DOI: 10.1016/s0081-1947(08)60678-5. The Quantum Defect Method

The Quantum Defect Method

10.1016/s0081-1947(08)60678-5

Crossref book-chapter

Elsevier

Solid State Physics (78)

Bibliography

Ham, F. S. (1955). The Quantum Defect Method. Solid State Physics, 127â192.

Authors 1

Frank S. Ham (first)

References 93 Referenced 111

{'key': '10.1016/S0081-1947(08)60678-5_bib1', 'series-title': 'Which has been submitted for publication to The Physical Review (1955)', 'author': 'Brooks', 'year': '1954'} / Which has been submitted for publication to The Physical Review (1955) by Brooks (1954)
National Research Council Postdoctoral Fellow, 1954–1955.
We shall refer to the Quantum Defect Method by the abbreviation QDM throughout most of this paper.
10.1103/PhysRev.79.382 / Phys. Rev. by Kuhn (1950)
10.1103/PhysRev.91.1027.2 / Phys. Rev. by Brooks (1953)
J. H. Van Vleck, Proc. 1953 Intern. Conf. Theoret. Phys., Tokyo, p. 640(1954).
F. S. Ham, Ph.D. thesis, Harvard University, Cambridge, Mass., 1954 (unpublished).
{'key': '10.1016/S0081-1947(08)60678-5_bib8', 'author': 'Brooks', 'year': '1955', 'journal-title': 'Phys. Rev.'} / Phys. Rev. by Brooks (1955)
10.1103/PhysRev.99.419 / Phys. Rev. by Kambe (1955)
{'key': '10.1016/S0081-1947(08)60678-5_bib10', 'series-title': '“Modern Theory of Solids,”', 'first-page': '234', 'author': 'Seitz', 'year': '1940'} / “Modern Theory of Solids,” by Seitz (1940)
10.1103/PhysRev.43.804 / Phys. Rev. by Wigner (1933)
10.1063/1.1750271 / J. Chem. Phys. by Bardeen (1938)
10.1088/0370-1298/65/5/308 / Proc. Phys. Soc. by Howarth (1952)
10.1103/PhysRev.94.1111 / Phys. Rev. by Kohn (1954)
10.1007/BF01340294 / Z. Physik by Fock (1930)
F. Seitz, p. 313 in ref. 10.
10.1017/S0305004100016108 / Proc. Cambridge Phil. Soc. by Dirac (1930)
{'key': '10.1016/S0081-1947(08)60678-5_bib18', 'first-page': '555', 'volume': '62', 'author': 'Bloch', 'year': '1928', 'journal-title': 'Z. Physik'} / Z. Physik by Bloch (1928)
10.1103/PhysRev.50.58 / Phys. Rev. by Bouckaert (1936)
10.1103/PhysRev.81.385 / Phys. Rev. by Slater (1951)
10.1103/PhysRev.46.1002 / Phys. Rev. by Wigner (1934)
10.1103/PhysRev.47.400 / Phys. Rev. by Seitz (1935)
10.1103/PhysRev.82.282 / Phys. Rev. by Herring (1951)
10.1103/PhysRev.92.626 / Phys. Rev. by Pines (1953)
10.1515/zna-1950-0402 / Z. Naturforsch. by Macke (1950)
S. Raimes, Proc. Symposium Radar Research Establishment Malvern, p. 16 (1954).
10.1007/BF01339048 / Z. Physik by Prokofjew (1929)
10.1103/PhysRev.47.400 / Phys. Rev. by Seitz (1935)
{'key': '10.1016/S0081-1947(08)60678-5_bib29', 'first-page': '328', 'volume': '9', 'author': 'Gorin', 'year': '1936', 'journal-title': 'Physik. Z. Sowjetunion'} / Physik. Z. Sowjetunion by Gorin (1936)
10.1090/qam/48637 / Quart. Appl. Math. by Kuhn (1951)
10.1103/PhysRev.64.358 / Phys. Rev. by Wannier (1943)
10.1103/PhysRev.73.60 / Phys. Rev. by Jastrow (1948)
{'key': '10.1016/S0081-1947(08)60678-5_bib33', 'author': 'Ham', 'year': '1955', 'journal-title': 'Phys. Rev.'} / Phys. Rev. by Ham (1955)
{'key': '10.1016/S0081-1947(08)60678-5_bib34', 'series-title': 'Tables for the Calculation of Coulomb Wave Functions', 'author': 'Ham', 'year': '1955'} / Tables for the Calculation of Coulomb Wave Functions by Ham (1955)
10.1103/PhysRev.79.515 / Phys. Rev. by Kuhn (1950)
{'key': '10.1016/S0081-1947(08)60678-5_bib36', 'series-title': '“Introduction to Atomic Spectra,”', 'first-page': '16', 'author': 'White', 'year': '1934'} / “Introduction to Atomic Spectra,” by White (1934)
10.1103/PhysRev.74.113.2 / Phys. Rev. by Imai (1948)
10.1103/PhysRev.57.1169 / Phys. Rev. by Herring (1940)
10.1103/PhysRev.85.227 / Phys. Rev. by Silverman (1952)
Brooks has given a proof (unpublished) based on Eq. (4.7) which shows that the sphere of equal volume yields the best result for the energy when the boundary conditions (4.2) are used, provided the wave function is an s function in the spherical approximation and provided both the correct wave function and the potential are nearly constant in the volume between the surface of the equivalent sphere and the surface of the polyhedral cell. The present author has used the spherical approximation and (4.2) to calculate the eigenvalues for the empty lattice41 when the vector k has the value (2π/a, 2π/ a,0), which is equivalent to k = (0,0,0) in the reduced zone scheme of the body-centered cubic lattice. He obtained the following approximate energy eigenvalues for the twelve functions which have the same energy as the possible combinations of twelve plane waves associated with this k vector and transform in the same way under the cubic symmetry operators.
