Basis set dependence of correlation corrections to protonation energies

The protonation energies of NH₃ and H₂O have been computed using a variety of basis sets. It is found that the effect of election correlation on these energies cannot reliably be determined without the use of large (triple-split and polarized) basis sets.     

Accurate energies calculated by empirical corrections to the local spin density approximation

We tested the efficacy of three empirical correction schemes on atomization energies calculated by the following theories: Kohn–Sham density functional theory (KS-DFT) with local spin density approximation (LSDA), two KS-DFT gradient-corrected methods, one hybrid Hartree–Fock/KS-DFT method similar to B3LYP, and the ab initio extrapolation procedures G1 and G2. Empirical corrections improved the LSDA results greatly, while the other theories were improved slightly or not at all. The best procedure for correcting LSDA atomization energies brings the mean absolute deviation from experiment from 38.3 to 4.0 kcal/mol on a subset of 44 molecules in the G1 dataset that were not used in deriving the empirical parameters. This corrected LSDA is interesting for three reasons: it could be a useful computational tool in some cases, it implies that the LSDA itself gives accurate energies for reactions where atomic coordinations stay unchanged, and it gives insight into the search of better functionals.     

Resolution of identity approach for the Kohn-Sham correlation energy within the exact-exchange random-phase approximation

Two related methods to calculate the Kohn-Sham correlation energy within the framework of the adiabatic-connection fluctuation-dissipation theorem are presented. The required coupling-strength-dependent density-density response functions are calculated within exact-exchange time-dependent density-functional theory, i.e., within time-dependent density-functional response theory using the full frequency-dependent exchange kernel in addition to the Coulomb kernel. The resulting resolution-of-identity exact-exchange random-phase approximation (RI-EXXRPA) methods in contrast to previous EXXRPA methods employ an auxiliary basis set (RI basis set) to improve the computational efficiency, in particular, to reduce the formal scaling of the computational effort with respect to the system size N from N(6) to N(5). Moreover, the presented RI-EXXRPA methods, in contrast to previous ones, do not treat products of occupied times unoccupied orbitals as if they were linearly independent. Finally, terms neglected in previous EXXRPA methods can be included, which leads to a method designated RI-EXXRPA+, while the method without these extra terms is simply referred to as RI-EXXRPA. Both EXXRPA methods are shown to yield total energies, reaction energies of small molecules, and binding energies of noncovalently bonded dimers of a quality that is similar and in some cases even better than that obtained with quantum chemistry methods such as Mo?ller-Plesset perturbation theory of second order (MP2) or with the coupled cluster singles doubles method. In contrast to MP2 and to conventional density-functional methods, the presented RI-EXXRPA methods are able to treat static correlation.     

Systematic corrections to the Born-Oppenheimer approximation

A method for providing systematic diabatic corrections to the Born-Oppenheimer approximation is presented. We begin with an adiabatic expansion of the exact vibronic wavefunctions and, via the molecular Hamiltonian, develop expressions for the diabatic terms in the Schrödinger equation. We then derive recursion relations which allow one to introduce the diabatic interactions to any desired degree of approximation. As an example, the first approximation (beyond the Born-Oppenheimer approximation) is discussed explicitly. In passing, we assess some of the common misconceptions associated with the Born-Oppenheimer approximation.     

Third-order energy derivative corrections to the Kohn-Sham orbital hardness tensor

The third term in the Taylor expansion of the total energy functional around the number of electronsN is evaluated as the second-order derivative of orbital Kohn-Sham energies with respect to orbital occupancy. Present approach is an extension of an efficient algorithm to compute densityfunctional based orbital reactivity indices. Various energy derivatives used to approximate orbital reactivity indices are defined within the space spanned by the orbital occupation numbers and the Kohn-Sham one-electron energies. The third-order energy functional derivative has to be considered for singular hardness tensor ([η]). On the contrary, this term has negligible influence on the reactivity index values for atomic or molecular systems with positively defined hardness tensors. In this context, stability of a system in equilibrium state estimated through the eigenvalues of [η] is discussed. Numerical illustration of the Kohn-Sham energy functional derivatives in orbital resolution up to the third order is shown for benchmark molecules such as H₂O, H₂S, and OH⁻.     

Third-order interaction approximation for linear polymer chains

Third-order interactions imposed by a pair of atoms separated by five bonds are taken into account in computations of the mean-square end-to-end distance and the mean-square radius of gyration for linear polymer chains. The statistical weight matrices are established on the basis of the rotational isomeric state model. The conformational energy of n-hexane is calculated as a function of the C? C bond rotation angles. The third-order interaction energy is obtained by comparison with that of n-pentane. The characteristic ratio of polymethylene is 6.6 in the third-order interaction approximation, which is in agreement with experimental data.     

