Ming-Hsien Lee's Ph.D. Thesis Chapter V

Pseudopotentials Generated

and

Tested for Collaborations

In the course of the research described in Chapter II to IV, over 50 pseudopotentials have been generated for a range of elements in a variety of applications, and the present Chapter reports on this aspect of the work. (The above number excludes of course pseudopotentials that were interim trials in the course of optimisation, or those found unsatisfactory on subsequent testing.)

On one hand the practical experience of generating a varity of pseudopotentials stimulated the ideas contained in Chapter II to IV and their gradual refinement. On the other hand there has been no better test of efficiency of these ideas than having them tested through a diversity of practical applications. I was actively involved in a few of these applicationss as a collaborator beyond generating and testing the pseudopotential. The other applications represent projects in the Cambridge resaerch group around Dr. M. C. Payne, and varius collaborations between others and this group.

Section V.1 lists various comments and tips relating to generating pseudopotentials, and Section V.2 gives details of the pseudopotentials with some indication of tests and applications which have been undertaken.

V.1. The Art of Generating Pseudopotentials

Although my work has been devolted to removing the "black magic" aspect of generating pseudopotentials, we give here some tips and comments which may be useful to a beginner.

On pre-generation research

Before generating a pseudopotential for an element, information on the physical and the chemical properties of the element will be useful. Information such as the ground state configuration of the free atom, the ionic radius and covalent radius of the element in molecules and solids, common compounds of the element (for a solid state test) with their inter-atomic spacings, all help in choosing generating parameters.

On choosing r_c

Avoid overlapping cores if possible. Usually r_c should be chose between the covalent radius and the first (lower) ionic radius of an elements. the r_c can be larger if in the application the element is an anion.

On choosing Q_c

Use the Q_c which give best agreement on the logarithmic derivative. When the same agreement is produced in a range of Q_c, the smallest should be used unless one is aiming for some special shape of the pseudopotential.

On choosing the local components of the pseudopotential

For 2p elements such as O, choosing minimum non-crossing local (i.e. making V^L(r) attractive among V_l so that dV_l(r) is alway positive) helps to improve the logarithmic derivative of all angular channels. But this only works for those elements. For d-valence elements, i.e transition metals, choosing V_l₌₀ as the local pseudopotentia is necessary for the Kleinman-Bylander construction. Choosing s local usually works for most cases, if there is no preference on which channel is to be taken as the local potential. A local potential which is a weighted mix from different components (usually two) is necessary for the Projector Reduction technique, a good mix producing a well balanced scattering property in all (both) channels.

On choosing the atomic electronic configurations

In most cases the atomic ground state works very well for using the occupied atomic orbitals. A special prescription to the configuration in the mimic solid state can be used but it may not change the result much. For example one might use the configuration 1s² 2s² 2p³ or 1s² 2s^1.5 2p^2.5 for carbon in diamond. (See also Cd000 and Cd001 in Section VI.2.) Configurations for excited states in the table published in BHS paper are good ones to start with. Avoid choosing extremely small occupation number of states, especially those states occupied in a neutral atomic configuration. The exchange-correlation interaction of core and valence electrons sampled by such a low density valence state is not correct and may cause problem. For a very large core radius r_c, a slightly ionised configuration can be used, but this is dangerous and the scattering tests on a neutral configuration must be performed to ensure that the potential is free from ghost states.

On relativistic calculations

The so-called "Scalar relativistic" wave equation is recommended for elements heavier than Xe. It usually lowers the potentials, i.e. makes them more attractive. The largest change appears on the s potential which is not surprising because the s electron penetrates most into the strong potential near the nucleus.

On interpreting logarithmic derivatives

Deviation of the logarithmic derivative in the higher energy range is less important. This is because at higher kinetic energy the scattering waves are more oscillatory, giving a larger variation of logarithmic derivative F'/F (i.e. [lnF]' ) with respect to energy but without changing the charge density (norm) doesn't change. Agreement near the atomic eigen level is important. An extra peak of logarithmic derivative (compared with an all-electron full potential calculation) near the atomic eigenvalue within the range of a typical band width is an indication of a ghost state, as there is one more node in the wave function that has been counted during the test when the energy scans across the range of solid state interest.

On the criterion for a error tolerable error in solid state tests

Compared to experimental results, an error in the lattice parameter within 2% for an insulator, 3% for a metal should be expected, while 15% error of the bulk modurus is acceptable. Be careful that some bulk moduli were measured at room temperature, and extrapolataion to 0^oK should be used.

On transferability

A pseudopotential should be designed to be transferable within one application, but not necessarily so between very different applications.

On making judgements

Always think of a physical interpretation, no matter whether it is elegant or speculative. Don't taketoo seriously that there is some Black Magic : but believe in trial and error.

V.2. A List of Working Pseudopotential

The following is a compilation of 40 to 50 reasonable pseudopotentials that have been generated by me using the methods described in Chapter II, III and IV. Generating parameters, figures for V_l(r) and logarithmic derivatives are presented for each pseudopotential.

Most of the pseudopotentials have been generated with suitability for general purpose use in mind. However some have been generated for specific purposes requireing special values of the parameters, e.g. an especially soft pseudopotental for a calculation with a very large basis set, a particular rc in a material with unusually small interatomic spacing.

