Diagnostic attribution of the cohesive-energy and phonon discrepancies of the Topp-Hopfield cosine pseudopotential for sodium

Topp and Hopfield (Phys. Rev. B 7, 1295 (1973)) introduced a local 'chemically motivated' pseudopotential for sodium - a cosine form inside a core radius matched to a Coulomb tail, fitted to atomic ground and excited states in order to reproduce the valence charge distribution - and applied it, within a uniform-electron-gas treatment with Nozières-Pines correlation and the Overhauser fit to the Singwi-Tosi-Land-Sjölander dielectric function, to the cohesive energy and the [111] phonon dispersion of bcc sodium. Two internal failures remained open: (a) a roughly 0.09 eV inconsistency between the virial-theorem and electrostatic self-energy routes to the cohesive energy, and (b) phonon errors of 5-10 percent versus the neutron data of Woods et al. that the improved potential did not reduce, with the candidate causes being the correlation-energy interpolation, the dielectric function (especially at small momentum transfer), or the breakdown of first-order (linear) screening for a potential that strongly depletes the core charge; the transferability of the potential to lower-symmetry environments such as molecules or strained sodium was left untested (background context only, not part of the required task below). The surrounding research program was superseded by ab initio norm-conserving pseudopotentials, nonlinear core corrections, and density-functional perturbation theory, which achieve accurate cohesive and vibrational properties of sodium by other means, while the specific diagnostic attribution for the cosine potential has never been carried out and remains open.

Background

In the pseudopotential approximation, the chemically inert inner electrons and the strong ionic Coulomb potential experienced by the outer (valence) electrons of an atom are replaced by a weak effective potential acting only on the valence electrons; because the effective potential is weak, the valence electrons in a simple metal can be treated nearly as a free-electron gas weakly perturbed by the periodic pseudopotential. Local 'model potentials' of the late 1960s, such as the flat-bottom optimized model potential of Shaw, were fitted to atomic ground-state energies but not to the excited-state spectrum; Topp and Hopfield (1973) noted a theorem (their Appendix) that for a local potential with a Coulomb tail, the valence charge inside the core radius is fixed by the energy derivative of the logarithmic derivative of the wave function at the core radius, so that fitting all discrete eigenvalues of an angular-momentum channel is what fixes the normalization of the wave function - and hence the charge distribution - in the outer, chemically relevant region. To exploit this they proposed a cosine potential $V(r)=V_0\cos(kr)+C$ for $r\le r_c$ matched in value and slope to $-1/r$ outside ($r_c=3.0$ a.u., $V_0=0.1790$, $k=1.224$ a.u.$^{-1}$, $C=-0.179$ in the units of the paper), fitted to the sodium term values (the discrete energies of the valence excited states; the core radius was chosen as a compromise between the fits to the $s$-like and $p$-like states of the atom), and showed that its ground-state wave function in the tail region is closer to the Hartree-Fock all-electron result than the Shaw potential's, with an atomic polarizability (155 a.u.) more accurate as well. For the solid they used the uniform-electron-gas (jellium) method: the valence electrons contribute a kinetic energy $1.105/r_s^2$, an exchange energy $-0.458/r_s$, and the Nozières-Pines interpolation formula for the correlation energy of the electron gas, $E_\text{corr}=-0.0575+0.0155\ln r_s$ hartree per electron (equivalently $(-0.115+0.031\ln r_s)$ rydberg), where $r_s$ is the mean inter-electron spacing; the periodic part of the ionic pseudopotential is screened in first-order (linear-response) fashion, the induced charge density being written in terms of the Fourier coefficients $V_G$ of the potential and a wave-vector-dependent static dielectric function $\epsilon_G$ containing exchange and correlation ('local-field') effects - specifically Overhauser's polynomial approximation to the dielectric function of Singwi, Tosi, Land, and Sjölander. Two formally equivalent routes to the cohesive energy of body-centered-cubic (bcc) sodium were evaluated: the virial theorem, which relates the total energy to the kinetic energy and a core-region integral over $[r\partial V/\partial r+V]\rho$, and the direct electrostatic self-energy of the lattice plus the redistribution energy of the screening charge. For the lattice dynamics, the dynamical matrix (the matrix whose eigenvalues at each wave vector give the squared phonon frequencies) was built from the same linearly screened interaction and compared with the inelastic-neutron-scattering dispersion curves of Woods, Brockhouse, March, Stewart, and Bowers. The outcome: a minimum of the self-energy curve giving a cohesive energy of about 1.09 eV, but a systematic disagreement of order 0.09 eV between the two routes - called by the authors a measure of the uncertainty in the then-current electron-gas understanding - and a [111] phonon spectrum with the same 5-10 percent errors as the Shaw potential, indicating that a correct equilibrium charge distribution does not by itself guarantee correct interatomic force constants. Prakash and Joshi (1969) had already shown, in contemporaneous work on normal metals, that the choice among rival dielectric screening treatments materially changes calculated phonon frequencies, and Brovman and Kagan's multiparticle theory (1974) showed that many-ion (effectively nonlinear) contributions beyond the pair interaction are quantitatively decisive for the phonons of metals including sodium; the diagnostic question for the specific cosine potential, however, was posed and left open by Topp and Hopfield. The later mainstream development abandoned fitted local model potentials screened by approximate dielectric functions in favor of ab initio pseudopotentials: Hamann, Schlüter, and Chiang's norm-conserving pseudopotentials (1979) build the charge-normalization constraint of Topp and Hopfield's appendix theorem into potentials derived self-consistently from the atom (making screening of the solid self-consistent within density-functional theory rather than linear in an approximate $\epsilon_G$); Louie, Froyen, and Cohen's nonlinear core corrections (1982) treat the nonlinear core-valence charge redistribution that first-order screening misses; Ceperley and Alder's quantum Monte Carlo benchmark (1980) - a stochastic, sampling-based solution of the many-electron problem - replaced interpolation formulas like Nozières-Pines as the source of electron-gas correlation energies; and density-functional perturbation theory, reviewed by Baroni, de Gironcoli, Dal Corso, and Giannozzi (2001), computes phonons from the self-consistent response to ionic displacements, as in the accurate modern lattice-dynamics calculations of sodium exemplified by Lepeshkin, Magnitskaya, and Maksimov (2009). A direct continuation of the 'good local pseudopotential for simple metals' agenda survives in orbital-free density functional theory - a formulation in which the electron kinetic energy is written as a functional of the electron density alone, with no Kohn-Sham orbitals, so that the electron-ion interaction enters only through a local potential - where local pseudopotentials for Li, Na, Mg, Al, and Si are constructed by inversion of the Kohn-Sham equations (the effective one-electron equations of density-functional theory) in a bulk crystal or by fitting (Zhou, Wang, and Carter 2004; Huang and Carter 2008; Legrain and Manzhos 2015), and historical reviews (Schwerdtfeger 2011) cite Topp and Hopfield as an early chemically motivated pseudopotential - but none of these works revisits the 1973 cosine potential itself.