10.1103/PhysRev.51.129 / Phys. Rev. by Shockley (1937)
10.1103/PhysRev.78.235 / Phys. Rev. by Sternheimer (1950)
10.1103/PhysRev.71.612 / Phys. Rev. by Von der Lage (1947)
10.1103/PhysRev.87.472 / Phys. Rev. by Kohn (1952)
This transformation actually is not necessary. Suppose one of the coefficients in the expansion (3.2) is arbitrarily set equal to unity. If a fixed set of N — 1 boundary conditions is used to determine N — 1 other coefficients for several values of the energy in the neighborhood of the eigenvalue, the resulting series of N terms may be substituted for ψ0 and ψ on the right side of (4.7). The energy ε, at which the right side of this expression, thus evaluated, is equal to zero, may be taken to be the “corrected” eigenvalue. This procedure is accurate because the expression for ψ0 obtained from the N — 1 boundary conditions plus an Nth one that yields ε0 = ε1 would have given ε = ε1 when substituted into the correction integral.
10.1103/PhysRev.92.603 / Phys. Rev. by Slater (1953)
10.1103/PhysRev.92.603 / Phys. Rev. by Slater (1953)
D. J. Howarth, Proc. Symposium Radar Research Establishment Malvern, p. 32 (1954).
R. S. Leigh, Proc. Symposium Radar Research Establishment Malvern, p. 40 (1954).
10.1103/PhysRev.51.846 / Phys. Rev. by Slater (1937)
Brooks has shown in unpublished work that it is possible to extend QDM to a calculation of the amplitude of the wave function at the nucleus if data on atomic hyperfine splitting are available or if an approximate Hartree-Fock potential is known. No calculations have as yet been carried out using this proposal. Details of the procedure will be reported at a future date. ε, we must show that UL, ε(r) is defined for every value of ε and that it has
{'key': '10.1016/S0081-1947(08)60678-5_bib52', 'series-title': '“A Course of Modern Analysis,”', 'author': 'Whittaker', 'year': '1952'} / “A Course of Modern Analysis,” by Whittaker (1952)
{'key': '10.1016/S0081-1947(08)60678-5_bib53', 'series-title': '“Ordinary Differential Equations,”', 'first-page': '73', 'author': 'Ince', 'year': '1927'} / “Ordinary Differential Equations,” by Ince (1927)
{'key': '10.1016/S0081-1947(08)60678-5_bib54', 'series-title': '“Theory of Functions,”', 'first-page': '65', 'author': 'Knopp', 'year': '1945'} / “Theory of Functions,” by Knopp (1945)
Whittaker and Watson, p. 201 in ref. 52.
{'key': '10.1016/S0081-1947(08)60678-5_bib56', 'series-title': '“The Theory of Atomic Collisions,”', 'first-page': '108', 'author': 'Mott', 'year': '1949'} / “The Theory of Atomic Collisions,” by Mott (1949)
Whittaker and Watson, Chapter 16 in ref. 52.
10.1103/PhysRev.87.977 / Phys. Rev. by Jost (1952)
{'key': '10.1016/S0081-1947(08)60678-5_bib59', 'series-title': '“The Fundamental Principles of Quantum Mechanics,”', 'author': 'Kemble', 'year': '1937'} / “The Fundamental Principles of Quantum Mechanics,” by Kemble (1937)
{'key': '10.1016/S0081-1947(08)60678-5_bib60', 'series-title': '“A Treatise on the Theory of Bessel Functions,”', 'first-page': '201', 'author': 'Watson', 'year': '1948'} / “A Treatise on the Theory of Bessel Functions,” by Watson (1948)
10.1103/PhysRev.71.360 / Phys. Rev. by Furry (1947)
{'key': '10.1016/S0081-1947(08)60678-5_bib62', 'first-page': '296', 'volume': '35', 'author': 'Bohr', 'year': '1923', 'journal-title': 'Proc. London Phil. Soc.'} / Proc. London Phil. Soc. by Bohr (1923)
Whittaker and Watson, p. 343 in ref. 52.
See Eqs. (A.10) and (A.11).
Had we not used the device of evaluating exp (2iz) in (6.20) by comparison with the exact solution, we should have obtained a factor (1 + Δ) on the right of (6.20) by direct integration of (6.10). Here A is a number of magnitude 0(l/n). This leads again to (6.24) and introduces into UL(r) an imaginary part which vanishes as |n| → ∞. Such an effect evidently is the result of inaccuracy introduced in evaluating the wave function by the WKB method and should not be taken seriously. However, it serves as a warning that the justification of QDM based on the WKB, while convincing at the higher energies, is not rigorous.