All the pseudopotentials have of course been tested as regards comparing their logarithmic derivatives against the all-electron calculations. (For those relativitstic ones we compare the logarithmic derivatives of KB and non-KB pseudopotentials). Most have also been tested by calculations on a simple solid or molecules, including always the lattice constant and sometimes bulk modulus, as indicated briefly below. The solid state test has been carried out either by myself, or otherwise by the named collaborator. Theough out this chapter, atomic unit (a.u.), i.e. Bohr radius, has been used as the unit for r_c. Lattice parameters are in Angstron in all tests.

Ag005

Electronic configuration: for all s, p, d : 4d^9.00 5s^0.75 5p^0.25

r_c(s) = r_c(p) = 2.5 a.u. (Bohr radius), r_c(d) = 2.7 a.u.

The s pseudowavefunction use only two sphereical Bessel functions to expand, the third term was set identical to the second one.

Q_c(s) = 1.00 *q₂(s), Q_c(p) = 1.016*q₃(p), Q_c(d) = 1.17*q₃(d)

Local potential: s

Tests :

Ag005 (smearing 1eV) (start from 4eV smearing)

Grid 18x18x18

E_cut = 495 eV (450 plane waves)

======================================================

Da/a₀ E_tot stress_xx stress_yy stress_zz

------------------------------------------------------------

-2% -930.4014202 -0.013532 -0.017884 -0.015376

-1% -930.4102006 0.008468 0.009065 0.009078

0% -930.3805164 0.034386 0.029028 0.029774

1% -930.3781922 0.046174 0.046512 0.047885

2% -930.3513368 0.061996 0.061053 0.063311

======================================================

Al013

Electronic configuration for s and p potential : 3s^2.00 3p^1.00. (No d potential)

r_c(s) = r_c(p) = 2.4 a.u., Q_c/q₃(s)=1.10, Q_c/q₃(p)=1.00 , Local potential is p.

Tests : See Chapter IV.

Al013a

A kinetic energy optimized Q_c-tuned Aluminum pseudopotential: Al013a

Gnerated with effect of using p-potential to immitate d-potential.

(i.e. the idea of Projector Reduction technique)

Configuration for s and p potential : 3s^2.00 3p^1.00(No d potential)

r_c(s) = r_c(p) = 2.4 a.u. , Q_c/q₃(s) = 1.10, Q_c/q₃(p) = 1.10 , Local potential is p.

Tests : See Section IV.3 of Chapter IV

As000

Electronic configurations : for s and p : 4s^2.00 4p^3.00: for d : 4s^1.00 4p 1.75 4d^0.25

r_c(s, p, d) = 2.0 a.u., Q_c/q₃(s, p, d) = (0.70, 0.90, 0.95)

Local : p

Au001

Atomic electronic configuration : [core = Xe + 4f ¹⁴] 5d^9.00 6s^0.75 6p^0.25 for all s, p and d components (note that f electrons are frozen into the pseudo-core)

r_c(s, p, d) = (2.5, 2.5, 2.8) a.u.,

The s and p waves only use two Sphereical Bessel function terms, d uses three.

Q_c/q₂(s) = 1.00, Q_c/q₂(p) = 1.00, Q_c/q₃(d) = 1.16

Local potential is s

B001

Electronic configurations : For s and p : 2s^2.00 2p^1.00 : For d : 2s^1.00 3d^0.20

r_c(s, p, d) = 1.4 a.u. , Q_c/q₃(s, p, d) = (0.80, 0.80, 1.00) , Local potential : p.

Tests :

BCl3 molecule in a 10 Angs cubic box : (use B001 with Cl000)

Grids : 64x64x64

E_cut = 300 eV

TOTAL ENERGY IS -1309.4623151

-------------------------------------------------------------------------

NSP ATOM a1 a2 a3 Fx Fy Fz

-------------------------------------------------------------------------

1 1 0.000000 0.000000 0.000000 -0.002578 0.000980 -0.001276

2 1 0.173099 1.000000 0.999987 -0.061288 0.000340 -0.000196

2 2 0.913453 0.149946 0.999984 0.033098 -0.055030 -0.000405

2 3 0.913472 0.850050 0.999992 0.032416 0.053032 0.000007

-------------------------------------------------------------------------

=> B-Cl bond length 1.09% smaller than expt. value (1.75 Angs)

E_cut = 350 eV

TOTAL ENERGY IS -1309.6483414

-------------------------------------------------------------------------

NSP ATOM a1 a2 a3 Fx Fy Fz

-------------------------------------------------------------------------

1 1 0.000000 0.000000 0.000000 0.002381 0.000014 -0.000915

2 1 0.172792 0.000002 0.999988 -0.010656 0.000304 0.000275

2 2 0.913596 0.149672 0.999984 0.003976 -0.010556 0.000255

2 3 0.913614 0.850319 0.999992 0.003575 0.010135 0.000290

-------------------------------------------------------------------------

=> B-Cl bond length about 1.26% smaller than expt. value

Tests (2) : (By Dr. M. Fearm at Material Modeling Lab, Oxford)

The E-V calculations for BP using B001 and the P001.

Converged at E_cut above 300-400 eV. The lattice constant

comes out around 1% too small, which is quite good.

32 k-pt calculations. q=4 from Monk-Pack scheme.

a(expt) = 4.538A

===================================================

a E_cut (300eV) E_cut (400eV) E_cut (600eV)

---------------------------------------------------

4.44 -258.91100 -259.15662 -259.19222

4.46 -258.92121 -259.16461 -259.19998

4.48 -258.92262 -259.16841 * -259.20304 *

4.50 -258.92658 * -259.16813 -259.20137

4.52 -258.92260 -259.16340 -259.19592

4.538 -258.91788 -259.15530 -259.18748

4.56 -258.90613 -259.14137 -259.17272

===========================================

* indicates energy minimum

B001a

Same as B001, but with d projector reduced.