Problem Statement

For the cosine pseudopotential of Topp and Hopfield for sodium, $V(r)=V_0\cos(kr)+C$ for $r\le r_c$ matched continuously in value and slope to $-1/r$ for $r\ge r_c$ with the parameters of the source paper ($r_c=3.0$ a.u., $V_0=0.1790$, $k=1.224$ a.u.$^{-1}$, $C=-0.179$ in the paper's units), determine the origin of the two internal failures of the 1973 electron-gas + linear-response treatment of bcc sodium: (a) the reported discrepancy of order 0.09 eV between the virial-theorem and electrostatic self-energy calculations of the cohesive energy as a function of lattice constant, and (b) the reported 5-10 percent errors of the calculated [111] phonon dispersion relative to the neutron data of Woods et al., which the cosine potential did not improve over the Shaw flat-bottom potential. The attribution must be established by controlled calculation rather than by plausibility argument, and must discriminate among the candidate causes identified in connection with the source: errors in the Nozières-Pines interpolation formula for the electron-gas correlation energy ($E_\text{corr}=-0.0575+0.0155\ln r_s$ hartree per electron, equivalently $(-0.115+0.031\ln r_s)$ rydberg), inaccuracies in the wave-vector-dependent dielectric function used for screening (Overhauser's polynomial approximation to the Singwi-Tosi-Land-Sjölander function, particularly at small momentum transfer where exchange and correlation are poorly constrained), and the breakdown of first-order (linear-response) screening itself for a potential that depletes the core-region charge as strongly as the cosine potential does - including the nonlinear electrostatic effects in the charge redistribution that the source left unexamined. This candidate list is not closed: attribution to other origins - for example the numerical convergence of the reciprocal-lattice sums, the treatment of the $G=0$/Ewald term, an inconsistent use of the linear-response density inside $r_c$ in the virial integral, or the uncertainty of the 90 K experimental reference - is equally admissible. Resolution requires a quantitative error budget that attributes each discrepancy to the dominant cause(s), whether among the named candidates or among the additional origins; any origin not found to be a dominant cause need be excluded only insofar as it could absorb the residual.

