In the past, basis sets for use in correlated molecular calculations have largely been taken from single configuration calculations. Recently, Almlof, Taylor, and co‐workers have found that basis sets of natural orbitals derived from correlated atomic calculations (ANOs) provide an excellent description of molecular correlation effects. We report here a careful study of correlation effects in the oxygen atom, establishing that compact sets of primitive Gaussian functions effectively and efficiently describe correlation effects i f the exponents of the functions are optimized in atomic correlated calculations, although the primitive (s p) functions for describing correlation effects can be taken from atomic Hartree–Fock calculations i f the appropriate primitive set is used. Test calculations on oxygen‐containing molecules indicate that these primitive basis sets describe molecular correlation effects as well as the ANO sets of Almlof and Taylor. Guided by the calculations on oxygen, basis sets for use in correlated atomic and molecular calculations were developed for all of the first row atoms from boron through neon and for hydrogen. As in the oxygen atom calculations, it was found that the incremental energy lowerings due to the addition of correlating functions fall into distinct groups. This leads to the concept of c o r r e l a t i o n c o n s i s t e n t b a s i s s e t s, i.e., sets which include all functions in a given group as well as all functions in any higher groups. Correlation consistent sets are given for all of the atoms considered. The most accurate sets determined in this way, [5s4p3d2f1g], consistently yield 99% of the correlation energy obtained with the corresponding ANO sets, even though the latter contains 50% more primitive functions and twice as many primitive polarization functions. It is estimated that this set yields 94%–97% of the total (HF+1+2) correlation energy for the atoms neon through boron.

Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen

https://easyspin.org/pubs/easyspin_jmr06.pdf

EasySpin, a comprehensive software package for spectral simulation and analysis in EPR.

The basis-set convergence of the electronic correlation energy in the water molecule is investigated at the second-order Mo/ller–Plesset level and at the coupled-cluster singles-and-doubles level with and without perturbative triples corrections applied. The basis-set limits of the correlation energy are established to within 2 mEh by means of (1) extrapolations from sequences of calculations using correlation-consistent basis sets and (2) from explicitly correlated calculations employing terms linear in the interelectronic distances rij. For the extrapolations to the basis-set limit of the correlation energies, fits of the form a+bX−3 (where X is two for double-zeta sets, three for triple-zeta sets, etc.) are found to be useful. CCSD(T) calculations involving as many as 492 atomic orbitals are reported.

Basis-set convergence of correlated calculations on water

Basis-set convergence in correlated calculations on Ne, N2, and H2O

We introduce density functional theory and review recent progress in its application to transition metal chemistry. Topics covered include local, meta, hybrid, hybrid meta, and range-separated functionals, band theory, software, validation tests, and applications to spin states, magnetic exchange coupling, spectra, structure, reactivity, and catalysis, including molecules, clusters, nanoparticles, surfaces, and solids.

Density functional theory for transition metals and transition metal chemistry

We report here an analysis of extensive configuration interaction (CI) wave functions for the 1s2 ground state of the helium atom using piecewise polynomial basis functions. Large numbers of natural radial orbitals (NROs) with l ranging from 0 to 11 have been treated accurately and analyzed systematically. The contribution of each NRO to the total energy is found to follow the formula ΔE∼−0.42(l+1/2) × (n−1/2)−6a.u., where n is the principal quantum number, and the expansion coefficient of the NRO configuration is found to follow the formula, c∼−0.35(l+1/2)2/3 (n−1/2)−4. (The constants for l=0 are a little different). From an examination of the tails of the NROs, we are able to suggest an l‐dependent universal asymptotic formula, rβ−l−1+δe−ζr, where β is a constant, where δ=δl0, and where ζ2/2 is the ionization potential. The nodes of the NROs are also found to behave in a systematic way that yields valuable information on the choice of basis functions. So‐called ’’L’’‐limit energies EL, more accurate tha...

Piecewise polynomial configuration interaction natural orbital study of 1 s2 helium

Long-range behavior of hartree--fock orbitals.