B002

A Boron pseudopotential with a smaller r_c for checking the B in bulk Si system.

Electronic configurations : for s and p : 2s^2.00 2p^1.00 : for d : 2s^1.00 3d^0.20 (from BHS)

r_c(s, p, d) = 1.25 a.u. , Q_c/q₃(s, p, d) = (0.80, 0.80, 1.00) , Local potential : p

Tests : Tested by Dr. M. Fearn whcih give similar results as in the case when B001 is used.

Ba015Rg

Q_c-tuned Optimised Ba Psuedopotential Ba015Rg (relativistic calculation)

The n = 5 core shell has been treated as valence.

Eletronic configuration : (Reletivistic) for all s, p, d : 5s^2.00 5p^5.75 5d^0.25

r_c(s, p, d) = 2.7 a.u. , Q_c/q₃(s, p, d) = (0.40, 0.80, 0.50) , Local : p.

Tests :

Test (1)

A quick BaO crystal test at E_cut = 500eV (2 k-points)

=======================================================

Da/a₀ E_tot stress_xx stress_yy stress_zz

--------------------------------------------------------------

0% -4535.5149685 0.030057 0.030044 0.030058

=======================================================

Test (2)

Ba015Rg has been used in the calculation of small molecule halides MX₂ [Ref.M.1]

Be000

Electronic configurations : for s : 2s^2.00 : for p and d : 2s^0.00 2p^0.25 3d^0.25

r_c(s, p, d) = 1.4 a.u. , Q_c/q₃(s, p, d) = (1.00, 1.00, 1.00) , Local : s.

Note:

Be000 might work just with s-potential, so in this Be000 the s-potential has been chosen as local. If p and d parts are not needed it can be swiched off.

Br000

Electronic configuration: for s and p : 4s^2.00 4p^5.00: for d : 4s^1.00 4p^3.75 4d^0.25

r_c(s, p, d) = 1.4 a.u. ,Q_c/q₃(s, p, d) = (1.00, 1.00, 0.90), Local : Mixed (s, p, d) = (0.0, 0.7, 0.3)

Projector Reduced for both p and d non-local parts. (Which means, s is the only non-local part, the local potential takes care of both p and d.)

C009

Electronic configuration: s and p components: 2s^2.0 2p^2.0 : for d : 2s^0.75 2p^1.00 3d^0.25

r_c(s, p, d) = (1.0, 1.4, 1.4) a.u. , Q_c(s,p,d) = q₃(s,p,d) + (1/8)*q₁(s,p,d).

Choice of local component : Minimum and no-crossing

Some informations on C009 (from the old info file of CASTEP data directory):

Purpose:

This Carbon potential was generated to describe organic molecules interact with surface, in which the C-C single, double and triple bonds of the molecules should be reasonably described without pre-asummption of the direction of charge transfer between molecules and surface. A highly transferable potential is required, so this is a tough test for any optimised potential.

Method:

We tackle this problem by using a r_c slightly smaller than the covalent radius of Carbon found from periodic table. 3-term scheme is used to make it as soft as possible. At r_c = 1.4 a.u., the kinetic energy filter Q_c is chosen to be q₃+(0.125)*q₁ to allow enough high wave-number components of pseudo wave function to be inside the region of r_c. By doing this we managed to remove the potential energy barrier at r_c which was the result of the optimization under norm-conservation but without the constraint of the wave function second derivative continuation.

d-component:

At the time C009 was generated we did not have any better method to decide whether to use d-component and how it should be generated, therefore we follow many people using the excitation configuration suggested in BHS paper.

The r_c of s-component:

The reason we use r_c(s) = 1.0 a.u. instead of 1.4 is that the scattering property (logarithmic derivative plot) of s potential looks not as good as those of the p and d potentials. However, we have not tested the r_c(s) = 1.4 a.u. case, maybe that r_c is already sufficient for most applications. (In many cases, it is more difficult for s potential than for p or d to reach the same degree of agreement to the all-electron logarithmic derivatives.)

Transferability:

An interesting feature worth mentioning for this C009 is that we did generate a similar one with 2s^2.0 2p^1.0 (ie, C⁺) configuration. We found that such C⁺ potential is almost identical (indistiguishable from graphics output) to C009, which give us some confidence on this C009 in ionized environment.

Some tests:

test 1:

Convergence test of C009: (diamond structure 2 atom unit cell)

With Grid 36x36x36 and 1 k-point (gamma point) (unit:eV)

=====================================================================

E_cut E_tot (C009) E_tot (C011) E_tot (c501)

---------------------------------------------------------------------

200 -259.4946695 -262.7598104 -259.4937132

300 -275.0913281 -277.7028793 -274.0124997

400 -277.9823971 -280.3250624 -276.8749429

500 -280.7073211 -282.8892206 -280.3055508

600 -281.3373075 -283.4969745 -281.2953102

700 -281.5868402 -283.7268133 -281.8431799

800 -281.5874702 -283.7287377 -281.8508776

900 -281.5913412 -283.7364188 -281.8767534

=====================================================================

( C011 is the same as C009 except r_c(s) = 1.5 a.u. )

( c501 is a Troulier & Martins potential generated by Dr Lin, with

reference configuration 2s 2.0 2p 2.0 3d 0.0, r_c(s,p,d) = 1.45,

and d component is chosen as local.)

