYP functionals [Phys. Rev. A 38, 3098 (1988); Phys. Rev. B 37, 785 (1988); J. Chem. Phys. 98, 5648 (1993); J. Phys. Chem. 98, 11623 (1994)]. The geometries of these systems, together with those of a set of transition metal compounds, are shown to be an improvement over the BLYP functional, while the reaction barriers for six hydrogen abstraction reactions are comparable to those of B3LYP. These improvements are achieved without introducing any fraction of exact orbital...

Left-right correlation energy

We first attempt to determine a local exchange functional Ex[p] which accurately reproduces the Hartree-Fock (HF) energies of the 18 first and second row atoms. Ex[p is determined from p and |δp|, and we find that we can improve significantly upon Beckes original generalized gradient approximation functional (commonly called B88X) by allowing the coefficient of the Dirac exchange term to be optimized (it is argued that molecules do not behave like the uniform electron gas). We call this new two parameter exchange functional OPTX. We find that neither δ p or t = Σ δ i |2 improve the fit to these atomic energies. These exchange functionals include not only exchange, but also left-right correlation. It is therefore proposed that this functional provides a definition for exchange energy plus left-right correlation energy when used in Kohn-Sham (KS) calculations. We call this energy the Kohn-Sham exchange (or KSX) energy. It is shown that for nearly all molecules studied these KSX energies are lower than the corresponding HF energies, thus giving values for the non-dynamic correlation energy. At stretched geometries, the KSX energies are always lower than the HF energies, and often substantially so. Furthermore all bond lengths from the KSX calculations are longer than HF bond lengths and experimental bond lengths, which again demonstrates the inclusion of left-right correlation effects in the functional. For these reasons we prefer to split the correlation energy into two parts: left-right correlation energy and dynamic correlation energy, arguing that the usage of the words ‘non-dynamic’ or ‘static’ or ‘near-degeneracy’ is less meaningful. We recognize that this definition of KSX is not precise, because the definition of a local Ex[p] can never be precise. We also recognize that these ideas are not new, but we think that their importance has been insufficiently recognized in functional determination. When we include third row atoms in our analysis, we are unable to find a local exchange functional which is a substantial improvement over B88X for the reproduction of HF energies. This must arise from the effects of the core orbitals, and therefore we do not consider that this detracts from the improved accuracy of OPTX. We report some MCSCF calculations constructed from bonding-antibonding configurations, from which we attempt to calculate ab initio left-right correlation. There is only moderate agreement between the two approaches. Finally we combine the OPTX functional with established correlation functionals (LYP, P86, P91) to form OLYP, OP86 and OP91; OLYP is a great improvement on BLYP for both energy and structure, and OP86, OP91 are an improvement over BP86, BP91 for structure. The importance of the exchange functional for molecular structure is therefore underlined.

On the evaluation of analytic energy derivatives for correlated wave functions

It is shown that to obtain the (2n)th and (2n+1)th energy gradients, it is only necessary to solve equations of the difficulty of the nth order coupled perturbed equations for the orbital and configuration interaction (CI) parameters. For example, to find analytic second and third energy derivatives for CI wave functions, it is only necessary to solve the first order coupled perturbed equations and some related equations, for the effects of orbital rotations. Similar results apply for gradients of energies derived using perturbation theory.

Improving virtual Kohn–Sham orbitals and eigenvalues: Application to excitation energies and static polarizabilities

Conventional continuum exchange-correlation functionals (e.g., local density approximation, generalized gradient approximation) offer a poor description of many response properties, such as static polarizabilities and single photon vertical excitation energies to Rydberg states. These deficiencies are related to errors in the virtual Kohn–Sham orbitals and eigenvalues, which arise due to a fundamental deficiency in the potentials of conventional continuum functionals. Namely, although these potentials approximately average over the exact integer discontinuity in energetically important regions, they fail to do so asymptotically, because they vanish. Our recent functional HCTH [J. Chem. Phys. 109, 6264 (1998)] was designed with this deficiency in mind, although its potential still fails to exhibit the appropriate asymptotic form. In this paper, we present a new procedure that explicitly corrects this asymptotic deficiency for any continuum functional. Self-consistent Kohn–Sham calculations are performed us...

A new parametrization of exchange–correlation generalized gradient approximation functionals

A new “HCTH” generalized gradient approximation (GGA) functional is presented. Its 15 parameters have been refined against data from a training set containing 407 atomic and molecular systems. We believe that the much enhanced training set means that the new functional HCTH/407 has a much greater universality than previous GGA functionals. Statistical data is presented for the 407 set for the new functional, as well as other functionals.

A new determinant-based full configuration interaction method

Abstract Siegbahn recently presented a new direct Cl method which is applicable to the CAS SCF method. Instead of using configuration state functions as adopted by Siegbahn, we propose that Slater determinants should be used. The approach then needs no formula tape, the one-electron coupling coefficients being easily generated as required. The method is totally vectorised, and applies as programmed to the lowest eigenvalue of any spin and space symmetry. The timings for Siegbahns examples are appropoximately halved using this procedure, and in a full CI calculation involving 107942 configurations, 18 CRAY-1S CPU seconds were required per iteration.

Size-consistent Brueckner theory limited to double substitutions

Abstract A size-consistent set of equations for electron correlation which are limited to double substitutions, based on Brueckner orbitals, is discussed. Called BD theory, it is shown that at fifth order of perturbation theory, BD incorporates more terms than CCSD and QCISD. The simplicity of the equations leads to an elegant gradient theory. Preliminary applications are reported.

Does density functional theory contribute to the understanding of excited states of unsaturated organic compounds

A comparative study has been performed on the electronic spectra of a number of unsaturated organic molecules, using on the one hand density functional linear response theory and on the other multiconfigurational second-order perturbation theory, in order to establish the accuracy that the density functional based methods can give for excitation energies and energy surfaces for excited states. The following molecules are included in the study: tetrazine; the five-membered ring systems cyclopentadiene, furan, pyrrole, and thiophene; acetone; and a dipeptide. The results show that DFT valence excited states have errors that vary between 0 and 1 eV, while Rydberg states are accurate to about 0.2eV in most cases. The use of an asymptotically corrected exchange-correlation potential was essential for the latter result. However, transitions which involve a considerable charge transfer have much larger errors. The results show in some cases a surprisingly strong interaction between valence and Rydberg excited st...

New exchange-correlation density functionals: The role of the kinetic-energy density

New density functionals, using the kinetic-energy density Tau are reported. The newly introduced variable enhances the performance of previous functionals, leading to highly accurate functionals with and without the use of exact exchange. All these functionals are compared to commonly used functionals for a large test set, looking also at reactions and hydrogen bonded systems. Furthermore, their physical plausibility is discussed.