References

1. W. C. Topp and J. J. Hopfield, 'Chemically Motivated Pseudopotential for Sodium', Physical Review B 7, 1295-1303 (1973). DOI: 10.1103/PhysRevB.7.1295, https://doi.org/10.1103/PhysRevB.7.1295
2. P. Nozières and D. Pines, 'Correlation Energy of a Free Electron Gas', Physical Review 111, 442-454 (1958). DOI: 10.1103/PhysRev.111.442, https://doi.org/10.1103/PhysRev.111.442
3. K. S. Singwi, M. P. Tosi, R. H. Land, and A. Sjölander, 'Electron Correlations at Metallic Densities', Physical Review 176, 589-599 (1968). DOI: 10.1103/PhysRev.176.589, https://doi.org/10.1103/PhysRev.176.589
4. A. W. Overhauser, 'Simplified Theory of Electron Correlations in Metals', Physical Review B 3, 1888-1898 (1971). DOI: 10.1103/PhysRevB.3.1888, https://doi.org/10.1103/PhysRevB.3.1888
5. R. W. Shaw, 'Local Model Potentials for Metals: Limitations and Applications', Physical Review B 5, 4742-4755 (1972). DOI: 10.1103/PhysRevB.5.4742, https://doi.org/10.1103/PhysRevB.5.4742
6. A. D. B. Woods, B. N. Brockhouse, R. H. March, A. T. Stewart, and R. Bowers, 'Crystal Dynamics of Sodium at 90°K', Physical Review 128, 1112-1120 (1962). DOI: 10.1103/PhysRev.128.1112, https://doi.org/10.1103/PhysRev.128.1112
7. A. Rothwarf, 'Virial theorem and cohesive energy of monovalent metals', Physics Letters A 29, 154-155 (1969). DOI: 10.1016/0375-9601(69)90086-3, https://doi.org/10.1016/0375-9601(69)90086-3
8. Satya Prakash and S. K. Joshi, 'Comparison of Dielectric Screening Methods Used in Phonon-Frequency Calculation of Normal Metals', Physical Review 185, 913-923 (1969). DOI: 10.1103/PhysRev.185.913, https://doi.org/10.1103/PhysRev.185.913
9. E. G. Brovman and Yu. M. Kagan, 'Phonons in nontransition metals', Soviet Physics Uspekhi 17, 125-152 (1974). DOI: 10.1070/PU1974v017n02ABEH004329, https://doi.org/10.1070/PU1974v017n02ABEH004329
10. D. R. Hamann, M. Schlüter, and C. Chiang, 'Norm-Conserving Pseudopotentials', Physical Review Letters 43, 1494-1497 (1979). DOI: 10.1103/PhysRevLett.43.1494, https://doi.org/10.1103/PhysRevLett.43.1494
11. D. M. Ceperley and B. J. Alder, 'Ground State of the Electron Gas by a Stochastic Method', Physical Review Letters 45, 566-569 (1980). DOI: 10.1103/PhysRevLett.45.566, https://doi.org/10.1103/PhysRevLett.45.566
12. S. G. Louie, S. Froyen, and M. L. Cohen, 'Nonlinear ionic pseudopotentials in spin-density-functional calculations', Physical Review B 26, 1738-1742 (1982). DOI: 10.1103/PhysRevB.26.1738, https://doi.org/10.1103/PhysRevB.26.1738
13. S. Baroni, S. de Gironcoli, A. Dal Corso, and P. Giannozzi, 'Phonons and related crystal properties from density-functional perturbation theory', Reviews of Modern Physics 73, 515-562 (2001). DOI: 10.1103/RevModPhys.73.515, https://doi.org/10.1103/RevModPhys.73.515
14. S. V. Lepeshkin, M. V. Magnitskaya, and E. G. Maksimov, 'Lattice dynamics and melting features of Li and Na', JETP Letters 89, 586-591 (2009). DOI: 10.1134/S0021364009110137, https://doi.org/10.1134/S0021364009110137
15. Baojing Zhou, Y. A. Wang, and Emily A. Carter, 'Transferable local pseudopotentials derived via inversion of the Kohn-Sham equations in a bulk environment', Physical Review B 69, 125109 (2004). DOI: 10.1103/PhysRevB.69.125109, https://doi.org/10.1103/PhysRevB.69.125109
16. Chen Huang and Emily A. Carter, 'Transferable local pseudopotentials for magnesium, aluminum and silicon', Physical Chemistry Chemical Physics 10, 7109 (2008). DOI: 10.1039/b810407g, https://doi.org/10.1039/b810407g
17. Fleur Legrain and Sergei Manzhos, 'Highly accurate local pseudopotentials of Li, Na, and Mg for orbital free density functional theory', Chemical Physics Letters 622, 99-103 (2015). DOI: 10.1016/j.cplett.2015.01.016, https://doi.org/10.1016/j.cplett.2015.01.016
18. Peter Schwerdtfeger, 'The Pseudopotential Approximation in Electronic Structure Theory', ChemPhysChem 12, 3143-3155 (2011). DOI: 10.1002/cphc.201100387, https://doi.org/10.1002/cphc.201100387
19. M. Fuchs and M. Scheffler, 'Ab initio pseudopotentials for electronic structure calculations of poly-atomic systems using density-functional theory', Computer Physics Communications 119, 67-98 (1999). DOI: 10.1016/S0010-4655(98)00201-X, https://doi.org/10.1016/S0010-4655(98)00201-X
20. M. Dolg and X. Cao, 'Relativistic Pseudopotentials: Their Development and Scope of Applications', Chemical Reviews 112, 403-480 (2012). DOI: 10.1021/cr2001383, https://doi.org/10.1021/cr2001383