(unit:eV)

with Grid 20x20x20 and 4 k-points (gamma point of 8 atom cell)

=================================================

E_cut E_tot (C009) E_tot (C011)

-------------------------------------------------

200 -289.6219489 -293.8776430

300 -301.4012288 -303.8535788

400 -304.7355822 -306.4620305

500 -306.5260928 -307.8590246

600 -306.9122977 -308.4647471

700 -307.0267655 -308.5219363

800 -307.0331341 -308.5328968

900 -307.0346273 -308.5335537

=================================================

[ Note : The strange 800 eV to 900 eV jump in 1 k-point tests disappear

when one use 4 k-points. ]

Test 2:

700 eV, one k point (gamma point) relaxed C₂ dimer in a 7 Ang cubic box

TOTAL ENERGY IS -300.3882883

-------------------------------------------------------------------------

NSP ATOM a1 a2 a3 Fx Fy Fz

-------------------------------------------------------------------------

1 1 .500000 .500000 .500000 -.005722 -.005608 -.005761

1 2 .602579 .602531 .602559 .005989 .005785 .005418

-------------------------------------------------------------------------

=> C-C distance = 1.243429482708 Angstrong

700 eV, 4 k points ( k=0.25 ) relaxed C₂ dimer in a 7 Ang cubic box

TOTAL ENERGY IS -300.4095048

-------------------------------------------------------------------------

NSP ATOM a1 a2 a3 Fx Fy Fz

-------------------------------------------------------------------------

1 1 .500000 .500000 .500000 .015179 .019426 .015562

1 2 .602687 .602827 .602625 -.022316 -.028882 -.011843

-------------------------------------------------------------------------

=> C-C distance = 1.245329363513 Angstrong

Test 3:

Diamond structure, 2 atom primitive unit cell.

Use 4 k-points which are equivalent to the gamma point of cubic 8 atom cell.

Plane wave cutoff energy is 700 eV.

1. Ideal Structure (lattice parameter 3.57 Angs)

TOTAL ENERGY IS -308.5224139

-------------------------------------------------------------------------

SIGTO .010419 .010450 .010905 -.000067 -.000062 .000044

-------------------------------------------------------------------------

NSP ATOM a1 a2 a3 Fx Fy Fz

-------------------------------------------------------------------------

1 1 .000000 .000000 .000000 .004692 -.004394 -.001716

1 2 .250000 .250000 .250000 .003574 -.004598 -.002641

-------------------------------------------------------------------------

2. Unit cell 0.3% smaller (lattice parameter 3.56 Angs)

TOTAL ENERGY IS -308.5221886

-------------------------------------------------------------------------

SIGTO -.014528 -.014734 -.014195 -.000099 -.000014 .000072

-------------------------------------------------------------------------

NSP ATOM a1 a2 a3 Fx Fy Fz

-------------------------------------------------------------------------

1 1 .000000 .000000 .000000 .001175 -.005514 .001180

1 2 .250000 .250000 .250000 .000398 -.004682 -.000157

-------------------------------------------------------------------------

3. Unit cell 0.3% larger (lattice parameter 3.58 Angs)

TOTAL ENERGY IS -308.5202617

-------------------------------------------------------------------------

SIGTO .034883 .034657 .035238 -.000097 -.000016 .000073

-------------------------------------------------------------------------

NSP ATOM a1 a2 a3 Fx Fy Fz

-------------------------------------------------------------------------

1 1 .000000 .000000 .000000 .001140 -.005841 .001095

1 2 .250000 .250000 .250000 .000045 -.004850 -.000267

-------------------------------------------------------------------------

[ Note : The stress changed its sign when lattice parameter went from +0.3%

to -0.3% ]

Interpretation of the results of tests:

1. Converge within 1 milli-Ryd (0.01 eV) at 700 eV, and 0.1 eV convergence at 600 eV.

2. The calculated C₂ dimer bond length is 1.243 or 1.245 depends on the k-points chosen.

We have found that there are two experimntal values for the interatomic distance of C₂ dimer,

one is 1.312 Angs. from [Ref.H.1], and the other being 1.243 from A. D. Becke,

Phys. Rev. A 33, 2786 (1986) also sited in p.179, Table 8.2 of [Ref.G.1]. If the later quote

is more reliable, then the calculated bond length is very good.

3. Lattice parameter at 700 eV, agree with experiment within 0.6% (Expt 3.57 Å).

Other sources of test :

Chirs Goringe and Dr. Adrian Sutton in Oxford have done extensive tests on C009 for C₂H₂, C₂H₄ and C₂H₆.

See also the CH₃OH molecule tests listed in C021.

C021

Electronic configuration : for s and p : 2s^2.00 2p^2.00 ; for d : 2s^0.75 2p^1.00 3d^0.25

r_c(s, p, d) = 1.4 (a.u.), Q_c/q₃(s, p, d) = (0.80, 1.05, 1.0325)

Local mixing ratio (s, p, d) = (0.0, 1.0, 0.0)

(i,e. V_p(r) is actually local but V_d(r) is so similar to V_p(r) such that this really doesn't matter !)

Application:

For organic molecule, ploymer, etc. Need 600 or 650 eV to converge the structure. For large systems.

Comments:

This C021 pseudopotential has the local part of potential reproduces almost exactly both p and d scattering of those from a full projector one. This technique avoids the uncertainty of whether one should use d-potential for Carbon and also redcuce the ambiguity of choosing the d-potential. There is only one non-local projector in this V_ps, which is s-potential. The advantage of using C021 for a better computing efficiency is expected.

