II

Optimised Pseudopotentials

with

Kinetic Energy Filter Tuning

We have developed an improved scheme for generating Optimised Pseudopotentials [1-3] which is more systematic and leads to a better insight. The control parameter Q_c connected with the kinetic energy of the pseudo wavefunction Y_l (r) is used in a new way to tune the pseudopotential. The scheme uses only three constraints and three spherical Bessel functions in the expansion of Y_l (r) inside the pseudising radius r_c, compared with four or more previously. The combined effect is that the fidelity of the pseudopotential as seen in the logarithmic derivative, can be improved in a simple and systematic way by tuning Q_c. The softness of the pseudopotential is also improved somewhat. The scheme opens the way to tailor-making pseudopotentials for specific requirements which will be particularly useful for large scale ab initio calculations.

II.1. Introduction

Soft and accurate pseudopotentials are essential for state-of-the-art large scale ab initio simulations for solids using plane-wave basis sets. This is in the context of periodic superlattice calculations based on Density Functional Theory with Local Density Approximation (LDA) for exchange and correlation with possibly a Generalised Gradient Approximation (GGA). Here, a "soft" pseudopotential means that a low energy cut-off, E_cut , can be used in the planewave expansion of the wavefunction, and the "accuracy" of a pseudopotential can be measured by the agreement between the logarithmic derivatives of pseudo and true wavefunctions in a certain energy range, which is an equivalent characterisation of the scattering property of a pseudopotential in terms of the phase shifts of different incoming waves. Even with the accuracy and efficiency achieved by a Car-Parrinello type of algorithm [4,5], the computational cost still requires a soft pseudopotential to make the largest calculations affordable.

While the advantage of the softness in a pseudopotential can not be over-emphasised, what is equally important is to have a flexible scheme to generate quickly and systematically a new pseudopotential appropriate to a new physical situation, such as short inter-atomic distances in some compound or at ultra high pressures. Another example would be some structural energy difference where errors cancel, so that one can compromise on convergence properties. A further situation might be one where the energy range over which the logarithmic derivative of the pseudo wavefunction has to be accurate is unusually narrow or wide. Whereas one standard transferable pseudopotential for a given chemical element may be satisfactory for many purposes, this will not be so for some calculations. On the contrary, the flexibility of balancing the accuracy (logarithmic derivative) and the efficiency (E_cut) of a pseudopotential in its construction will be useful, because then computing effort will not be wasted in achieving unnecessary precision. This utility can not be replaced by simply using an insufficient E_cut in exchange for a less accurate result because the quality of computed physical quantities falls very rapidly when decreasing E_cut below some point which is essentially determined by the way the pseudopotential is generated. A flexible scheme is therefore to be welcomed if it allows one to regulate the degree of approximation in different parts of a pseudopotential, such as the logarithmic derivative, E_cut and a large pseudising radius ("cut-off" radius) r_c to maximise the pseudopotential performance in a given problem without compromising the science.

In order to generate a new pseudopotential efficiently, it is a great help to understand how changes in the input parameters affect the pseudopotential that results. Such understanding also helps to one avoid an unrealistic choice of parameters. We present in this chapter a robust way of constructing pseudopotentials which addresses those essential points mentioned above, namely accuracy, softness, flexibility and understanding, through further improvements to the well established "Optimised Pseudopotentials" approach. [1-3]. We regard our method as an alternative to those very popular schemes proposed by Troullier and Martins [9], by Vanderbilt [19], and by Blöchl [20]. These schemes all aim to improve the softness and/or the accuracy of pseudopotentials.

Before we describe the technical detail of our current scheme, it is necessary to outline previous pseudopotential optimisation methods. The "Optimised Pseudopotentials" proposed by Rappe, Rabe, Kaxiras and Joannopoulos (RRKJ) [1,2] are recognised as very soft pseudopotentials. Their scheme is very suitable for transition metals and first-row elements which we are interested in and which usually need higher E_cut then other elements due to their very localised 3d (or 4d) or 2p valence electrons. Based on the RRKJ idea, a modified strategy of generating optimised pseudopotentials was suggested by Lin, Qteish, Payne and Heine (LQPH) [3] in order to simplify the numerical procedure. Although using different options and procedures, the basic formulations in LQPH and the current work are the same as those in the original RRKJ. In all these schemes the pseudo wavefunction Y_l (r) of angular momentum l is generated first, and then the pseudopotential V_l (r) is derived from it by inverting the Schrödinger equation [6]. The Y_l (r) is expressed in terms of some specially chosen spherical Bessel functions as follows :