Adiabatic corrections to density functional theory energies and wave functions

The adiabatic finite-nuclear-mass-correction (FNMC) to the electronic energies and wave functions of atoms and molecules is formulated for density-functional theory and implemented in the deMon code. The approach is tested for a series of local and gradient corrected density functionals, using MP2 results and diagonal-Born-Oppenheimer corrections from the literature for comparison. In the evaluation of absolute energy corrections of nonorganic molecules the LDA PZ81 functional works surprisingly better than the others. For organic molecules the GGA BLYP functional has the best performance. FNMC with GGA functionals, mainly BLYP, show a good performance in the evaluation of relative corrections, except for nonorganic molecules containing H atoms. The PW86 functional stands out with the best evaluation of the barrier of linearity of H2O and the isotopic dipole moment of HDO. In general, DFT functionals display an accuracy superior than the common belief and because the corrections are based on a change of the electronic kinetic energy they are here ranked in a new appropriate way. The approach is applied to obtain the adiabatic correction for full atomization of alcanes C(n)H(2n+2), n = 4-10. The barrier of 1 mHartree is approached for adiabatic corrections, justifying its insertion into DFT.     

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.     

Additive density functional correlation corrections to single particle theories

The correlation energy functional E_c of the Hartree-Fock density is investigated. It was previously established that E_c produces the exact ground-state energy when added to the Hartree-Fock energy. Except when certain degeneracies occur, it is here shown that E_c is bounded from below when the coordinates of the Hartree-Fock density are scaled uniformly by λ, as λ → ∞. Consequently, approximations to E_c should display this bounded property, which is a second-order perturbation energy. It is also shown that a corresponding result applies to that correlation energy functional, Ë_c, which is to be added to a completed exact exchange-only (in the OPM sense) density functional calculation. Scaling requirements are presented for each order in the perturbation expansion for Ë_c. For instance, the second-order term is dimensionless. © 1997 John Wiley & Sons, Inc.     

Incomplete basis set problem. IV. Perturbative corrections to pair energies

A partitioning of the molecular Hamiltonian into the occupied and virtual orbital spaces and their orthogonal complement is introduced and used to develop a perturbation expansion of the exact ground-state energy relative to the Hartree–Fock energy computed using an incomplete basis set. The leading perturbation corrections to pair energies due to using the incomplete basis set are considered in detail. Summations of certain classes of pair contributions are discussed and a resummed correction is obtained.     

Third- and fourth-order perturbation corrections to excitation energies from configuration interaction singles

Complete third-order and partial fourth-order Rayleigh-Schrodinger perturbation corrections to excitation energies from configuration interaction singles (CIS) have been derived and termed CIS(3) and CIS(4)(P). They have been implemented by the automated system TENSOR CONTRACTION ENGINE into parallel-execution programs taking advantage of spin, spatial, and index permutation symmetries and applicable to closed- and open-shell molecules. The consistent use of factorization, first introduced by Head-Gordon et al. in the second-order correction to CIS denoted CIS(D), has reduced the computational cost of CIS(3) and CIS(4)(P) from O(n(8)) and O(n(6)) to O(n(6)) and O(n(5)), respectively, with n being the number of orbitals. It has also guaranteed the size extensivity of excited-state energies of these methods, which are in turn the sum of size-intensive excitation energies and the ground-state energies from the standard M?ller-Plesset perturbation theory at the respective orders. The series CIS(D), CIS(3), and CIS(4)(P) are usually monotonically convergent at values close to the accurate results predicted by coupled-cluster singles and doubles (CCSD) with a small fraction of computational costs of CCSD for predominantly singly excited states characterized by a 90%-100% overlap between the CIS and CCSD wave functions. When the overlap is smaller, the perturbation theory is incapable of adequately accounting for the mixing of the CIS states through higher-than-singles sectors of the Hamiltonian matrix, resulting in wildly oscillating series with often very large errors in CIS(4)(P). Hence, CIS(3) and CIS(4)(P) have a rather small radius of convergence and a limited range of applicability, but within that range they can be an inexpensive alternative to CCSD.     

Comments on the geometric approximation to the second-order perturbed energies

It is shown that the ideas similar to those behind the ordinary geometric approximation can be efficiently used to speed up the convergence of the coupled Hartree-Fock iterative procedure in the density matrix formalism. The formulation of the geometric approximation in the case of multiple perturbation is also considered and an alternative second-order energy formula is proposed. The validity of the upper bound formula for the error of the geometric approximation denied by Burrows, is discussed.     

