SUMMARY Ch. 13 "THE METHOD OF PSEUDOPOTENTIALS"
"Inventors": Phillips and Kleinman 1959;
Harrison 1962, 1963, 1965, 1966;
13.1 OPW-based pseudopotentials
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Starting point of the theory is the expansion of a Bloch wave by the use
of orthogonalized plane waves [Eq. (13.1)].
The convergence of OPW is in many cases very good: a relatively small number
of OPW's is able to properly describe Bloch states (wavefunctions and energies).
The reason for that is that the OPW coefficients a_t(k) in Eq. (13.1) decrease
much faster with inceasing t than the corresponding PW coefficients
(see Fig. 13.1: "Comparison of the convergence of the 2s Bloch state in Li
at the Gamma point:
black curve: Moduli of expansion coefficients a_t^{PW} as a function of t
= the number of reciprocal-lattice vectors,
red curve: Moduli of expansion coefficients a_t^{OPW} as a function of t
= the number of reciprocal-lattice vectors.
That means: the first term of Eq. (13.1) represents a fast-converging plane-
wave expansion.
This expansion [Eq. (13.2)] is called the "pseudo wavefunction of the problem
on an OPW basis". The correspondence of this pseudo wavefunction chi-OPW
and the corresponding "true wavefunction" is given in Eq. (13.3):
this equation gives the principal idea of the pseudopotential method:
in order to get the "true wavefunction" psi_k, the pseudo wavefunction
chi_k(OPW) is reduced by all its "core function components":
Remember: |k,j>
defined by Eq. (12.1).
Comparisons of "true" wavefunctions with their corresponding "pseudo" wave-
functions are given in Figs. 13.2 and 13.3: especially in the neighborhood
of the atom, the pseudo wavefunction chi_k is significantly smoother than
the true wavefunction psi_k :
this is the reason for the fast convergence of chi_k.
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The main question is now the following: if |psi_k> is an eigenstate of the
Hamiltonian H to the eigenvalue E_k , is it possible to define a
"pseudo-Hamiltonian" H_P which has the pseudo-eigenstate |chi_k>
TO THE SAME EIGENVALUE?
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The derivation of such a pseudo-Hamiltonian is given on page 164 of the lecture
notes; the result of the "pseudo-Hamiltonian in the OPW representation" is given
in Eq. (13.4), and the corresponding OPW pseudopotential W^{OPW} (often called
the "Phillips-Kleinman potential") can be found in Eq. (13.5).
However, such a pseudopotential has some displeasing properties:
(*) it is an operator, not simply a function,
(*) it is non-local,
(*) it contains the eigenenergies E(k) to be calculated.
(*) The OPW pseudopotential is at least self-adjoint; in other cases, even this
property might not be valid.
(*) What concerns the OPW pseudopotential, its correspondence to a pseudo
wavefunction is not unique (see Appendix A):
"If |chi_k^{OPW}> is an eigenstate of H_P^{OPW} for the eigenvalue E(k),
all vectors Eq. (13.6) are also eigenstates of this pseudo-Hamiltonian
to E(k)."
Caption of Fig. 13.3: Schematic comparison of a wavefunction with a
corespondic pseudo wavefunction (above),
schematic comparison of a true potential with
a corresponding pseudopotential.
Quotation: R.M. Martin, Pseudopotentials in Electronic
Structure Theory (2004).
13.2 A general condition for pseudopotentials
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is derived in this section: in Eqs. (13.79) and (13.8), the true and the
pseudo Schroedinger equation are given, leading to the relation (13.9)
for the pseudo-Hamiltonian.
Starting from these equations, it is easy to prove the GENERAL CONDITION
< psi_k | W-V(r) | chi_k > = 0 . (13.10)
It is also easy to show that - by inserting Eq. (13.5) into Eq. (13.10) -
the Phillips-Kleinman potential fulfills this condition.
13.3 Pseudopotentials of the "OPW family"
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The enormous importance of the method of pseudopotentials, especially in
metal and semiconductor physics, is primarly not based on the OPW pseudo-
potential Eq. (13.5) but on the possibility to develop numerous versions
of this pseudopotential.
In a paper of R.M. Martin (Pseudopotentials in Electronic Structure Theory,
2004), one reads:
"A pseudopotential is any potential which does not have bound core states
but gives the same valence state energies than the "real" potential."
All pseudopotentials defined in the following have the same basis:
(*) The electrons can be seperated in the group of "core electrons" |k,j>
and the group of "valence electrons" |psi_k>,
(*) the |k,j> and |psi_k> belong to the same Hamiltonian, with the consequence
Eq. (13.11).
Starting from the OPW results (13.3) and (13.5), one defines now the more
general definitions for the pseudo wavefunction (13.12) and the corresponding
pseudopotential (13.13): in both equations, the bra = sum_j (E_k-E_j) =0
which results from the orthogonality Eq. (13.11).
(*) Is the pseudo wavefunction chi_k [Eq. (13.12)] an eigensolution
of the Schroedinger equation including the pseudopotential (13.13)?
This is proved on page 168 of the lecture notes.