These conclusions can be demonstrated algebraically by expressing α(n) as an integral of the form (5.18) with (z/2)N2L+1n(z) replacing 1Ũc(L, n)(r), and considering the limiting form of the ratio α(n)/γ(n) from these integrals as |n| → 0.
10.1098/rspa.1948.0047 / Proc. Roy. Soc. by Hartree (1948)
10.1017/S0305004100020557 / Proc. Cambridge Phil. Soc. (1938)
Of course, the actual crystal potential will be slightly different from — 2/r near the surface of the cell because of exchange and correlation effects and the influence of nearby ions. This difference from the ion potential is difficult both to estimate and to include in calculating the wave functions, and it is usually omitted. The additional uncertainty concerning the proper choice of ri causes an error in the wave functions and eigenvalues that should be no larger than that arising from this omission. If such errors are significant, the corresponding energy band calculations cannot be considered reliable. There is good reason to believe that the ground-state energy and wave function are not very sensitive to small changes in the potential near the surface of the cell, but calculations for states of higher energy are probably less reliable. The author intends to examine this sensitivity in future work.
Although the core eigenvalues and, consequently, the value of η(n) at the eigenvalues are not altered significantly by the change in the potential outside the core, the change does affect η(n) at energies intermediate to the core eigenvalues. At such energies, the function UL(r), which satisfies the boundary condition at the origin, increases exponentially for large values of r, instead of decreasing exponentially, as at the eigenvalues. If we consider the process of integrating (3.3) numerically outward from the origin, it is clear that the combination of Coulomb functions, which is obtained in a Coulomb region r >r0, will depend on the nature of V (r) for rir r0, even though ri lies in the region in which the core eigenfunctions decrease exponentially. This conclusion follows also from Eqs. (5.17), (5.18), and (7.2). It was suggested by the WKB formula (6.24). Evidently η(n) may oscillate about a smooth curve through the eigenvalues at these low energies, for a change in the potential outside the core may change η(n) at energies between the eigenvalues. This oscillation makes the extrapolation of η(n) far below the valence eigenvalue range uncertain. However, from the data on the alkali metal atoms presented in Section X, it does not appear that this uncertainty is appreciable for s and p functions at interesting energies less than 0.5 Rydberg units below the lowest valence eigenvalue. The uncertainty for the d functions, most notably those in potassium, is considerably greater.
Seitz (p. 357 in ref. 10) has calculated the wave functions and cohesive energies of lithium, sodium, and potassium when an r2 term, arising from a uniform distribution of the charge of the conduction electrons, is included in the potential. The wave functions of occupied states differ by at most a few per cent from those obtained with the omission of this term, and the cohesive energies are changed by less than 1 kg cal/mole. However, we may expect the effect of this term in the potential to be greater in a multivalent atom, for the corresponding charge density is greater. Moreover, wave functions at points in k-space near the surface of the Brillouin zone should be more strongly perturbed by such a term than those near the energy minimum. Such wave functions correspond to occupied states in multivalent metals.
This view is contrary to that expressed by Kuhn and Van Vleck,4 who asserted that the method assumes a “best central field” but avoids the difficulties of determining this field and of using it in the band calculations.
10.1103/PhysRev.95.371 / Phys. Rev. by Herman (1954)
Ψ should be antisymmetrical under permutations of all the electron coordinates. Hence, if Ψc in (9.2) is antisymmetrical, Ψ should contain N — 1 additional terms in which r1 is interchanged with r2 … rN. When r1 lies outside the core, however, as is assumed in writing (9.2), the additional terms have negligible amplitude.