, (1.1) ,

in which the j_l (q_i r) are spherical Bessel functions with (i-1) zeros for r < r_c, and j_l' (q_i r) their first derivative with respect to r. The f_l (r) is the proper all-electron atomic wavefunction and f_l' (r) its first derivative. Since we start with f_l (r_c) when generating a new pseudopotential, all the q_i are fixed once the r_c is chosen. The portion of the kinetic energy of the pseudo wavefunction due to the q > Q_c part of its Fourier components is denoted by DE_k (in atomic-Rydberg unit) :

(1.2a)

, (1.2b)

in which the Y_l (q) is the Fourier transform of the Y_l (r). The central idea of optimising a pseudopotential is that for a given Q_c, the coefficients a_i of Y_l (r) in (1.1) can be obtained by minimising DE_k in (1.2) with Lagrange multipliers constraining the normalisation and the continuity of the first and second derivatives of the pseudo wavefunction at r_c. Thus a smooth and norm-conserving pseudo wavefunction Y_l (r) can be determined. Incidentally, the continuity of the pseudo wavefunction Y_l (r) at r_c is not imposed explicitly because the optimisation procedure results in this condition being fulfilled automatically. This can be understood from the definition of the j_l (q_i r) in (1.1). We know that when the constrained minimisation is successful, one has Y_l'(r_c) = f_l'(r_c) : therefore

, (1.3)

which gives the required continuity of Y_l (r) from the continuity of Y_l'(r_c) due to the special choice of expansion functions in (1.1). Although the mathematical scheme is essentially the same, there are some differences in the way the procedure is used by the previous authors and in the present work. In the RRKJ method, typically ten or more spherical Bessel function terms in (1.1) are used, with Q_c being varied iteratively such that the DE_k is minimised to a pre-chosen tolerance, say 1 mRyd. It appears that using Q_c in that way has the advantage of controlling the quality of the total energy convergence with respect to the energy cut-off used in the calculations. In the LQPH method, the number of spherical Bessel function terms is fixed to be four so that there are four a_i coefficients for the Y_l (r) to be determined, making the number of free parameters equal to the number of constraints, namely norm-conservation, continuity of the first and second derivatives of Y_l (r) at r_c, and minimisation of DE_k. Efficient numerical routines exist for such a problem, and together with the reasonably small number (four) of terms in Y_l (r), it helps to stabilise the numerical procedure. In addition to using just four spherical Bessel function terms in Y_l (r), the LQPH method always sets Q_c equal to the largest q_n , i.e. q₄ , which avoids the variation of Q_c and therefore makes the numerical procedure significantly simpler.

We recognise the success of the above mentioned schemes, but the consequences of some of their detailed assumptions, such as the value of Q_c, the number of terms and the choice of constraints, were not fully clear to us. In particular, the role of Q_c in optimising pseudopotentials attracted our attention. From the definition in (1.2), Q_c can be regarded as a kinetic energy filter controlling the constrained minimisation of the kinetic energy of Y_l in the range q > Q_c. If the minimisation is effective, the resulting k-space pseudo wavefunction Y_l (q) will be restricted as far as possible to the range 0 < q < Q_c, which will subsequently determine the analogous behaviour of the pseudopotential V_l^ps(q) in k-space in solid state applications. A unique correspondence is therefore likely to exist between Q_c and the pseudopotential in k-space V_l^ps(q,Q_c), which of course also applies to Q_c and V_l^ps(r,Q_c) in r-space due to the duality of r and k spaces. Most importantly, the scattering property of such a pseudopotential should also depends on Q_c in some simple manner because it is all in the characteristics of V_l^ps(q,Q_c). In the current scheme we therefore vary Q_c to control the phase shift, i.e. logarithmic derivative, as will be demonstrated in Section 2. We shall call this "Q_c tuning" and it will be the crux of the present work. Additionally, from the argument above we expect the Q_c to correspond roughly to E_cut. The Q_c therefore controls both the scattering property and the energy convergence of our new optimised pseudopotential.

