psp8
Information on the format 8 for pseudopotentials.¶
(Note: the implementation of format 8 was done by D. R. Hamann).
The format 8 for ABINIT pseudopotentials is designed to allow users who wish to experiment with pseudopotentials, possibly with nonstandard features, to have great flexibility in doing so. It does not correspond to any publicly available tabulation. The opensource ONCVPSP package available at www.matsimresearch.com produces pseudopotentials in this format.
An annotated example is presented in detail below, followed by a second example incorporating spinorbit coupling. An extended discussion follows below the examples, with a separate section discussing spinorbit as produced by ONCVPSP.
When first producing a pseudopotential in this format, it would be a very good idea to do a wellconverged calculation of an isolated atom in a big box to confirm that you are correctly recreating the atomic valence levels you intend.
The format is best explained by an example, which is presented in
detail below. (The example is excerpted from Psps_for_tests/20ca_sic.drh
.
Another example of pspcod=8 is Psps_for_tests/8o_sic.drh
.)
All but the last data block of this file are read in subroutine psp8in.f.
The “comment” (#) lines below are not permitted in the actual file. Nor are blank lines.
Header line 1: Title line 2: atomic number, pseudoion charge, date line 3: pspcod=8 identifies this format pspxc=2 identifies the exchangecorrelation functional (see ixc list in the input variables documentation) lmax=2 gives the largest angular momentum potential present lloc=4 (which is >lmax) indicates that the local potential is independent of that for any angular momentum. In other cases, it can be <=lmax, and the local potential can be that of a particular angular momentum channel. (The choice of 4 is arbitrary, but the index of the corresponding data block below must be consistent with the value assigned in the header.) mmax=313, giving the number of radial grid points (arbitrary). r2well=0 is an historical header holdover, and is not used line 4: rchrg=6.18.. is the radius beyond which the model core charge (if used) is zero or negligible. The maximum radial mesh point must equal or exceed this value. fchrg=1.0 signals that a model core charge is present. Any value >0.0 will do, and the value is not used otherwise. qchrg=0.0 is a historical holdover, not used. line 5: number of BloechlKleinmanBylander projectors nproj for each angular momentum l=0 to l=lmax. The value of nproj for lloc must be 0 (if lloc <= lmax). line 6: extension_switch=0, not initially used, but values >0 but may be utilized to signal the presence of additonal lines of header information for specialpurpose use in the future. The value 2 is now used to indicate the presence os spinorbit projectors (see SPINORBIT section below).
EXPERIMENTAL selfintractioncorrected psp for calcium (D. R. Hamann):
20.0000 2.0000 040701 zatom,zion,pspd 8 2 2 4 313 0 pspcod,pspxc,lmax,lloc,mmax,r2well 6.18295050 1.00000000 0.00000000 rchrg fchrg qchrg 2 1 1 0 0 nproj 0 extension_switch First data block first line: angular momentum l=0, ekb(ii) for ii=1,2(=nproj(1)) following lines labeled 1,2,...,mmax  1st column: radial grid index 2nd column: radial grid mesh point (LINEAR mesh starting at 0.0) 3rd column: first BKB projector for l=0 4th column: second BKB projector for l=0 The projectors should go smoothly to zero, or decay to negligible values within the range of the radial grid. In general there are nproj ekb and projector columns for each l, as many as you want. If nproj is zero, AND lloc/= the current l, this block is omitted. While some may be absent, l blocks must be in ascending order. 0 1.8442549429363D+01 1.5467656752626D+01 1 0.0000000000000D+00 0.0000000000000D+00 0.0000000000000D+00 2 2.0000000000000D02 3.2695476840515D03 7.7803072439537D05 3 4.0000000000000D02 6.5413292698932D03 1.5573364986856D04 ..... ..... Second data block first line  angular momentum l=1, ekb(ii) for ii=1,1(=nproj(2)) following lines labeled 1,2,...,mmax  1st column: radial grid index 2nd column: radial grid mesh point 3rd column: first (only) BKB projector for l=1 1 1.0250215833500D+01 1 0.0000000000000D+00 0.0000000000000D+00 2 2.0000000000000D02 4.1259561971920D05 3 4.0000000000000D02 1.6501618805883D04 ..... .....