We may include the effect of core polarization by assuming that Ψc depends parametrically on r1. With the appropriate parametric dependence, we find from the variational principle that φ(r1) satisfies the Schrodinger equation for a Coulomb potential to which a polarization term has been added.
A similar approximation of using core wave functions appropriate to the ground state of the free atom or ion is usually employed in calculating energy bands by methods which make use of an explicit crystalline potential. These methods take less accurate account than does QDM of exchange and correlation interaction between valence and core electrons, however.
This section has been taken almost verbatim from ref. 8.
{'key': '10.1016/S0081-1947(08)60678-5_bib77_1', 'volume': '467', 'author': 'Moore', 'year': '1949', 'journal-title': 'Natl. Bur. Standards (U.S.) Circ.'} / Natl. Bur. Standards (U.S.) Circ. by Moore (1949)
{'key': '10.1016/S0081-1947(08)60678-5_bib77_2', 'series-title': '“Atomic Energy States.”', 'author': 'Bacher', 'year': '1932'} / “Atomic Energy States.” by Bacher (1932)
10.1098/rspa.1938.0103 / Proc. Roy. Soc. by Hartree (1938)
{'key': '10.1016/S0081-1947(08)60678-5_bib79', 'first-page': '398', 'volume': '6', 'author': 'Fock', 'year': '1934', 'journal-title': 'Physik. Z. Sowjetunion'} / Physik. Z. Sowjetunion by Fock (1934)
10.1098/rspa.1935.0135 / Proc. Roy. Soc. by Hartree (1935)
White, Chapter 19 in ref. 36.
{'key': '10.1016/S0081-1947(08)60678-5_bib82', 'series-title': '“The Theory of Electric and Magnetic Susceptibilities,”', 'author': 'Van Vleck', 'year': '1932'} / “The Theory of Electric and Magnetic Susceptibilities,” by Van Vleck (1932)
10.1103/PhysRev.92.890 / Phys. Rev. by Tessman (1953)
J. H. Van Vleck, p. 216, Eq. (12), in ref. 76.
The revision of the calculations is being carried out by H. Brooks, and his results will be published in The Physical Review.
10.1103/PhysRev.80.912 / Phys. Rev. by Silverman (1950)
10.1103/PhysRev.99.423 / Phys. Rev. by Swenson (1955)
The values for the theoretical pressure were calculated by the author using data for the constants in Eq. (11.1) kindly supplied by Brooks. The values of the lattice constants and cohesive energies of Li, K, and Rb were given by Brooks,5 and those for Na and Cs were calculated by Brooks in an unpublished revision of that work. In these revised calculations, account was taken of errors discovered7,84 in Kuhn's tables.30
The revised calculations using the form of QDM presented in this paper and the recent values of correlation and exchange energy calculated by Pines24 undoubtedly will change slightly the theoretical values of Table V. The values of the theoretical pressure will be altered the most, for these values are particularly sensitive to small changes in the constants in Eq. (11.1). Hence in Table V the theoretical pressure values are probably significant only to the extent that they are correct in order of magnitude. A more exacting comparison of theory and experiment must await the completion of the revised calculations. These will include the contribution of the k4 term in the energy and will make use of a more accurate expression for the pressure than that derived from (11.1), which is not very satisfactory from either a theoretical or experimental80 point of view.
{'key': '10.1016/S0081-1947(08)60678-5_bib90', 'first-page': '1411', 'volume': '94', 'author': 'Brooks', 'year': '1954', 'journal-title': 'Phys. Rev.'} / Phys. Rev. by Brooks (1954)
This section is based upon Kuhn's analysis of the Coulomb wave functions30 and upon a recent paper by the author33 which corrects an oversight in Kuhn's work. The second paper develops both convergent and asymptotic series in (1/n2) for the irregular functions. The reader is referred to the papers for proofs of statements made above. Tables of the coefficients Uk(L)(z) in (A.2) have been prepared by the author for the series expansions of (z/2)J21+1n and (z/2) N2L+1n(z). These are valid for L = 0 and for values of z of interest in band calculations. The tables, together with recurrence relations connecting functions of different L, make possible accurate and relatively simple computations of the Coulomb functions needed in such work.34 The tables are more detailed than those published by Kuhn. They may be obtained from the author or from Cruft Laboratory, Harvard University.