Tests:

Test I (by Rajiv Shah)

Equilibrium bond lengths, in Å, for different

pseudopotentials. 550eV cutoff.

=============================================================

Full Redu

proj Reduced projector proj Expt.

=============================================================

C009 ------------C021------------- C022

O027 O046 O049 O050 O051 O051 Expt.

=============================================================

C---H 1.097 1.095 1.095 1.094 1.096 1.097 1.094

C---O 1.423 1.453 1.460 1.455 1.460 1.457 1.437

O---H 0.972 0.979 0.980 0.978 0.979 0.978 0.978

=============================================================

Test II (by Matthew D. Segall)

All simulations were run with a cut-off energy of 600ev, using GGAs

for 10 iterations without relaxation followed by 20 iterations with

relaxation. The h2001, C021 and n2001N1 pseaudopotentials were used.

Cubic supercells were used and the side length was increased until

the bond length of interest had converged.

The results are as follows:

Methylamine

~~~~~~~~~~

H H

| /

H-C--N

| \

H H

Source C--N bond length

Expt. 1.483

6A side 1.463

7A side 1.460

8A side 1.460

The calculated bond length was 1.55% too short.

Ethane

~~~~~

H H

| |

H-C--C-H

| |

H H

Source C--C bond length

Expt. 1.532

6A side 1.509

7A side 1.516

8A side 1.516

The calculated bond length was 1.04% too short.

Test III

7 Angs. box, 1 k-point. 600 eV, 54x54x54 FFT grid

(Using pseudopotentials C021, N010, h2001) LDA k-space

TOTAL KINETIC ENERGY 377.0564616

LOCAL POTENTIAL ENERGY -590.1426327

NONLOCAL POTENTIAL ENERGY 55.7420618

HARTREE ENERGY CORRECTION -561.4103683

EX-CORR ENERGY CORRECTION 49.6492433

EWALD ENERGY 164.2169554

TOTAL ENERGY IS -504.6039709

-------------------------------------------------------------------------

NSP ATOM a1 a2 a3 Fx Fy Fz

-------------------------------------------------------------------------

1 1 0.989533 0.985081 0.924390 -0.015919 -0.010734 0.006742

2 1 0.020078 0.032737 0.125474 0.015158 0.017455 -0.004553

3 1 0.011097 0.177348 0.148314 0.019823 0.007451 -0.028711

3 2 0.153005 0.989888 0.170329 0.006489 0.001885 0.005931

3 3 0.100238 0.039158 0.823880 -0.010656 0.005027 -0.003085

3 4 0.982468 0.828355 0.908003 0.002626 -0.002726 0.000070

3 5 0.850438 0.042015 0.877170 -0.004533 -0.010354 0.021526

-------------------------------------------------------------------------

-> 1.4595 Ang (-1.5%)

(Using pseudopotentials C021, N010, h2001) GGA k-space

TOTAL KINETIC ENERGY 383.5600152

LOCAL POTENTIAL ENERGY -596.8865826

NONLOCAL POTENTIAL ENERGY 55.5306252

HARTREE ENERGY CORRECTION -567.3534514

EX-CORR ENERGY CORRECTION 49.3671008

EWALD ENERGY 166.8291592

TOTAL ENERGY IS -508.6688256

-------------------------------------------------------------------------

NSP ATOM a1 a2 a3 Fx Fy Fz

-------------------------------------------------------------------------

1 1 0.989613 0.985125 0.923399 -0.019320 -0.018566 0.001300

2 1 0.020626 0.032923 0.125226 0.018893 -0.002341 0.002824

3 1 0.011006 0.176120 0.148505 0.018646 0.004903 -0.032083

3 2 0.152230 0.990158 0.169784 0.004881 0.001932 0.007333

3 3 0.099749 0.038900 0.824586 -0.007569 0.007232 -0.006358

3 4 0.982791 0.829904 0.907572 0.002578 0.000627 -0.006353

3 5 0.851994 0.042023 0.876893 -0.001223 -0.008178 0.023481

-------------------------------------------------------------------------

-> C-N distance : 1.468 Angs (-0.9%)

(Using pseudopotentials C021, N010, h2001) GGA r-space

TOTAL KINETIC ENERGY 383.5608976

LOCAL POTENTIAL ENERGY -596.8881185

NONLOCAL POTENTIAL ENERGY 55.5304068

HARTREE ENERGY CORRECTION -567.3564348

EX-CORR ENERGY CORRECTION 49.3671624

EWALD ENERGY 166.8329500

TOTAL ENERGY IS -508.6688284

-------------------------------------------------------------------------

NSP ATOM a1 a2 a3 Fx Fy Fz

-------------------------------------------------------------------------

1 1 0.989614 0.985127 0.923403 -0.018928 -0.018398 0.001347

2 1 0.020629 0.032924 0.125226 0.019222 -0.002243 0.002764

3 1 0.011011 0.176123 0.148500 0.018666 0.004936 -0.031998

3 2 0.152232 0.990160 0.169786 0.004700 0.001969 0.007202

3 3 0.099747 0.038903 0.824587 -0.007697 0.007167 -0.006294

3 4 0.982792 0.829905 0.907572 0.002562 0.000507 -0.006309

3 5 0.851993 0.042024 0.876900 -0.001423 -0.008138 0.023363

-------------------------------------------------------------------------

-> C-N distance : 1.468 Angs (-0.9%)

Ca005

Electronic configuration: for s and p : 3s^2.00 3p ^6.00 (This adds up to Ca²⁺.)

r_c(s, p) = 2.0 a.u. ; Q_c/q₃(s, p) = (0.90, 1.06), Local : p.