On investigating the choice of constraints, we realised that it is not necessary to impose strictly the continuity of Y_l''(r) at r_c because minimisation of DE_k already more or less constrains the higher derivatives of Y_l (r) by reducing its high q amplitude. Moreover, dropping unnecessary constraints means that a less restricted and more efficient minimisation can be performed. Thus not only can a softer pseudopotential be obtained but also the resulting pseudopotential will be more sensitive to the choice of Q_c, which enhances the controllability of the pseudopotential by Q_c. We have, in fact, tried applying the Q_c tuning within the four-term/four-constraint framework, and found that both logarithmic derivative and the shape of the pseudopotential do not vary systematically with respect to Q_c, which is presumably due to the extra (unnecessary) constraint which somehow restricts the effect of Q_c tuning.

However with the use of three terms and three constraints, we found that the logarithmic derivative and pseudopotential varied smoothly with Q_c so that one can use Q_c tuning efficiently. This is the main reason for preferring the three-term/three-constraint framework. Incidentally, keeping the number of terms and unknown coefficients in (1.1) equal to the number of constraints is very helpful in maintaining a stable numerical procedure, as found by LQPH. We can also see why the pseudopotential becomes somewhat softer. To a first approximation, E_cut = Q_c² Ryd. if Q_c is in atomic units as will be assumed hereafter, and Q_c is approximately the maximum q_i. Thus omitting the term j_l(q₄r) reduces E_cut, because j_l(q₄r) has the maximum number of nodes and hence the highest Fourier components of the j_l(q_ir).

To summarise : using a three-term expansion in (1.1) and three constraints gives a stable numerical procedure allowing a good flexibility in the pseudo wavefunction. It combines well with tuning Q_c to optimise the accuracy of the pseudopotential, which is the main purpose of the present work. The effect of Q_c tuning and relevant technical points are discussed in Section 2. In Section 3 some solid state tests of the pseudopotentials will be presented, which is followed by a discussion and conclusion as Section 4.

II.2. The Q_c Tuning Method

To generate a pseudopotential in the current scheme, as in all ab initio pseudopotential generating procedures, an all-electron LDA or GGA atomic calculation is first performed to obtain all the atomic orbitals of a selected configuration : in the present work we just use the LDA. The procedure described in Section 1 is then implemented with three terms in (1.1) and the three constraints already discussed, while Q_c remains as an adjustable input parameter. In Fig.1 we demonstrate the effect of varying Q_c in the current scheme on the oxygen 2p pseudopotential. Three different Q_c were used to generate the corresponding pseudopotentials, and the logarithmic derivative was tested on these pseudopotentials. We can see that for a given atomic configuration and pseudising radius, there is a certain value of Q_c which yields the best agreement with the logarithmic derivative, in this case that shown in Fig.1(b). We note that the logarithmic derivative curve of the pseudo wavefunction for a larger and a smaller Q_c deviate from the curve with the best possible Q_c in opposite directions, as shown in Fig.1(a) and Fig.1(c). In Fig.1 we also see that the shape of the pseudopotential changes with Q_c, which can be regarded as the reason why the scattering properties of the resulting pseudopotentials are different. The monotonic correspondence between the variation of (a) the Q_c, (b) the shape of the pseudopotential and (c) the logarithmic derivative of the pseudopotential is the most important feature in the current scheme. This feature enables us to establish a systematic procedure for updating Q_c towards the best results judged by the following criterion. As mentioned in Section 1, Q_c controls the softness of a pseudopotential as well as its accuracy because it affects both the E_cut and the logarithmic derivative of the wavefunction. If transferability is a higher priority in a particular application, then Q_c should be tuned to obtain the best match between the logarithmic derivatives of the pseudo and true wavefunctions. Depending on the application, one may require a good match over a wide range of energy for broad bands or only a narrower range in the case of narrow bands. If a satisfactory match can be obtained for a range of Q_c, then the smallest Q_c should be used to achieve the lowest E_cut.

