(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 non-standard features, to have great flexibility in doing so. It does not correspond to any publicly available tabulation. The open-source ONCVPSP package available at www.mat-simresearch.com produces pseudopotentials in this format.
An annotated example is presented in detail below, followed by a second example incorporating spin-orbit coupling. An extended discussion follows below the examples, with a separate section discussing spin-orbit as produced by ONCVPSP.
When first producing a pseudopotential in this format, it would be a very good idea to do a well-converged 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 exchange-correlation 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 Bloechl-Kleinman-Bylander 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 special-purpose use in the future. # The value 2 is now used to indicate the presence os spin-orbit # projectors (see SPIN-ORBIT section below).
self-intraction-corrected 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 2.0000000000000D-02 3.2695476840515D-03 -7.7803072439537D-05
3 4.0000000000000D-02 6.5413292698932D-03 -1.5573364986856D-04
.....
.....
#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 2.0000000000000D-02 4.1259561971920D-05
3 4.0000000000000D-02 1.6501618805883D-04
.....
.....
# Third data block # as above, for l=2
2 -4.9318560092991D-01 1 0.0000000000000D+00 0.0000000000000D+00 2 2.0000000000000D-02 -2.8426374194219D-04 3 4.0000000000000D-02 -2.2672423014226D-03 ..... .....
#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 # small-amplitude Gibbs oscillation as a function of the Q of the # Fourier-transformed local potenial with a period of 2*Pi/rad(mmax).
4 1 0.0000000000000D+00 -9.3285752771727D-01 2 2.0000000000000D-02 -9.3284556214441D-01 3 4.0000000000000D-02 -9.3280966542583D-01 ..... ..... 313 6.2400000000000D+00 -3.3275722650983D-01
#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 -1.3822038662342D-06 -3.4248477651502D-05 2.2949926081983D-05 -1.3721984594931D+01 2 2.0000000000000D-02 1.8263581757935D+00 -1.8290946429655D-05 -2.7430630291457D-03 -2.7416158741933D-01 -1.3679142685104D+01 3 4.0000000000000D-02 1.8263568047539D+00 -1.4618869292620D-04 -1.0954897796014D-02 -5.4658870398836D-01 -1.3549201500559D+01 ..... .....
SPIN-ORBIT
Spin-orbit 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 spin-orbit portion of the discussion section
#Header - lines 1-4 as above # line 5 - number of Vanderbilt-Kleinman-Bylander projectors nproj for # the scalar-relativistic non-local 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 spin-orbit projectors are present # line 7 - number of Vanderbilt-Kleinman-Bylander projectors nprojso for # the spin-orbit non-local 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 1-4 are as-above for the scalar-relativistic non-local projector
# coefficients ("ekb"), grid indices, grid points, and projectors.
#Data block 5 is the local potential lloc=4 as above.
#Data blocks 6-8 are as-above for the spin-orbit non-local projector
# coefficients ("ekb"), grid indices, grid points, and projectors.
#Non-linear 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 norm-conserving 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:
(1) 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.
(2) 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.
(3) Allowance for any number of nonlocal projectors for each anugular momentum channel, previously available for several formats but without the current flexibility.
(4) Direct input of the Bloechl-Kleinman-Bylander projectors rather than wave functions and semi-local potentials. This allows greater flexibility important for some exploratory applications such as self-interaction-corrected psp's. The projectors are applied to the wave functions as the following operators
Sum_(nlm) |fkb(n,l),m> ekb(n,l) <fkb(n,l),m|
The conventional Kleinman-Bylander 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 semi-local norm-conserving pseudopotential for angular momentum l, V_loc® is the local potential, and u(r,l) is the radial Schroedinger equation solution (bound or scattering).[1,2] Note that u® has the conventional “r” factor, u®=r*psi®. In this case, the “energies” ekb (which have dimensions 1/energy for the above definition of fkb) are
ekb(l) = 1 / [Integral_(0, inf) fkb(r,l)*u(r,l) dr]
(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 equation in its fully nonlocal form and compare logarithmic derivatives of the pseudo and all-electron 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.
(5) 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 d-channel (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 core-orthogonalization 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 tail-region all-electron potential to the origin. While some other Abinit psp formats include the ability to form the local potential as a simple average over semi-local potentials, the d potential would likely still cause problems in this case.
(6) 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.
(7) The ability to experiment with various model core charges for the non-linear 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 4th derivative, and piecewise-constructed 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.
(8) Release 7.x introduces the ablity to include spin-orbit coupling. The Pt pseudopotential used as an example above was initially generated with two non-local projectors each from Dirac wave functions with j=l+1/2 and j=l-1/2 using Vanderbilt generalized norm conservation, which gives much greater accuracy than the Bloechl construction discussed above.[7] Weighted sums and differences of these non-local operators were separately re-diagonalized to produce efficient orthonormal projectors for the scalar-relativistic and spin-orbit 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_(nlm) |fkb(n,l),m> ekb(n,l) <fkb(n,l),m|
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 (<2E-5 Ha at present) are neglected. There is full flexibiltiy to use other methods to generate the SR and SO terms.
REFERENCES
[1] "Generalized Norm-Conserving pseudopotentials," D. R. Hamann,
Phys. Rev. B 40, 2980 (1989); "Norm-Conserving 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 Electronic-Structure
Calculations," P. E. Bloechl, Phys. Rev. B 41, 5414 (1990).
[4] "Ab Initio Electronic-Structure Calculations for II-VI
Semiconductors Using Self-Interaction-Corrected Pseudopotentials,"
D. Vogel, P. Krueger, and J. Pollmann, Phys. Rev. B 52, 14316 (1995).
[5] "Ghost States for Separable, Norm-Conserving, 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 Spin-Density-Functional
Calculations," S. G. Louie, S. Froyen, and M. L. Cohen, Phys.
Rev. B26, 1738 (1982).
[7] "Optimized norm-conserving Vanderbilt pseudopotentials", D. R.
Hamann, Phys. Rev. B 88, 085177 (1913), and references therein.