A theory of electron correlation in nonclosed‐shell states, such as excited states of atoms and molecules, triplet states, free radicals, and transient species, is developed. A general (multiconfiguration) restricted Hartree—Fock (GRHF) wavefunction is taken as the starting point. When near degeneracies are not strong enough to cause large orbital and charge deformations, Roothaan‐type restricted Hartree—Fock (RHF) open‐shell orbital theories are used. The first‐order Schrodinger equation with RHF as the zeroth‐order wavefunction is solved. Various correlation processes (near‐degeneracy‐type internal correlations, semi‐internal pair correlations in which only one electron is ejected out of the Hartree—Fock sea, external pair correlations as in closed shells, and spin and symmetry orbital average polarization effects) are obtained in a form separated from one another. The second‐order energy also includes ``cross‐pair correlation effects,'' an open‐shell phenomenon that need be taken into account in the ev...

Many‐Electron Theory of Nonclosed‐Shell Atoms and Molecules. I. Orbital Wavefunction and Perturbation Theory

The theory of the field ionization of the hydrogen atom is developed analytically. The leading term of an asymptotic expansion for the ionization rate---the reciprocal lifetime---is derived. In the weak-field limit, the formula for the ionization rate reduces to $\frac{1}{\ensuremath{\tau}}={n}^{\ensuremath{-}3}{[{n}_{2}!({n}_{2}+|m|)!]}^{\ensuremath{-}1} {(n\frac{^{3}F}{4})}^{\ensuremath{-}2{n}_{2}\ensuremath{-}{|m|}_{\ensuremath{-}1}}\ifmmode\times\else\texttimes\fi{}\mathrm{exp}[3({n}_{1}\ensuremath{-}{n}_{2})\frac{\ensuremath{-}2}{(3n^{3}F)}]$, where $n$, $m$, and ${n}_{1}$ and ${n}_{2}$ are the usual principal, magnetic, and parabolic quantum numbers, respectively, and $F$ is the field strength in atomic units. For the ground state, this formula agrees with that of Landau and Lifshitz. For all ${m}^{2}=1$ states, the formula agrees asymptotically for large ${n}_{2}$ with that of Lanczos after correction of the latter for an error. It is in disagreement with the result of Oppenheimer and with the low-field result of Rice and Good. Significantly better agreement with numerical calculations of Alexander, of Hehenberger, McIntosh, and Br\"andas, of Damburg and Kolosov, and of Bailey, Hiskes, and Riviere is obtained with a formula for not quite such small $F$, $\frac{1}{\ensuremath{\tau}}={(\ensuremath{-}2\mathrm{Re}E)}^{\frac{3}{2}}{[{n}_{2}!({n}_{2}+|m|)!]}^{\ensuremath{-}1}{f}^{\ensuremath{-}B}\mathrm{exp}[\ensuremath{-}\frac{1}{(6f)}]$, where $E$ is the perturbed energy, where $f=\frac{[{(\ensuremath{-}2E)}^{\ensuremath{-}\frac{3}{2}}F]}{4}$, and where $\frac{B}{2}={\ensuremath{\beta}}_{2,{n}_{2}}$ is the usual perturbed separation constant [${\ensuremath{\beta}}_{2}\ensuremath{\rightarrow}\frac{{n}_{2}+|m|}{2}+\frac{1}{2}$, as $F\ensuremath{\rightarrow}0$].

Theory of the ionization of the hydrogen atom by an external electrostatic field

The improvement of Hartree–Fock wavefunctions by perturbation theory is hampered by slow convergence with the usual choice of H0 =  ∑ i = 1N F(i), where F denotes the one‐electron Hartree–Fock Hamiltonian. The slow convergence is usually attributed to the unphysical nature of the excited states of F, which do not describe electrons moving in the field of the nuclei, shielded by N‐1 electrons. It is shown how to redefine F so that the zeroth‐order wavefunction still remains the Hartree–Fock wavefunction but so that the excited states of F correspond to an electron moving in the field of the nuclei screened by N‐1 electrons. The redefined F thus results in a more appropriate H0. Calculations of some second‐order polarization and semi‐internal correlation energies in first‐row atoms are given to illustrate the use of the redefined F.

Harris J. Silverstone

Papers

Piecewise polynomial configuration interaction natural orbital study of 1 s2 helium

Long-range behavior of hartree--fock orbitals.

Many‐Electron Theory of Nonclosed‐Shell Atoms and Molecules. I. Orbital Wavefunction and Perturbation Theory

Theory of the ionization of the hydrogen atom by an external electrostatic field

Modified Perturbation Theory for Atoms and Molecules Based on a Hartree–Fock φ0