By removing the constraint on the second derivative of the wavefunction at r_c , we allow our pseudopotential to have a discontinuity there because the kinetic energy, which is proportional to Y ^{' '}(r), is discontinuous across r_c, and hence so is the potential. Although a large discontinuity in a pseudopotential can damage its scattering property, in the current scheme the best possible Q_c is chosen to give the best fit of the logarithmic derivative, which thus guarantees that the discontinuity is harmless. This is also consistent with our observation that whenever the logarithmic derivative agreement is satisfactory, the discontinuity is always small. This can also be understood from the fact that in the current scheme the high q components of the pseudo wavefunction are reduced as much as possible, both because they are expanded using the least possible number of spherical Bessel functions and also because of the minimising procedure imposed on DE_k in (1.2). A small discontinuity at r_c will give the pseudopotential some very high Fourier components of small weight spread over a wide range of q, which will not affect calculation significantly if they are cut off by E_cut. Incidentally, it is generally the case that the Q_c yielding the best fit to the logarithmic derivative of the true potential need not be the Q_c that minimises the discontinuity of the pseudopotential, even though these two Q_c are usually close. We regard the quality of scattering being optimised by the Q_c as being more significant than the existence of the discontinuity.

The harmlessness of a small discontinuity is further confirmed by our experience that good agreement is obtained between the results of super-cell calculations using both k-space and r-space versions [7] of the same pseudopotential expressed in Kleinman-Bylander form [8]. To convert the k-space pseudopotential to one in r-space we use the method of King-Smith et al. [7]. It will modify the original pseudopotential in a way that depends on E_cut in minimising the aliasing error of the Fast Fourier Transform in planewave supercell calculations. The discontinuity at r_c in the original pseudopotential is smoothed out by the transformation. The fact that both the original and the transformed pseudopotentials gave almost identical results for the relaxation and energy of structures shows that the high q feature at r_c is irrelevant to the super-cell results when a reasonable E_cut is used.

II.3. Generation and Test of Some Pseudopotentials

Although the logarithmic derivative test gives a useful indication about the quality of a pseudopotential, there is no precise criterion of how good the agreement should be for a particular physical application. Also the test is evaluated at a given r outside the pseudising radius r_c, which does not give us the information whether the r_c is small enough for the frozen core approximation to be valid for the given application. A solid state calculation is therefore always necessary for a serious test of a pseudopotential.

To test the pseudopotential generated by our current scheme, we have chosen some bulk properties of Cu metal because it is a popular case tested by other authors [1, 9]. We follow the RRKJ paper in using a slightly ionised Cu configuration 3d ⁹ 4s ^0.75 4p ^0.25 from which to generate the pseudopotential. After generating the pseudopotential for each l as described in Section 2, it was converted to Kleinman-Bylander [8] form with the s-potential chosen as the local potential. Two Cu pseudopotentials were prepared (Fig.2), one with smaller d-core and the other a larger d-core, with r_c(s,p,d) = (2.0, 2.0, 2.0) a.u. and r_c(s,p,d) = (2.0, 2.0, 2,4) a.u. respectively. The Q_c for these two potentials are Q_c(s,p,d) = (3.17, 4.66, 6.47) and Q_c(s,p,d) = (3.17, 4.66, 5.17). In most cases, we found it useful to choose q₃ as the initial guess for Q_c from which to start the tuning, so that it is convenient to express the final Q_c in terms of the ratio Q_c/q₃. For the Cu pseudopotentials in this section, this becomes Q_c/q₃(s,p,d) = (0.8, 1.0, 1.175) and Q_c/q₃(s,p,d) = (0.8, 1.0, 1.2).

The Cu pseudopotential with the smaller d-pseudocore (r_c = 2.0 a.u.) allows our results to be compared directly with those of other popular schemes in the literature [1,9], while we shall use the one with a big d-pseudocore to demonstrate the flexibility of using Q_c-tuning to generate a pseudopotential with a larger r_c. Although a pseudopotential with larger r_c is always softer, it may not be accurate enough. In our current scheme we can tune the value of Q_c so that we obtain a good logarithmic derivative even for such a large r_c.