Dates

Type	When
Created	17 years, 2 months ago (May 30, 2008, 4:45 a.m.)
Deposited	3 years, 11 months ago (Sept. 11, 2021, 5:26 a.m.)
Indexed	1 month, 3 weeks ago (July 2, 2025, 3:29 p.m.)
Issued	70 years, 7 months ago (Jan. 1, 1955)
Published	70 years, 7 months ago (Jan. 1, 1955)
Published Print	70 years, 7 months ago (Jan. 1, 1955)

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BibTeX

@inbook{Ham_1955, title={The Quantum Defect Method}, ISBN={9780126077018}, ISSN={0081-1947}, url={http://dx.doi.org/10.1016/s0081-1947(08)60678-5}, DOI={10.1016/s0081-1947(08)60678-5}, booktitle={Solid State Physics}, publisher={Elsevier}, author={Ham, Frank S.}, year={1955}, pages={127–192} }

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      "volume": "92",
      "author": "Pines",
      "year": "1953",
      "journal-title": "Phys. Rev."
    },
    {
      "key": "10.1016/S0081-1947(08)60678-5_bib25",
      "doi-asserted-by": "crossref",
      "first-page": "192",
      "DOI": "10.1515/zna-1950-0402",
      "volume": "5a",
      "author": "Macke",
      "year": "1950",
      "journal-title": "Z. Naturforsch."
    },
    {
      "key": "10.1016/S0081-1947(08)60678-5_bib26",
      "unstructured": "S. Raimes, Proc. Symposium Radar Research Establishment Malvern, p. 16 (1954)."
    },
    {
      "key": "10.1016/S0081-1947(08)60678-5_bib27",
      "doi-asserted-by": "crossref",
      "first-page": "255",
      "DOI": "10.1007/BF01339048",
      "volume": "68",
      "author": "Prokofjew",
      "year": "1929",
      "journal-title": "Z. Physik"
    },
    {
      "key": "10.1016/S0081-1947(08)60678-5_bib28",
      "doi-asserted-by": "crossref",
      "first-page": "400",
      "DOI": "10.1103/PhysRev.47.400",
      "volume": "47",
      "author": "Seitz",
      "year": "1935",
      "journal-title": "Phys. Rev."
    },
    {
      "key": "10.1016/S0081-1947(08)60678-5_bib29",
      "first-page": "328",
      "volume": "9",
      "author": "Gorin",
      "year": "1936",
      "journal-title": "Physik. Z. Sowjetunion"
    },
    {
      "key": "10.1016/S0081-1947(08)60678-5_bib30",
      "doi-asserted-by": "crossref",
      "first-page": "1",
      "DOI": "10.1090/qam/48637",
      "volume": "9",
      "author": "Kuhn",
      "year": "1951",
      "journal-title": "Quart. Appl. Math."
    },
    {
      "key": "10.1016/S0081-1947(08)60678-5_bib31",
      "doi-asserted-by": "crossref",
      "first-page": "358",
      "DOI": "10.1103/PhysRev.64.358",
      "volume": "64",
      "author": "Wannier",
      "year": "1943",
      "journal-title": "Phys. Rev."
    },
    {
      "key": "10.1016/S0081-1947(08)60678-5_bib32",
      "doi-asserted-by": "crossref",
      "first-page": "70",
      "DOI": "10.1103/PhysRev.73.60",
      "volume": "73",
      "author": "Jastrow",
      "year": "1948",
      "journal-title": "Phys. Rev."
    },
    {
      "key": "10.1016/S0081-1947(08)60678-5_bib33",
      "author": "Ham",
      "year": "1955",
      "journal-title": "Phys. Rev."
    },
    {
      "key": "10.1016/S0081-1947(08)60678-5_bib34",
      "series-title": "Tables for the Calculation of Coulomb Wave Functions",
      "author": "Ham",
      "year": "1955"
    },
    {
      "key": "10.1016/S0081-1947(08)60678-5_bib35",
      "doi-asserted-by": "crossref",
      "first-page": "515",
      "DOI": "10.1103/PhysRev.79.515",
      "volume": "79",
      "author": "Kuhn",
      "year": "1950",
      "journal-title": "Phys. Rev."
    },
    {
      "key": "10.1016/S0081-1947(08)60678-5_bib36",
      "series-title": "\u201cIntroduction to Atomic Spectra,\u201d",
      "first-page": "16",
      "author": "White",
      "year": "1934"
    },
    {
      "key": "10.1016/S0081-1947(08)60678-5_bib37",
      "doi-asserted-by": "crossref",
      "first-page": "113",
      "DOI": "10.1103/PhysRev.74.113.2",
      "volume": "74",
      "author": "Imai",
      "year": "1948",
      "journal-title": "Phys. Rev."
    },
    {
      "key": "10.1016/S0081-1947(08)60678-5_bib38",
      "doi-asserted-by": "crossref",
      "first-page": "1169",
      "DOI": "10.1103/PhysRev.57.1169",
      "volume": "67",
      "author": "Herring",
      "year": "1940",
      "journal-title": "Phys. Rev."
    },
    {
      "key": "10.1016/S0081-1947(08)60678-5_bib39",
      "doi-asserted-by": "crossref",
      "first-page": "227",
      "DOI": "10.1103/PhysRev.85.227",
      "volume": "85",
      "author": "Silverman",
      "year": "1952",
      "journal-title": "Phys. Rev."
    },
    {
      "key": "10.1016/S0081-1947(08)60678-5_bib40",
      "unstructured": "Brooks has given a proof (unpublished) based on Eq. (4.7) which shows that the sphere of equal volume yields the best result for the energy when the boundary conditions (4.2) are used, provided the wave function is an s function in the spherical approximation and provided both the correct wave function and the potential are nearly constant in the volume between the surface of the equivalent sphere and the surface of the polyhedral cell. The present author has used the spherical approximation and (4.2) to calculate the eigenvalues for the empty lattice41 when the vector k has the value (2\u03c0/a, 2\u03c0/ a,0), which is equivalent to k = (0,0,0) in the reduced zone scheme of the body-centered cubic lattice. He obtained the following approximate energy eigenvalues for the twelve functions which have the same energy as the possible combinations of twelve plane waves associated with this k vector and transform in the same way under the cubic symmetry operators."