Pseudopotentials defined by Eq. (13.13) are called "of the OPW family".
It is obvious that these potential operators are in general not self-
adjoint (with exception of the Phillips-Kleinman potential):
|k,j> < k,j|
13.4 General pseudopotentials
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In practice, we know from bandstructure calculations that pseudopotentials
can be formulated even more general than in case of the "OPW family",
according to the short definition of Phillips and Kleinman:
"Empirical pseudopotentials: Define potentials which give the desired bands".
In this lecture, it is not possible to give detailled answers to all
questions concerning the pseudopotenial method. Only a few important
remarks shall be given in the following:
A very general formulation for a pseudopotential has been proposed by
Austin-Heine-Sham [Eq. (13.14)]; the connections to the OPW family are
given by Eq. (13.15).
Now it can be asked for an "optimum pseudopotential" which is as smooth
as possible, leading to a PW expansion with best-possible convergence.
In fact such a potential can be found by miminalization of
int d^3r |nabla chi|^2 / int d^3r |chi|^2 ,
leading to the pseudopotential of Cohen and Heine (1961) [see Eq. (13.16)].
Although Cohen and Heine's potential is too complicated to be used in
practice, it has become useful as a starting point for numerous
developments on the pseudopotential field.
"Norm-conserving pseudopotentials" (Hamann, Schlueter, and Chiang 1979,
Kleinman and Bylander 1982) are very important up to our days:
their basic idea is demonstrated on Fig. 13.3 of this lecture note.
r_c is a so-called "cut-off radius", and both the pseudo-wavefunction
chi(r) and the pseudopotential have to obay the following rules:
(1) chi(r) = psi(r) for r >= r_c . (13.17)
(2) within the sphere with radius r_c , the pseudo-wavefunction
and the "true" wavefunction should have the same normalization
integral [see Eq. (13.18)].
The advantages of such a potential are obvious: in order to understand
this, have a further look on Fig. 13.2: the OPW function is correctly
normalized according to quantum mechanics, i.e., the integral of the
squared modulus of this function over the crystal volume Omega amounts
1.0.
This is clearly not the case for the "pseudo-OPW": if one normalizes
this function with respect to Omega (which is mathematically no
problem), this "normalized pseudo-OPW" does not agree with the true
OPW function for all valus of r .
In Fig. 13.3, the situation is completely different: as defined above,
the "norm-conserving pseudo-wavefunction" is identical to the true
wavefunction for all r>= r_c . Therefore, together with Eq. (13.18),
this function represents a quantummechanically correct approximation to
the true wavefunction.
Of course, chi(r) is only an approximation: as it can be seen in Fig.
13.4 , for r close to the nucleus, the true wavefunctions of electrons
in a silicon atom are quite different to the corresponding norm-conserving
pseudo-wavefunctions. Consider the l-dependent values of the cut-off
radii: for l=0 (3s), l=1 (3p), and l=2 (3d) electrons, r_c amounts to
1.73, 2.22, and 5.25 atomic units, respectively.
Such differences between true and pseudo wavefunctions will, of course,
also lead to different results if wavefunctions are used for calculating
various physical quantities. To study this, see Fig. 13.5 with
theoretically determined Compton profiles in silicon:
Fig. 13.5: Comparison of Compton profiles in Si along [100] and [110]
in k space. The dashed curves have been obtained by using
pseudo wavefunctions, the solid curves by using
"reconstructed wavefunctions".
The squares and triangles are experimental results, taken
from papers of Sakai (1989) and Gilat and Bharatiya (1975).
This figure can be found in Delaney et al., PRB 58, 4320 (1998).
As it can be seen, there are significant differences between the
measured and the theoretical Compton profiles obtained by the use of
pseudo wavefunctions.
However, during the last decades, various authors [see, e.g., Meyer
et al., J. Condens. Matter 7, 9201 (1995)] developed mathematical
methods that enable very effective reconstructions of true wavefunctions
from pseudo wavefunctions (compare the solid lines in Fig. 13.5).
One of the most important progresses on the field of pseudo potentials is
the development of so-called "ultrasoft pseudopotentials", based on the
work of Vanderbilt and co-workers [see, e.g. D. Vanderbilt, PRB 41, 7892
(1990)].
It is not possible to discuss these two important topics "reconstruction
of true wavefunctions from pseudo wavefunctions" and "ultrasoft pseudo-
potentials" in this lecture of the basics of bandstructure methods.
If you are interested in more details, have a look into the publications
of the following list (partly available from the website of my lecture):
[1] Vanderbilt_06.pdf (D. Vanderbilt, Rutgers Univ., 2006, at a
summerschool in Bangalore)
[2] Martin_04.pdf ("Pseudopotentials in Electronic Structure Theory")
Further lectures of Prof. Martin, Univ. of Illinois, about different
topics of Solid State Physics can be found on
www.mcc.uiuc.edu/summerschool/2001/Richard%20Martin/martin.htm
[3] Fuchs_03.pdf ("Pseudopotentials for ab initio electronic structure
calculations", 2003)
[4] Meyer_xx.pdf ("B. Meyer, ???, "The Pseudopotential PW Approach").