For calculating the bulk properties of Cu metal, an 8 by 8 by 8 Monkhorst-Pack k-point grid [10] was used for a simple-cubic unit cell containing four atoms. With such a coarse grid of k-points, smearing of the occupation function at the Fermi level of 1eV was needed, and the energy was corrected appropriately [11]. In the case of the small d-core pseudopotential, the convergence test was done and the sudden drop of the total energy in a super-cell calculation was found to occur at 650 eV where absolute convergence to about 0.1 eV per atom is reached (Fig.3). To justify the results obtained at E_cut = 650 eV, a similar calculation was also performed at 1000 eV where the total energy converged to within 0.01 eV per atom, and the results for the bulk properties, as shown in Table.1, were found to be essentially the same. This is consistent with our experience that the E_cut that gives the calculated total energy converged to around 0.1 eV per atom is usually high enough for reliable solid state bulk properties. In the case of the pseudopotential with a large d-core, the convergence test was also done (Fig.3) and we chose E_cut = 500 eV to run the simple bulk property tests which are shown as the third line in Table.1. As one can see from the table, the overall result is satisfactory in comparison with experiment and other computational methods. We understand that the valid comparison is with the all-electron calculation because we are testing the pseudising, not the accuracy of LDA.

II.4. Discussion and Conclusion

In Section 2 we described how Q_c may be varied to obtain the best fit to the logarithmic derivative of the original potential, and we turn now to look at how the variation of Q_c manifests itself in the resultant pseudopotential. Fig.1 shows that the main effect of varying the Kinetic Energy Filter parameter Q_c is to change the depth of the pseudopotential in r-space. One can interpret qualitatively the effect of the optimisation on the shape of a pseudopotential from an r-space view point, which is useful when using Q_c to regulate the shape of the pseudopotential. If Q_c is set to be relatively small, this pushes Y_l (r) in the direction of having lower Fourier components, which means having lower kinetic energy inside r_c. Since the energy eigenvalue is fixed and is equal to the kinetic energy plus potential energy, the low kinetic energy implies a rather shallow (weak) pseudopotential. On the other hand using a higher Q_c results in a deeper pseudopotential as shown in Fig.1(c). If Q_c is reduced even further, the pseudopotential becomes even shallower (weaker) and a barrier will be raised near r_c as a result of the norm-conserving constraint so that the pseudopotential preserves the correct amount of charge within the pseudo-core region. Such a barrier may look strange but experience shows it does not affect E_cut or the accuracy of the pseudopotential in solid state tests provided the logarithmic derivative fits well.

The effect of Q_c-tuning on the shape of a pseudopotential also depends on other factors. In the case of nodeless orbitals such as 2p and 3d, the pseudopotentials are highly attractive because there is no "Cancellation Effect" from inner shells in the sense of conventional pseudopotential theory [12]. Optimising these pseudopotentials therefore means shifting the electrons outward from the centres of the atoms. On the other hand, in the case of (soft) pseudopotentials that do have a cancellation effect from inner shells, using a smaller Q_c means spreading the charge distribution inwards towards the centres of the atoms, which serves to lower the magnitude of the originally repulsive (or weakly attractive) pseudopotentials at r = 0, but has less effect on their shape near r_c. Such a trend can be used to systematically regulate the shape of a pseudopotential by tuning Q_c.

The current scheme has been used to generate a significant number of pseudopotentials for a wide range of applications, some of which have already been published, namely those for Co [13]; Ge [14]; C, O and Pd [15]; Cu and Cl [16].

In summary, therefore, we have introduced an improved scheme for generating Optimised Pseudopotentials. The Q_c parameter is used in a new way, and is tuned to give as accurate a pseudopotential as possible, meaning a good match to the all-electron logarithmic derivative of the wave function over a suitably wide range of energy. The continuity constraint of Y_l''(r) at r_c is dropped and the number of terms in the expansion (1.1) is also reduced to three, to remain equal to the number of constraints in order to give a numerically stable algorithm. Dropping the constraint on continuity of the Y_l''(r) means the pseudopotential has a discontinuity at r_c, but in practice the Q_c is tuned in our scheme to match the logarithmic derivative which always makes the discontinuity small, so that it does not adversely affect the accuracy or the softness of the pseudopotential. In some sense the dropping of one constraint allows the pseudo wavefunction (and hence pseudopotential) greater freedom for optimisation with regard to accuracy and convergence properties.

A most important point is that the generated pseudopotential and the corresponding logarithmic derivative vary with the chosen Q_c parameter in a systematic way. One therefore has a well controlled situation for generating and improving a pseudopotential for any given physical application, depending on the required balance between E_cut, the accuracy of the pseudopotential and the width of the energy range over which it has to be accurate. This is important for many calculations. Moreover we have shown how one can physically understand the connection between Q_c and the shape of the pseudopotential, which helps one to operate the scheme systematically and efficiently. The scheme represents a further significant step toward generating systematically good pseudopotentials for a wide variety of physical systems.

