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 norm-conserving separable pseudopotentials ([Hamann1979], [Hamann1989] and [Kleinman1982]) 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 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 of spin-orbit
projectors (see SPIN-ORBIT section below).
EXPERIMENTAL self-interaction-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 is not precisely satisfied for the experimental 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 potential 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 subroutine psp8cc)
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:
-
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 Bloechl-Kleinman-Bylander projectors rather than wave functions and semi-local potenials. 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
The conventional Kleinman-Bylander [2] choice for a single projector (single n) would beSum |fkb(n,l),m> ekb(n,l) <fkb(n,l),m| (nlm)where V(r,l) is the semi-local norm-conserving 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 conventionalfkb(r,l) = (V(r,l) - V_loc(r))u(r,l),rfactor, u(r)=r psi(r). In this case, the “energies” ekb (which have dimensions 1/energy for the above definition of fkb) are(no 4*Pi factor). For the Bloechl generalization, see [Bloechl1990]. Other possible generalizations include the treatment of self- interaction corrections as additional projectors, see [Vogel1995]. 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 [Gonze1990a], [Gonze1991], but can lead to poor results.ekb(l) = 1 / [Integral fkb(r,l)*u(r,l) dr], (0,inf) -
Ability to use an arbitrary local potential rather than that for one particular angular momentum channel. This can be helpful in eliminating ghost states [Gonze1990a], [Gonze1991]. 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.
-
Spherical bessel functions for Fourier transforms in the
psp8*.froutines 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 non-linear core correction [Louie1982] 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.
-
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+½ and j=l-½ using Vanderbilt generalized norm conservation, which gives much greater accuracy than the Bloechl construction discussed above.[Hamann2013] 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
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 flexibility to use other methods to generate the SR and SO terms.Sum |fkb(n,l),m> ekb(n,l) <fkb(n,l),m| (nlm)