Third data block as above, for l=2
2 4.9318560092991D01 1 0.0000000000000D+00 0.0000000000000D+00 2 2.0000000000000D02 2.8426374194219D04 3 4.0000000000000D02 2.2672423014226D03 ..... ..... Fourth data block first line  index corresponding to lloc in header. In this case, the local potential does not correspond to that for any angular momentum. lloc can be arbitrary in this case, but MUST be >lmax, and this block must follow the blocks for projectors with l <=lmax. following lines labeled 1,2,...,mmax  1st column: radial grid index 2nd column: radial grid mesh point 3rd column: local potential If lloc<lmax, the corresponding block must occur in its proper l order among the blocks of nonlocal projectors. The last line should, ideally, be equal to zion/rad(mmax), and the numerical data should approach zion/r as continuously as possible, since the Fourier transform is extended to infinity analytically assumming this functional form. This isn't precisely satisfied for the expeimental potential in the example, and the result will be a smallamplitude Gibbs oscillation as a function of the Q of the Fouriertransformed local potenial with a period of 2*Pi/rad(mmax). 4 1 0.0000000000000D+00 9.3285752771727D01 2 2.0000000000000D02 9.3284556214441D01 3 4.0000000000000D02 9.3280966542583D01 ..... ..... 313 6.2400000000000D+00 3.3275722650983D01 Fifth (last) data block (read in psp8cc.f) lines labeled 1,2,...,mmax  1st column: radial grid index 2nd column: radial grid mesh point (LINEAR mesh starting at 0.0) 3rd column: rhoc = 4*Pi*model core charge density 4th column: d rhoc / dr 5th column: d^2 rhoc / dr^2 6th column: d^3 rhoc / dr^3 7th column: d^4 rhoc / dr^4 This block is only read if the header variable fchrg >0.0. 1 0.0000000000000D+00 1.8263604328455D+00 1.3822038662342D06 3.4248477651502D05 2.2949926081983D05 1.3721984594931D+01 2 2.0000000000000D02 1.8263581757935D+00 1.8290946429655D05 2.7430630291457D03 2.7416158741933D01 1.3679142685104D+01 3 4.0000000000000D02 1.8263568047539D+00 1.4618869292620D04 1.0954897796014D02 5.4658870398836D01 1.3549201500559D+01 ..... .....
SPINORBIT¶
Spinorbit coupling is present in the file Psps_for_tests/78_Pt_r.oncvpsp.psp8
.
This pseudopotential treats the 5s and 5p core states as valence.
More details are given in the spinorbit portion of the discussion section
Header: lines 14 as above line 5: number of VanderbiltKleinmanBylander projectors nproj for the scalarrelativistic nonlocal potential for each angular momentum l=0 to l=lmax. The value of nproj for lloc must be 0 (if lloc <= lmax). line 6: extension_switch=2 indicating spinorbit projectors are present line 7: number of VanderbiltKleinmanBylander projectors nprojso for the spinorbit nonlocal potential for each angular momentum l=1 to l=lmax. Pt NOPTPSP r_core= 2.21 2.52 2.40 3.02 78.0000 18.0000 140202 zatom,zion,pspd 8 2 3 4 500 0 pspcod,pspxc,lmax,lloc,mmax,r2well 4.99000000 0.00000000 0.00000000 rchrg fchrg qchrg 2 4 3 3 nproj 2 extension_switch 4 4 3 nprojso Data blocks 14 are asabove for the scalarrelativistic nonlocal projector coefficients ("ekb"), grid indices, grid points, and projectors. Data block 5 is the local potential lloc=4 as above. Data blocks 68 are asabove for the spinorbit nonlocal projector coefficients ("ekb"), grid indices, grid points, and projectors. Nonlinear core corrections are not used in this example, but if they were, the model core charge and its derivatives would be the last data block
DISCUSSION¶
This introduces a normconserving pseudopotential input file format designed by D. R. Hamann which offers additional flexibility over previously available Abinit psp formats, as well as possible improvements in performance in certain respects. Its new features are:

Numerical psp’s, which offer the ability to experiment beyond existing tabulated collections. Available psp’s given as basis function sums are easily convertible to this format.

Linear radial grids, which have the following advantages:
(a) Computational efficiency compared to log grids because the core region of those grids is highly redundant for psp’s.
(b) Increased accuracy at large ecut values. The spacing of mesh points for log grids at larger radii, where there are usually still significant contributions to the psp’s, becomes too large and introduces aliasing noise into the Fourier transform psp’s. In fact, this code is written to double, triple, etc. the linear grid and interpolate prior to performing the transform to ensure accuracy, based on ecut.
(c) Storage economy (probably not important in modern computers, but the files are easier to read).
(d) While redundant, repeating the grid in the data block with each function facilitates inspection and plotting of the various inputs.

Allowance for any number of nonlocal projectors for each anugular momentum channel, previously available for several formats but without the current flexibility.

Direct input of the BloechlKleinmanBylander projectors rather than wave functions and semilocal potenials. This allows greater flexibility important for some exploratory applications such as selfinteractioncorrected psp’s. The projectors are applied to the wave functions as the following operators
Sum fkb(n,l),m> ekb(n,l) <fkb(n,l),m (nlm)
The conventional KleinmanBylander choice for a single projector (single n) would be
fkb(r,l) = (V(r,l)  V_loc(r))u(r,l),
where V(r,l) is the semilocal normconserving pseudopotential for
angular momentum l, V_loc(r) is the local potential, and u(r,l) is
the radial Schroedinger equation solution (bound or scattering).[1,2]
Note that u(r) has the conventional r
factor, u(r)=r psi(r).
In this case, the “energies” ekb (which have dimensions 1/energy
for the above definition of fkb) are
ekb(l) = 1 / [Integral fkb(r,l)*u(r,l) dr], (0,inf)
(no 4*Pi factor). For the Bloechl generalization, see [3]. Other possible generalizations include the treatment of self interaction corrections as additional projectors.[4]
Finally, creating numerical projectors as part of the psp generation process (rather than within Abinit) allows one to solve the radial Schrodinger eqution in its fully nonlocal form and compare logarithmic derivatives of the pseudo and allelectron wave functions over a wide energy range. This is a superior test of transferability, and will locate shallow “ghost resonances” above the reference valence states, which would be missed in the standard ghost tests [5] but can lead to poor results.
 Ability to use an arbitrary local potential rather than that for one particular angular momentum channel. This can be helpful in eliminating ghost states.[5]
In the example above, a calcium pseudopotential, the “conventional” choice of the dchannel (lloc=2) for the local potential is particuarly bad because the incipient d shell represented by a shallow d resonance leads to a very deep d potential. With this potential as local, it is nearly impossible to avoid ghost states. Using p as the local potential would avoid this, but would place a spurious coreorthogonalization repulsive term in channels for higher angular momenta (f, g, h, …) which could lead to errors in, for example, lattice constants. The local potential here was formed by a smooth (polynomial) extrapolation of the tailregion allelectron potential to the origin. While some other Abinit psp formats include the ability to form the local potential as a simple average over semilocal potentials, the d potential would likely still cause problems in this case.

Spherical bessel functions for Fourier transforms in the
psp8*.f
routines are computed by a highly accurate recursion method in sbf8.f, and all angular momentum channels are treated simultaneously, increasing accuracy and efficiency compared to sin/cos formulations, especially for large and small arguments. 
The ability to experiment with various model core charges for the nonlinear core correction[6] is advantageous. Functions that are not sufficiently smooth can lead to potentially severe convergence errors, especially in response function calculations. For other formats, Abinit internally generates higher derivatives of the model core charge up to the 4^{th} derivative, and piecewiseconstructed model functions that LOOK smooth can cause havoc. The requirement that all the derivatives to be present in the input file permits early detection of such problems, and aids their remediation.

Release 7.x introduces the ablity to include spinorbit coupling. The Pt pseudopotential used as an example above was initially generated with two nonlocal projectors each from Dirac wave functions with j=l+½ and j=l½ using Vanderbilt generalized norm conservation, which gives much greater accuracy than the Bloechl construction discussed above.[7] Weighted sums and differences of these nonlocal operators were separately rediagonalized to produce efficient orthonormal projectors for the scalarrelativistic and spinorbit terms which are required internally by abinit. For a detailed description of the projectors, see Ref. 7. The operation of SR and SO terms on the spinor wave functions is a straightforward generalization of
Sum fkb(n,l),m> ekb(n,l) <fkb(n,l),m (nlm)
where “fkb” and “ekb” are appropriately reinterpreted. Since the “fkb” for the Pt example are orthonormal, the “ekb” have the dimensions energy. Projectors with negligibly small energies (<2E5 Ha at present) are neglected. There is full flexibiltiy to use other methods to generate the SR and SO terms.
REFERENCES¶
[1] “Generalized NormConserving pseudopotentials,” D. R. Hamann, Phys. Rev. B 40, 2980 (1989); “NormConserving Pseudopotentials,” D. R. Hamann, M. Schlueter, and C. Chiang, Phys. Rev. Lett. 43, 1494 (1979).
[2] “Efficacious Form for Model Pseudopotentials,” L. Kleinman and D. M. Bylander, Phys. Rev. Lett. 48, 1425 (1982).
[3] “Generalized Separable Potentials for ElectronicStructure Calculations,” P. E. Bloechl, Phys. Rev. B 41, 5414 (1990).
[4] “Ab Initio ElectronicStructure Calculations for IIVI Semiconductors Using SelfInteractionCorrected Pseudopotentials,” D. Vogel, P. Krueger, and J. Pollmann, Phys. Rev. B 52, 14316 (1995).
[5] “Ghost States for Separable, NormConserving, ab initio Pseudopotentials,” X. Gonze, P. Kaeckell, and M. Scheffler, Phys. Rev. B 41, 12,264 (1990); “Analysis of Separable Potentials,” X. Gonze, Phys. Rev. B 44, 8503 (1991).
[6] “Nonlinear Ionic Pseudopotentials in SpinDensityFunctional Calculations,” S. G. Louie, S. Froyen, and M. L. Cohen, Phys. Rev. B26, 1738 (1982).
[7] “Optimized normconserving Vanderbilt pseudopotentials”, D. R. Hamann, Phys. Rev. B 88, 085177 (1913), and references therein.