Comments :

We expect that using this Ca005 will help to reduce most problems related to core correction and

core polarization because the core is unfreezed to one shell below valence shell. (However, depends on systems, there might be no problem at all.) Having in mind that this Ca will be used in a ionic (i.e. Ca²⁺) system, we prefere Ca²⁺ [core] 3s² 3p⁶ instead of Ca[neutal] [core] 3p⁶ 4s². We believe optimisation is essential for such kind of stratage, because the highly ionised pseudopotential is extreme hard.

Tests :

A lot of tests has been done by Dr. K. Refson (Earth Sciences, Oxford).

Cd000

Electrinc configuration : for s, p and d : 4d^10.00 5s^0.75 5p^0.25

r_c(s, p, d) = 2.5 a.u. , Q_c/q₃(s, p, d) = (1.00,1.00,1.18) , Local potential : s.

Cd001

Same as Cd000 expect for the electronic configuration : for s, p and d : 4d^10.00 5s^1.27 5p^0.37

Cl000

Electronic configurations : for s and p : 3s^2.00 3p^5.00 ; for d : 3s^1.00 3p^3.75 3d^0.25

r_c(s, p, d) = 1.7 a.u., Q_c/q₃(s, p, d) = 1.00 ; Local potential : p.

Tests:

Cl₂ molecule in a 10 Å cubic box :

E_cut : 300 eV

Grids : 64x64x64 (60x60x60 is sufficient)

TOTAL ENERGY IS -818.6694545

-------------------------------------------------------------------------

NSP ATOM a1 a2 a3 Fx Fy Fz

-------------------------------------------------------------------------

1 1 0.000000 0.000000 0.000000 0.038538 0.003198 -0.002190

1 2 0.197806 0.999982 0.999959 -0.083952 0.003397 0.002644

-------------------------------------------------------------------------

TOTAL ENERGY IS -818.6694939

-------------------------------------------------------------------------

NSP ATOM a1 a2 a3 Fx Fy Fz

-------------------------------------------------------------------------

1 1 0.000000 0.000000 0.000000 0.028101 0.000897 -0.000453

1 2 0.197739 0.999983 0.999960 -0.041045 0.000211 0.000553

-------------------------------------------------------------------------

The bond length is about 0.65% smaller than experiment (1.99 Å)

Co013_v2

Configuration: for all s p d : 3d^7.00 4s^1.00 4p^0.25

r_c(s, p, d) = (2.0, 2.0, 2.4) a.u. , Q_c/q₃(s, p, d) = (0.70, 0.965, 1.18), Local (mixed) 0.2 0.8 0.0

Projector reduced for s and p projector.

Tests : (By Dr. V. Milman)

First - CoSi2. It has fluorite structure, primitive unit cell has an fcc basis vectors and atoms are at Co(000), Si (1/4,1/4,1/4) and (3/4,3/4,3/4). Tests were done for this primitive cell with symmetrisation and they are pretty well converged with respect to sampling.

Convergence test.

Co013_v2 and Si801s_only, a₀= 5.4 (Angs)Å.

==================================

500 eV Etot= -884.4999 SIGT=-0.0413

700 -884.5590 -0.0407

==================================

Bands are the same within 0.001 eV or better for these two cutoffs.

Bands differ by 0.1 eV between Co006 and Co013_v2 potentials.

--------------------------------------------------------------------

Equilibrium a₀ with Co013_v2 potential, 500 eV, q = 4 set (10 points with

symmetry). DEL from 4.0 to 0.05 eV with NDEL = 4. Si is s-only.

========================================

a0 E SIGT EFERMI

----------------------------------------------

5.3 -884.3316 -0.1179 1.644

5.4 -884.4999 -0.0413 1.355

5.45 -884.5281 -0.0097 1.216

5.45* -884.5278 -0.0079 1.325

5.5 -884.5236 0.0306 1.080

5.6 -884.3828 0.0913 0.790

========================================

* - q=8 set, 60 k-points, corresponds to 512 points in full BZ

converged within 3e-4 eV for Etot and 2e-3 for SIGT

Results:

===========================================

from P(V) from E(V) exp

-------------------------------------------

a0 5.465 5.465 5.36

B 205 203 169

===========================================

The same Co013_v2 potential with the Si012 (i.e. si501).

Assume convergence is the same (same Co, after all).

Lattice constant. Ecut=500 eV, q=4.

=============================================

a0 E_tot SIGT EFERMI

---------------------------------------------

5.26 -885.68090 -0.05867 -0.209

5.28 -885.70311 -0.04321 -0.244

5.30 -885.71892 -0.02859 -0.278

5.32 -885.72896 -0.01455 -0.313

5.34 -885.73307 -0.00140 -0.347

5.35 -885.73322 0.00504 -0.364

5.36 -885.73186 0.01117 -0.380

5.38 -885.72526 0.02314 -0.414

5.42 -885.69907 0.04422 -0.466

5.45 -885.66314 0.05979 -0.530

=============================================

Results (Murnaghan EOS):

================================================

from P(V) from E(V) exp (300K)

------------------------------------------------

a0 5.342 5.345 5.36

B 181.8 181.3 169

dB/dP 4.5 4.4

==========================================

The CoSi2 work has been published in [Ref.M.1]

Cs006b

(Relativistic atomic calculation) Electronic configuration : for all s, p, d : 5s^2.00 5p^6.00 5d^0.25

r_c(s, p, d) = (2.0, 1.7, 2.0) a.u. , Q_c/q₃(s, p, d) = (0.80, 0.80, 1.00) , Local : p

Cu006a

Electronic configuration : for all s, p, d : 3d^9.00 4s^0.75 4p^0.25

r_c(s, p, d) = (2.0, 2.0, 2.5) a.u. , Q_c/q₃(s, p, d) = (1.15, 1.00, 1.20) , Local : s.