    },
    {
      "key": "10.1016/S0081-1947(08)60678-5_bib41",
      "doi-asserted-by": "crossref",
      "first-page": "129",
      "DOI": "10.1103/PhysRev.51.129",
      "volume": "61",
      "author": "Shockley",
      "year": "1937",
      "journal-title": "Phys. Rev."
    },
    {
      "key": "10.1016/S0081-1947(08)60678-5_bib42",
      "doi-asserted-by": "crossref",
      "first-page": "238",
      "DOI": "10.1103/PhysRev.78.235",
      "volume": "78",
      "author": "Sternheimer",
      "year": "1950",
      "journal-title": "Phys. Rev."
    },
    {
      "key": "10.1016/S0081-1947(08)60678-5_bib43",
      "doi-asserted-by": "crossref",
      "first-page": "612",
      "DOI": "10.1103/PhysRev.71.612",
      "volume": "71",
      "author": "Von der Lage",
      "year": "1947",
      "journal-title": "Phys. Rev."
    },
    {
      "key": "10.1016/S0081-1947(08)60678-5_bib44",
      "doi-asserted-by": "crossref",
      "first-page": "472",
      "DOI": "10.1103/PhysRev.87.472",
      "volume": "87",
      "author": "Kohn",
      "year": "1952",
      "journal-title": "Phys. Rev."
    },
    {
      "key": "10.1016/S0081-1947(08)60678-5_bib45",
      "unstructured": "This transformation actually is not necessary. Suppose one of the coefficients in the expansion (3.2) is arbitrarily set equal to unity. If a fixed set of N \u2014 1 boundary conditions is used to determine N \u2014 1 other coefficients for several values of the energy in the neighborhood of the eigenvalue, the resulting series of N terms may be substituted for \u03c80 and \u03c8 on the right side of (4.7). The energy \u03b5, at which the right side of this expression, thus evaluated, is equal to zero, may be taken to be the \u201ccorrected\u201d eigenvalue. This procedure is accurate because the expression for \u03c80 obtained from the N \u2014 1 boundary conditions plus an Nth one that yields \u03b50 = \u03b51 would have given \u03b5 = \u03b51 when substituted into the correction integral."
    },
    {
      "key": "10.1016/S0081-1947(08)60678-5_bib46",
      "doi-asserted-by": "crossref",
      "first-page": "603",
      "DOI": "10.1103/PhysRev.92.603",
      "volume": "92",
      "author": "Slater",
      "year": "1953",
      "journal-title": "Phys. Rev."
    },
    {
      "key": "10.1016/S0081-1947(08)60678-5_bib47",
      "doi-asserted-by": "crossref",
      "first-page": "1126",
      "DOI": "10.1103/PhysRev.92.603",
      "volume": "92",
      "author": "Slater",
      "year": "1953",
      "journal-title": "Phys. Rev."
    },
    {
      "key": "10.1016/S0081-1947(08)60678-5_bib48",
      "unstructured": "D. J. Howarth, Proc. Symposium Radar Research Establishment Malvern, p. 32 (1954)."
    },
    {
      "key": "10.1016/S0081-1947(08)60678-5_bib49",
      "unstructured": "R. S. Leigh, Proc. Symposium Radar Research Establishment Malvern, p. 40 (1954)."
    },
    {
      "key": "10.1016/S0081-1947(08)60678-5_bib50",
      "doi-asserted-by": "crossref",
      "first-page": "846",
      "DOI": "10.1103/PhysRev.51.846",
      "volume": "61",
      "author": "Slater",
      "year": "1937",
      "journal-title": "Phys. Rev."
    },
    {
      "key": "10.1016/S0081-1947(08)60678-5_bib51",
      "unstructured": "Brooks has shown in unpublished work that it is possible to extend QDM to a calculation of the amplitude of the wave function at the nucleus if data on atomic hyperfine splitting are available or if an approximate Hartree-Fock potential is known. No calculations have as yet been carried out using this proposal. Details of the procedure will be reported at a future date. \u03b5, we must show that UL, \u03b5(r) is defined for every value of \u03b5 and that it has"
    },
    {
      "key": "10.1016/S0081-1947(08)60678-5_bib52",
      "series-title": "\u201cA Course of Modern Analysis,\u201d",
      "author": "Whittaker",
      "year": "1952"
    },
    {
      "key": "10.1016/S0081-1947(08)60678-5_bib53",
      "series-title": "\u201cOrdinary Differential Equations,\u201d",
      "first-page": "73",
      "author": "Ince",
      "year": "1927"
    },
    {
      "key": "10.1016/S0081-1947(08)60678-5_bib54",
      "series-title": "\u201cTheory of Functions,\u201d",
      "first-page": "65",
      "author": "Knopp",
      "year": "1945"
    },
    {
      "key": "10.1016/S0081-1947(08)60678-5_bib55",
      "unstructured": "Whittaker and Watson, p. 201 in ref. 52."
    },
    {
      "key": "10.1016/S0081-1947(08)60678-5_bib56",
      "series-title": "\u201cThe Theory of Atomic Collisions,\u201d",
      "first-page": "108",
      "author": "Mott",
      "year": "1949"
    },
    {
      "key": "10.1016/S0081-1947(08)60678-5_bib57",
      "unstructured": "Whittaker and Watson, Chapter 16 in ref. 52."
    },
    {
      "key": "10.1016/S0081-1947(08)60678-5_bib58",
      "doi-asserted-by": "crossref",
      "first-page": "977",
      "DOI": "10.1103/PhysRev.87.977",
      "volume": "87",
      "author": "Jost",
      "year": "1952",
      "journal-title": "Phys. Rev."
    },
    {
      "key": "10.1016/S0081-1947(08)60678-5_bib59",
      "series-title": "\u201cThe Fundamental Principles of Quantum Mechanics,\u201d",
      "author": "Kemble",