Acknowledgements

We are grateful to Dr. K. Rabe for the very useful discussion on the use Q_c at an early stage of this work. We also want to thank Dr. R. J. Needs for his interesting comments on basis set expansion for pseudo wavefunctions. I also want to thank Dr V. Milman and Dr S. Crampin for collaborating with me in doing the Cu tests.

References

[1] A. M. Rappe, K. M. Rabe, K. Kaxiras and J. D. Joannopoulos, Phys Rev B 41, 1227 (1990)

[2] A. M. Pappe and J. D. Joannopoulos, in Computer Simulation in Material Science, edited by M. Mayer and V. Pontikis, pp. 409-422 (Kluwer, Dordrecht, 1991)

[3] J. -S. Lin, A. Qteish, M. C. Payne and V. Heine, Phys Rev B 47, 4174 (1993)

[4] R. Car and M. Parrinello, Phys. Rev. Lett 55, 2471 (1985)

[5] M. C. Payne, M. P. Teter, D. C. Allen, T. A. Arias and J. D. Joannopoulos, Rev. Mod. Phys. 64, 1045 (1992)

[6] D. R. Hamann, M. Schluter and C. Chiang, Phys. Rev. Lett 43, 1494 (1979)

[7] R.D. King-Smith, M.C. Payne and J-S. Lin, Phys. Rev. B 44, 13063 (1991)

[8] L. Kleinman and D. M. Bylander, Phys Rev. Lett. 4, 1425 (1978)

[9] N. Troullier and J. L. Martins, Phys. Rev. B 43, 1993 (1991)

[10] H. J. Monkhorst and J. D. Pack, Phys. Rev. B 13, 5188 (1976)

[11] M. J. Gillan, J. Phys : Condensed Matter 1, 689 (1989);

A. De Vita, Ph.D. Thesis, Keele University, 1992, and A. De Vita et al., to be published.

[12] V. Heine, Pseudopotential Concept, Solid State Physics Vol.24 (1970)

[13] V. Milman, M. H. Lee, and M. C. Payne, Phys Rev B 49, 16300 (1994)

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Tables

TABLE.1. The solid state bulk test of Cu pseudopotentials, comparing the results of the present calculations (first three lines) with those from another pseduopotential (TM [9]) and from an all-electron calculation (LAPW), and with experiment : for lattice constant a, bulk modulus B and B' the pressure derivative of bulk modulus fitted from the equation of state. Percentage errors relative to experimental value are given in brackets.

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

Type E_cut(eV) a(Å) B(GPa) B'

____________________________________________________________________________

r_c(d)=2.0 a.u. 1000 3.60 <-0.3%> 166 <17%> 5.0 <-5.3%>

r_c(d)=2.0 a.u. 650 3.59 <-0.6%> 163 <15%> 5.4 < 2.3%>

r_c(d)=2.4 a.u. 500 3.66 < 1.4%> 145 < 2%> 4.8 <-9.1%>

r_c(d)=2.3 a.u.^a 982 3.60 160 5.1

LAPW^b 3.61 162

Experiment^c 3.61 142 5.28

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

a Ref.9

b Ref.17

c Ref.18

Figure Captions

FIG.1. Oxygen 2p pseudopotential showing its variation with Q_c. Upper panels, V_l=1(r) : lower panels, logarithmic derivatives of the true potential (dashed line) and pseudopotential (solid line). (a) Q_c/q₃=0.98 (b) Q_c/q₃=1.15 (c) Q_c/q₃=1.20.

FIG.2. The s, p, d pseudopotentials for Cu. (a) Small r_c=2.0 a.u. for l=0,1,2. (b) r_c=2.0 a.u. for l=0,1 but larger r_c=2.4 a.u. for l=2. (s: dashed line, p: dot-dashed line, d: solid line)

FIG.3. Convergence of total energy per atom of copper metal with respective to the cut-off energy for two pseudopotentials with r_c=2.5 a.u. (solid line) and r_c=2.0 a.u. (dashed line).