Test : See Cu006g

Cu006g

Electronic configuration : for all s, p, d : 3d^9.00 4s^0.75 4p^0.25

r_c(s, p, d) = (2.0, 2.0, 2.5) a.u. , Q_c/q₃(s, p, d) = (0.80, 0.95, 1.20) , Local : mixed (0.5, 0.5, 0.0)

Both s and p projectors reduced.

Tests :

Converged at 600 eV - q = 12 (MP points in the primitive cell).

Can be used safely at 500eV and q = 8, perhaps even below that.

Lattice parameters and bulk modulii

====================================

a0 B dB/dP

-----------------------------------------

Cu006a 3.650 153 4.8+-0.5

Cu006g 3.647 150 5.1+-0.4

Cu006i 3.658 145 4.8+-0.2

exp. 3.61 142 5.59

====================================

+1-1.5% in a0, +6% in B - pretty good.

Cu006i

Configuration : 3d^9.00 4s^0.75 5p^0.25

r_c(s, p, d) = (2.0, 2.0, 2.5) a.u. , Q_c/q₃(s, p, d) = (1.00, 1.20, 1.20) , Local : s

Comments :

This potential is used to demostrate the relexibility of choosing the r_c for a optimized Q_c-tuned pseudototential. For a given r_c, Q_c can be tuned in the way to allow the potential to deliver the best scatering under that r_c. This Cu006i is serve as a typical example of optimised Q_c-tuned potential with "standard" Kleiman-Bylander projectors construction.

Tests :

See Cu006g

Cu027a

Electronic configuration : 3d^9.00 4s^0.75 5p^0.25 (for all s, p, d potentials)

r_c(s, p, d) = 2.0 a.u. , Q_c/q₃(s, p, d) = (0.80, 1.00, 1.175) , Local : s

Tests : (Done by Dr S. Crampin.)

An 8x8x8 k point grid was used, with a simple-cubic

4 atoms unit cell, and a smearing of 1eV.

============================================

Ecut-off a(Angst) B(GPa) B'

--------------------------------------------

1000 3.60 166 5.0

650 3.59 163 5.4

LAPW 3.61 162

TM 3.60 160 5.1

expt. 3.61 142 5.28

============================================

The LAPW (all electron calc.), TM (Trouiller and Martin) and experimental

data are from the Trouiller Martin paper PRB43 1993 (1991).

There is little error introduced when one reduces the cut-off from 1000eV, when the absolute energy is converged to about 0.01eV, down to 650eV, when the absolute error is about 0.1eV.

F000

Electronic configurations : for s and p : 2s^2.00 2p^5.00 ; for d : 2s^1.25 2p^2.50 3d^0.25

r_c(s, p, d) = 1.8 a.u. , Q_c/q₃(s, p, d) = (0.95, 1.15, 1.00) , Local : d.

Fe002

Electronic configuration : for s, p and d : 3d^6.00 5s^1.00 5p^0.25

r_c(s, p, d) = 2.4 a.u. , Q_c/q₃(s, p, d) = (0.48, 0.87, 1.18), Local : mixed, with coefficients (0.3, 0.7, 0.0), sp projectors reducced

Hg000R

(Relativistic atomic calculation) Electronic configuration : for all s, p, d : 5d^9.00 6s^2.00 6p^0.25

r_c(s, p, d) = 2.4 a.u. , Q_c/q₃(s, p, d) = (0.40, 1.10, 1.14) , Local : s.

I000

Electronic configurations: for s and p : 5s^2.00 5p^5.00 ; for d : 5s^1.00 5p^3.75 5d^0.25

r_c(s, p, d) = 2.2 a.u. , Q_c/q₃(s, p, d) = (0.8, 0.9, 1.0) , Local component : (0.5, 0.5, 0.0)

s and p projectors reduced, d can be switched off.

In000

The In000 is designed to represent In⁺ character, in which case the r_c should not be too much larger than ionic radius of In in it's 1+ state. In order to achieve this we found the neutral configuration is not suitable because the reasonable r_c for it is too large. So we increase the inonicity of In till the r_c fall back to 2.5 a.u..

Electronic configuration for all s, p, d potentials: 5s^1.50 5p^0.25 5d^0.25

r_c(s, p, d) = 2.5 a.u. , Q_c/q₃(s, p, d) = (0.90, 0.95, 0.85) , the local potential is s.

K003

Electronic configurations : for s and p : 4s^0.75 2p^0.25

r_c(s, p) = 3.3 a.u. , Q_c/q₃(s, p) = (0.65, 0.65) , Local : s , non-local p component omitted.

K003 is a 3-term optimised pseudopotential. It was designed to be very economical when used in real space (non-local operation) code. This is because that it has only local potential (which is s), the s potential was made to be as close to p as possible, so that we may reasonably discard p projector under such approximation because the local (s) potential resemble (not very much) the p potential. The advantage of using such local-only potential in real-space code is that there is no need real space projector at all, which saves memory and speeds up the calculation.

Mg006

Configuration : 3s^2.0 (for s potential only)

r_c(s) = 2.0 a.u. , Q_c/q₃(s)=1.00 , Local potential : s.