      "year": "1937"
    },
    {
      "key": "10.1016/S0081-1947(08)60678-5_bib60",
      "series-title": "\u201cA Treatise on the Theory of Bessel Functions,\u201d",
      "first-page": "201",
      "author": "Watson",
      "year": "1948"
    },
    {
      "key": "10.1016/S0081-1947(08)60678-5_bib61",
      "doi-asserted-by": "crossref",
      "first-page": "360",
      "DOI": "10.1103/PhysRev.71.360",
      "volume": "71",
      "author": "Furry",
      "year": "1947",
      "journal-title": "Phys. Rev."
    },
    {
      "key": "10.1016/S0081-1947(08)60678-5_bib62",
      "first-page": "296",
      "volume": "35",
      "author": "Bohr",
      "year": "1923",
      "journal-title": "Proc. London Phil. Soc."
    },
    {
      "key": "10.1016/S0081-1947(08)60678-5_bib63",
      "unstructured": "Whittaker and Watson, p. 343 in ref. 52."
    },
    {
      "key": "10.1016/S0081-1947(08)60678-5_bib64",
      "unstructured": "See Eqs. (A.10) and (A.11)."
    },
    {
      "key": "10.1016/S0081-1947(08)60678-5_bib65",
      "unstructured": "Had we not used the device of evaluating exp (2iz) in (6.20) by comparison with the exact solution, we should have obtained a factor (1 + \u0394) on the right of (6.20) by direct integration of (6.10). Here A is a number of magnitude 0(l/n). This leads again to (6.24) and introduces into UL(r) an imaginary part which vanishes as |n| \u2192 \u221e. Such an effect evidently is the result of inaccuracy introduced in evaluating the wave function by the WKB method and should not be taken seriously. However, it serves as a warning that the justification of QDM based on the WKB, while convincing at the higher energies, is not rigorous."
    },
    {
      "key": "10.1016/S0081-1947(08)60678-5_bib66",
      "unstructured": "These conclusions can be demonstrated algebraically by expressing \u03b1(n) as an integral of the form (5.18) with (z/2)N2L+1n(z) replacing 1\u0168c(L, n)(r), and considering the limiting form of the ratio \u03b1(n)/\u03b3(n) from these integrals as |n| \u2192 0."
    },
    {
      "key": "10.1016/S0081-1947(08)60678-5_bib67_1",
      "doi-asserted-by": "crossref",
      "first-page": "299",
      "DOI": "10.1098/rspa.1948.0047",
      "volume": "A193",
      "author": "Hartree",
      "year": "1948",
      "journal-title": "Proc. Roy. Soc."
    },
    {
      "key": "10.1016/S0081-1947(08)60678-5_bib67_2",
      "doi-asserted-by": "crossref",
      "first-page": "550",
      "DOI": "10.1017/S0305004100020557",
      "volume": "34",
      "year": "1938",
      "journal-title": "Proc. Cambridge Phil. Soc."
    },
    {
      "key": "10.1016/S0081-1947(08)60678-5_bib68",
      "unstructured": "Of course, the actual crystal potential will be slightly different from \u2014 2/r near the surface of the cell because of exchange and correlation effects and the influence of nearby ions. This difference from the ion potential is difficult both to estimate and to include in calculating the wave functions, and it is usually omitted. The additional uncertainty concerning the proper choice of ri causes an error in the wave functions and eigenvalues that should be no larger than that arising from this omission. If such errors are significant, the corresponding energy band calculations cannot be considered reliable. There is good reason to believe that the ground-state energy and wave function are not very sensitive to small changes in the potential near the surface of the cell, but calculations for states of higher energy are probably less reliable. The author intends to examine this sensitivity in future work."
    },
    {
      "key": "10.1016/S0081-1947(08)60678-5_bib69",
      "unstructured": "Although the core eigenvalues and, consequently, the value of \u03b7(n) at the eigenvalues are not altered significantly by the change in the potential outside the core, the change does affect \u03b7(n) at energies intermediate to the core eigenvalues. At such energies, the function UL(r), which satisfies the boundary condition at the origin, increases exponentially for large values of r, instead of decreasing exponentially, as at the eigenvalues. If we consider the process of integrating (3.3) numerically outward from the origin, it is clear that the combination of Coulomb functions, which is obtained in a Coulomb region r >r0, will depend on the nature of V (r) for rir r0, even though ri lies in the region in which the core eigenfunctions decrease exponentially. This conclusion follows also from Eqs. (5.17), (5.18), and (7.2). It was suggested by the WKB formula (6.24). Evidently \u03b7(n) may oscillate about a smooth curve through the eigenvalues at these low energies, for a change in the potential outside the core may change \u03b7(n) at energies between the eigenvalues. This oscillation makes the extrapolation of \u03b7(n) far below the valence eigenvalue range uncertain. However, from the data on the alkali metal atoms presented in Section X, it does not appear that this uncertainty is appreciable for s and p functions at interesting energies less than 0.5 Rydberg units below the lowest valence eigenvalue. The uncertainty for the d functions, most notably those in potassium, is considerably greater."
    },
    {
      "key": "10.1016/S0081-1947(08)60678-5_bib70",