Effect of semi-empirical parameters on the simple random-phase approximation calculations of conjugated hydrocarbons

Singlet-singlet transition energies, oscillator strengths, triplet energy levels, and the ground state correlation energy of a number of conjugated hydrocarbons have been calculated by the simple random-phase approximation (RPA ) within the framework of the Pariser-Parr-Pople (PPP ) model. The effect of semi-empirical parameters in such calculations has been examined in detail. A set of parameters has been deduced from these parametric studies which is found to yield results for the singlet spectra of the molecules in excellent agreement with experiment. It is, however, not possible to treat the triplet states using these same parameters, since they produce triplet instabilities in all the molecules. The triplet instability problem associated with semi-empirical RPA calculations has been discussed in detail.     

A block variational procedure for the iterative diagonalization of non-Hermitian random-phase approximation matrices

We present a technique for the iterative diagonalization of random-phase approximation (RPA) matrices, which are encountered in the framework of time-dependent density-functional theory (TDDFT) and the Bethe-Salpeter equation. The non-Hermitian character of these matrices does not permit a straightforward application of standard iterative techniques used, i.e., for the diagonalization of ground state Hamiltonians. We first introduce a new block variational principle for RPA matrices. We then develop an algorithm for the simultaneous calculation of multiple eigenvalues and eigenvectors, with convergence and stability properties similar to techniques used to iteratively diagonalize Hermitian matrices. The algorithm is validated for simple systems (Na(2) and Na(4)) and then used to compute multiple low-lying TDDFT excitation energies of the benzene molecule.     

Simple molecular wavefunctions with correlation corrections

It is shown that the main deficiencies of wavefunctions of Hartree-Fock type (wrong dissociation behavior and absence of correlation between electrons of unlike spin) may be corrected by a simple method. Just sufficient CI is admitted to ensure qualitatively correct dissociation, while the short-range correlation energy is estimated with the Colle-Salvetti functional. Potential energy curves for H₂ and LiH are computed at various levels of approximation and the main features of the method are discussed.     

Accurate all-electron correlation energies for the closed-shell atoms from Ar to Rn and their relationship to the corresponding MP2 correlation energies

All-electron correlation energies E(c) are not very well-known for atoms with more than 18 electrons. Hence, coupled-cluster calculations in carefully designed basis sets are combined with fully converged second-order M?ller-Plesset perturbation theory (MP2) computations to obtain fairly accurate, nonrelativistic E(c) values for the 12 closed-shell atoms from Ar to Rn. These energies will be useful for the evaluation and parameterization of density functionals. The results show that MP2 overestimates ∣E(c)∣ for heavy atoms. Spin-component scaling of the MP2 correlation energy is used to provide a simple explanation for this overestimation.     

Orbit-orbit relativistic corrections to the pure vibrational non-Born-Oppenheimer energies of H(2)

We report the derivation of the orbit-orbit relativistic correction for calculating pure vibrational states of diatomic molecular systems with sigma electrons within the framework that does not assume the Born-Oppenheimer (BO) approximation. The correction is calculated as the expectation value of the orbit-orbit interaction operator with the non-BO wave function expressed in terms of explicitly correlated Gaussian functions multiplied by even powers of the internuclear distance. With that we can now calculate the complete relativistic correction of the order of alpha(2) (where alpha=1/c). The new algorithm is applied to determine the full set of the rotationless vibrational levels and the corresponding transition frequencies of the H(2) molecule. The results are compared with the previous calculations, as well as with the frequencies obtained from the experimental spectra. The comparison shows the need to include corrections higher than second order in alpha to further improve the agreement between the theory and the experiment.     

Delocalization corrections to the strictly localized molecular orbitals: A linearized SCF approximation

An explicit formula is derived for calculating the delocalization corrections (tails) to be added to the strictly localized bond orbitals. It was obtained by solving analytically the SCF problem for the interbond interactions in a linearized approximation. The model calculations at the CNDO/2 level show that this simple approach is sufficient to account for the molecular conformations.     

Bremsstrahlung cross-section with screening and Coulomb corrections at high energies

Using analytical formulae which are exact in Born approximation, the doubly differential bremsstrahlung cross-section with form-factor screening is calculated. For the atomic form factor parameters are applied which approximate self-consistent Dirac–Hartree–Fock–Slater calculations. The evaluation of the bremsstrahlung spectrum requires a single numerical integration. The results are superior to the customary Bethe approximation, in particular at the high-energy part of the spectrum. At high energies the screening correction can be added to any Coulomb-corrected cross-section without screening. In the present work, we are using a cross-section calculated by means of Sommerfeld–Maue functions with additional higher-order terms.