The periodic table above gives Hartree-Fock and Dirac-Fock Average Relativistic Effective (AREP) pseudopotentials (effective core potentials) for the atoms H to Ba and Lu to Hg. These pseudopotentials are finite at the origin, which makes them particularly suitable for use in QMC calculations, although they are also suitable for many other methods, including quantum chemistry calculations. These pseudopotentials were generated by John Trail and, if you use them, please cite this website and both of the following papers:
1. “Smooth relativistic Hartree-Fock pseudopotentials for H to Ba and Lu to Hg”
J.R. Trail and R.J. Needs, J. Chem. Phys. 122, 174109 (2005)[link]
2. “Norm-conserving Hartree-Fock pseudopotentials and their asymptotic behaviour”
J.R. Trail and R.J. Needs, J. Chem. Phys. 122, 014112 (2005)[link]
A reasonable compact reference would be:
J.R. Trail and R.J. Needs, J. Chem. Phys. 122, 174109 (2005); J.R. Trail and R.J. Needs,
J. Chem. Phys. 122, 014112 (2005), see also www.vallico.net/casinoqmc/pplib/.
There are currently three types of pseudopotential available for each of the atoms H to Ba and Lu to Hg: a Dirac-Fock Average Relativistic Effective Potential (AREP) with small core radii*, a Hartree-Fock pseudopotential generated with small core radii, and an AREP generated with larger core radii. The pseudopotentials are designed for use with methods in which the non-relativistic Schrödinger equation is solved. Solving the Schrödinger equation with Dirac-Fock AREP pseudopotentials will result in the inclusion of scalar relativistic effects (spin-orbit potentials are also included in the table, and when these are included the calculations will also include spin-orbit effects). For most purposes Dirac-Fock AREPs are to be preferred because they contain important relativistic effects, but we also provide Hartree-Fock pseudopotentials, which may be useful in some circumstances.
The pseudopotentials have s, p, and d angular momentum channels. The pseudopotentials are finite at the origin, which is very important for QMC applications and may also be advantageous in other methods. We envisage the small core Dirac-Fock and Hartree-Fock pseudopotentials being used with localized basis sets such as Gaussian functions. These pseudopotentials are given tabulated on a grid, and as fits to Gaussian basis sets for use with various quantum chemistry packages. The large core (or “softer”) Dirac-Fock pseudopotentials are designed for use with plane-wave basis sets. The larger core radii improve convergence with the size of the plane-wave basis, but the region over which the pseudopotential is non-local is then slightly larger, so they are more costly to use within QMC calculations.
The table also gives plots of each pseudopotential (for Gaussian fits the original tabulated representation is also plotted in the same figure for comparison), a summary of the properties of each pseudopotential, and atomic orbitals for some typical configurations. Each of the pseudopotentials has been tested in atomic calculations.
* The “core radii” are the radii outside of which we demand that the atomic ground state valence orbitals resulting from all-electron and pseudopotential calculations agree.
Description of Gaussian fits
The method of fitting the pseudopotentials to a Gaussian basis set is described in Ref. 1 above [link]. Fits are reported in formats suitable for the CRYSTAL and GAUSSIAN codes, and for the GAMESS code. It is important to realise that the CRYSTAL and GAUSSIAN fits are identical, and only the format of the files differs. Unfortunately not all quantum chemistry codes can deal with the large powers of r used to multiply the Gaussian functions. GAMESS is one of the codes which does not currently support all of the powers we would like to use and we have had to fit the pseudopotentials with a more restricted basis set for use in GAMESS. This leads to some loss of quality and the CRYSTAL/GAUSSIAN fits should be used whenever possible.
Spin orbit potentials are also provided, in the standard Gaussian parameterization.
For some of the atoms parameters for core polarization potentials are also listed, from
“Many-body core-valence partitioning” E.L. Shirley and R.M. Martin, Phys. Rev. B 47, 15413 (1993).
Description of the summary file
The core radii are the radii outside of which we demand that the pseudo-orbitals and all-electron orbitals agree for the initial pseudopotential. Pseudopotentials generated by inversion of the Schrödinger or Dirac-Fock equations have long ranged non-local tails which must be cutoff (as described in Ref. 1 [link] and Ref. 2[link] above). The cutoff procedure forces the pseudopotentials to be local outside of the localisation radius rloc whilst correcting the pseudopotentials inside of rloc to conserve ground state properties*. This pseudopotential is the “tabulated” pseudopotential given in the periodic table. Parameterising such a pseudopotential in terms of a Gaussian basis set normally increases rloc.