(So there is no non-local component in this potential.)

Tests:

Chales Randel has used this Mg006 together with O020 in inner shell spectroscopy calculation.

Also see Chapter VI.

N001

Electronic configuration : for s and p : 2s 2.00 2p 3.00 ; for d : 2s 1.00 2p 1.75 3d 0.25

r_c(s, p, d) = (1.0, 1.4, 1.4) a.u. , Q_c/q₃(s, p, d) = (0.95, 1.065, 1.00) ,

Choice of local component : Minimum and no-crossing

Some extra infomations about N001:

Purpose:

This N001 is generated to be used for NO molecule interacting with metal surface and also we want to see if it can also be used in Ba2NH crystal. Ionicity (charge state) of N in both

cases are the thing one will be interested, so we can not use a large core which makes potential more reference configuration dependent.

Method :

Following the experience of generating and testing Carbon potential C009, and also due to the fact that they are both 2p element, we use the same strategy used in generating C009 (for more detail, please see relavent part of C009), namely 3-term optimisation with "minimum curvature" treatment at r_c.

Tests :

See C021.

Expectation :

We expect this N001 works reasonably good in melocule, and hope it give good decsription of Nitrogen at least in the range of N- to N+.

N010

Electronic configuration : for s and p : 2s 2.00 2p 3.00 ; for d : 2s 0.75 2p 2.00 3d 0.25

r_c(s, p, d) = 1.4 a.u. , Q_c/q₃(s, p, d) = (0.80, 1.05, 1.035) , Local : p . The d-projector is reduced.

Tests :

See C021

Na001

Electronic configuration : for s : 3s^1.00 ; for p and d : 3p^0.25 3d^0.25

r_c(s, p, d) = (2.2, 2.2, 2.5) a.u. , Q_c/q₃(s, p, d) = (0.80, 1.05, 0.60) , Local : s

The p and d non-local parts can be turned off (which is Na001a).

Ni001

Electronic configuration : for all s, p, d : 3d^8.00 4s^1.27 5p^0.73

r_c(s, p, d) = 2.0 a.u. ,Q_c/q₃(s, p, d) = (0.74, 0.97, 1.17) , Local : mixed (0.2, 0.8, 0.0)

The s and p projectors reduced

O020

Configuration: for s and p : 2s^2.00 2p^4.00 ; for d : 2s^1.00 2p^1.75 3d^0.25

r_c(s, p, d) = 1.8 a.u. , Q_c(s, p, d) = q₃ + 0.25*q₁ . Local : 0 0 0 (Mixed)

O020c

Electronic configuration : for s and p : 2s^2.00 2p^4.00 ; for d : 2s^1.00 2p^1.75 3d^0.25

r_c(s, p, d) = 1.8 a.u. , Q_c/q₃(s, p, d) = (1.00,1.085,1.075) . Local: p.

Projector reduced, this pseduopotential has only s non-local projector

O020a

Electronic configuration : for s and p : 2s^2.00 2p^4.00 ; for d : 2s^1.00 2p^1.75 3d^0.25

r_c(s, p, d) = 1.8 a.u. , Q_c/q₃(s, p, d) = (1.00,1.085,1.075) . Local: p.

O027

Electronic configurations: For s and p: 2s^2.00 2p^4.00 ; For d : 2s^1.00 2p^1.75 3d^0.25

r_c(s, p, d) = 1.4 a.u. , Q_c/q₃(s, p, d) = (0.40, 1.10, 1.00).

Loncal potential: mix of s, p, d, minimum. (0.0 0.0 0.0)

Tests :

Convergence test: MgO (premitive cell, 1 k-point, ion fixed)

=================================================

E_cut E_tot(o2001Mg006) E_tot(O027Mg006)

-------------------------------------------------

200 -416.6554883 -407.7530943

300 -438.0174302 -427.9129811

400 -443.3917391 -439.7956222

500 -444.3537604 -448.6898144

600 -444.3706999 -450.0756426

700 -444.3731004 -451.3899518

800 -444.3732176 -452.0890654

900 -444.3734519 -452.2232181

=================================================

The (suggested) lowest reliable E_cut for O027 is 500 eV.

(for o2001, 350~400 eV)

Molecule test: CO (in a 7A box, 500 eV) (use C009 with O027)

Expremental bond length (1.128 A):

TOTAL ENERGY IS -584.2688296

-------------------------------------------------------------------------

SIGTO 0.041554 0.041526 0.041475 0.000307 0.000302 0.000297

-------------------------------------------------------------------------

NSP ATOM a1 a2 a3 Fx Fy Fz

-------------------------------------------------------------------------

1 1 0.000000 0.000000 0.000000 -0.018158 -0.014563 -0.008307

2 1 0.093039 0.093039 0.093039 0.037726 0.018490 0.043561

-------------------------------------------------------------------------

Bond length 1% larger:

TOTAL ENERGY IS -584.2615490

-------------------------------------------------------------------------

SIGTO 0.042617 0.042740 0.042582 0.001592 0.001554 0.001613

-------------------------------------------------------------------------

NSP ATOM a1 a2 a3 Fx Fy Fz

-------------------------------------------------------------------------

1 1 0.000000 0.000000 0.000000 0.756317 0.756195 0.757012

2 1 0.093969 0.093969 0.093969 -0.756386 -0.766932 -0.753488

-------------------------------------------------------------------------

From above CO tests in box, one sees the converged force changes sign

across the zero, so teh equilibrium bond length should be between 0%~1%.

[This is just luck! 2~3% bond length error is rather common.]