The all-electron and pseudo-HF LS-coupling eigenvalues should agree to high precision for a Hartree-Fock pseudopotential, but not necessarily for an AREP pseudopotential that is generated within Dirac-Fock theory.
The terms in the energy for a pseudo-HF calculation of the ground state of the atom are given. These are useful for checking purposes, for example these energies should be reproduced by a variational Monte Carlo calculation using the pseudopotential and the atomic orbitals resulting from a Hartree-Fock calculation using the same
Data for a few excitation energies are also included in the summary. It should be noted that a pseudo-HF calculation with an AREP pseudopotential will not reproduce the same relativistic effects as those present in the original all-electron DF atom, even if scalar relativistic corrections and spin-orbit coupling are included. This implies that excitation energies for the pseudo-HF atom would differ from those of the all-electron DF atom even if the pseudopotential were able to reproduce the influence of the core electrons exactly.
Please note that there is a seperate summary file for each representation of each type of pseudopotential. These give the energies resulting from pseudo-HF calculations carried out using the tabulated representation, the CRYSTAL/GAUSSIAN Gaussian expansion, and the GAMESS Gaussian expansion of the pseudopotential. Small differences between the energies resulting from the tabulated and parameterised representations reflect the accuracy of the Gaussian fits.
In addition please note that the atomic orbitals provided in the periodic table have been calculated from the tabulated pseudopotentials, not the parameterised pseudopotentials.
* Without this procedure |Vl(r) – Vl’(r)| remains finite as r→∞ for some l and l’, and for all atoms except a few special cases.
Results for atomic ground and excited states obtained using these pseudopotentials are also available, together with an error analysis:
¤ Ground state eigenvalues resulting from Dirac-Fock AE calculations (spin averaged for each l) compared with Hartree-Fock calculations using the AREP pseudopotentials. These calculations are not strictly speaking Hartree-Fock, but are the result of taking the c→∞ limit of Dirac-Fock theory (see Ref. 1 above [link]).
¤ Excitation energies resulting from AE Hartree-Fock calculations (with scalar relativistic corrections), compared with those resulting from Hartree-Fock calculations with AREP pseudopotentials.
¤ Ground state eigenvalues resulting from AE Hartree-Fock calculations, compared with those resulting from Hartree-Fock calculations with HF pseudopotentials.
¤ Excitation energies resulting from AE Hartree-Fock calculations, compared with those resulting from Hartree-Fock calculations with HF pseudopotentials.
In the data given above the “parameterised pseudopotential” refers to the GAUSSIAN/CRYSTAL Gaussian fit for each pseudopotential. Equivalent analyses of the data resulting from the GAMESS parameterisations are given in:
¤ Ground state eigenvalues from AE Dirac-Fock calculations, compared with those resulting from Hartree-Fock calculations using the AREP pseudopotentials.
¤ Excitation energies from AE Hartree-Fock calculations, compared with those resulting from Hartree-Fock calculations using the AREP pseudopotentials.
¤ Ground state eigenvalues from AE Hartree-Fock calculations, compared with those resulting from Hartree-Fock calculations using the HF pseudopotentials.
¤ Excitation energies from AE Hartree-Fock calculations, compared with those resulting from Hartree-Fock calculations using the HF pseudopotential.
The pseudopotentials fitted to Gaussians are meant for use in quantum chemistry codes, which are often used to generate orbitals for QMC trial wave functions. The tabulated pseudopotentials can be used in other codes, such as plane-wave DFT codes. CASINO uses tabulated pseudopotentials, and the periodic table provides input files in the correct format. Note that the tabulated pseudopotentials are those which were generated on radial grids which have not been fitted to Gaussians. We recommend that these tabulated pseudopotentials be used in QMC calculations even when the orbitals were generated using the (very similar, but not exactly the same) pseudopotentials fitted to Gaussians. The reasons for this are:
(1) The tabulated pseudopotentials reproduce the orbitals and eigenvalues they were calculated from slightly more accurately than the Gaussian fits.
(2) The tabulated pseudopotentials have smaller localisation radii (the radius outside of which the components of a pseudopotential differ by less than some tolerance). The cost of evaluating the non-local energy in QMC calculations is lower if the localisation
radius is smaller.
The orbitals from Hartree-Fock calculations using the tabulated pseudopotentials are also given. These are in the correct format for CASINO, and are given for a number of configurations, including the ground state.
Look in the utils/pseudo_converters directory in the CASINO distribution for utilities that convert pseudopotentials formatted for other codes into the correct format for CASINO. Utilities are currently available for ABINIT, CASTEP, CHAMP, PWSCF, and GP.