      "unstructured": "Seitz (p. 357 in ref. 10) has calculated the wave functions and cohesive energies of lithium, sodium, and potassium when an r2 term, arising from a uniform distribution of the charge of the conduction electrons, is included in the potential. The wave functions of occupied states differ by at most a few per cent from those obtained with the omission of this term, and the cohesive energies are changed by less than 1 kg cal/mole. However, we may expect the effect of this term in the potential to be greater in a multivalent atom, for the corresponding charge density is greater. Moreover, wave functions at points in k-space near the surface of the Brillouin zone should be more strongly perturbed by such a term than those near the energy minimum. Such wave functions correspond to occupied states in multivalent metals."
    },
    {
      "key": "10.1016/S0081-1947(08)60678-5_bib71",
      "unstructured": "This view is contrary to that expressed by Kuhn and Van Vleck,4 who asserted that the method assumes a \u201cbest central field\u201d but avoids the difficulties of determining this field and of using it in the band calculations."
    },
    {
      "key": "10.1016/S0081-1947(08)60678-5_bib72",
      "doi-asserted-by": "crossref",
      "first-page": "371",
      "DOI": "10.1103/PhysRev.95.371",
      "volume": "95",
      "author": "Herman",
      "year": "1954",
      "journal-title": "Phys. Rev."
    },
    {
      "key": "10.1016/S0081-1947(08)60678-5_bib73",
      "unstructured": "\u03a8 should be antisymmetrical under permutations of all the electron coordinates. Hence, if \u03a8c in (9.2) is antisymmetrical, \u03a8 should contain N \u2014 1 additional terms in which r1 is interchanged with r2 \u2026 rN. When r1 lies outside the core, however, as is assumed in writing (9.2), the additional terms have negligible amplitude."
    },
    {
      "key": "10.1016/S0081-1947(08)60678-5_bib74",
      "unstructured": "We may include the effect of core polarization by assuming that \u03a8c depends parametrically on r1. With the appropriate parametric dependence, we find from the variational principle that \u03c6(r1) satisfies the Schrodinger equation for a Coulomb potential to which a polarization term has been added."
    },
    {
      "key": "10.1016/S0081-1947(08)60678-5_bib75",
      "unstructured": "A similar approximation of using core wave functions appropriate to the ground state of the free atom or ion is usually employed in calculating energy bands by methods which make use of an explicit crystalline potential. These methods take less accurate account than does QDM of exchange and correlation interaction between valence and core electrons, however."
    },
    {
      "key": "10.1016/S0081-1947(08)60678-5_bib76",
      "unstructured": "This section has been taken almost verbatim from ref. 8."
    },
    {
      "key": "10.1016/S0081-1947(08)60678-5_bib77_1",
      "volume": "467",
      "author": "Moore",
      "year": "1949",
      "journal-title": "Natl. Bur. Standards (U.S.) Circ."
    },
    {
      "key": "10.1016/S0081-1947(08)60678-5_bib77_2",
      "series-title": "\u201cAtomic Energy States.\u201d",
      "author": "Bacher",
      "year": "1932"
    },
    {
      "key": "10.1016/S0081-1947(08)60678-5_bib78",
      "doi-asserted-by": "crossref",
      "first-page": "450",
      "DOI": "10.1098/rspa.1938.0103",
      "volume": "A166",
      "author": "Hartree",
      "year": "1938",
      "journal-title": "Proc. Roy. Soc."
    },
    {
      "key": "10.1016/S0081-1947(08)60678-5_bib79",
      "first-page": "398",
      "volume": "6",
      "author": "Fock",
      "year": "1934",
      "journal-title": "Physik. Z. Sowjetunion"
    },
    {
      "key": "10.1016/S0081-1947(08)60678-5_bib80",
      "doi-asserted-by": "crossref",
      "first-page": "96",
      "DOI": "10.1098/rspa.1935.0135",
      "volume": "A151",
      "author": "Hartree",
      "year": "1935",
      "journal-title": "Proc. Roy. Soc."
    },
    {
      "key": "10.1016/S0081-1947(08)60678-5_bib81",
      "unstructured": "White, Chapter 19 in ref. 36."
    },
    {
      "key": "10.1016/S0081-1947(08)60678-5_bib82",
      "series-title": "\u201cThe Theory of Electric and Magnetic Susceptibilities,\u201d",
      "author": "Van Vleck",
      "year": "1932"
    },
    {
      "key": "10.1016/S0081-1947(08)60678-5_bib83",
      "doi-asserted-by": "crossref",
      "first-page": "890",
      "DOI": "10.1103/PhysRev.92.890",
      "volume": "92",
      "author": "Tessman",
      "year": "1953",
      "journal-title": "Phys. Rev."
    },
    {
      "key": "10.1016/S0081-1947(08)60678-5_bib84",
      "unstructured": "J. H. Van Vleck, p. 216, Eq. (12), in ref. 76."
    },
    {
      "key": "10.1016/S0081-1947(08)60678-5_bib85",
      "unstructured": "The revision of the calculations is being carried out by H. Brooks, and his results will be published in The Physical Review."
    },
    {
      "key": "10.1016/S0081-1947(08)60678-5_bib86",
      "doi-asserted-by": "crossref",
      "first-page": "912",
      "DOI": "10.1103/PhysRev.80.912",
      "volume": "80",
      "author": "Silverman",
      "year": "1950",
      "journal-title": "Phys. Rev."
    },
    {
      "key": "10.1016/S0081-1947(08)60678-5_bib87",
      "doi-asserted-by": "crossref",
      "first-page": "423",
      "DOI": "10.1103/PhysRev.99.423",
      "volume": "99",
      "author": "Swenson",
      "year": "1955",
      "journal-title": "Phys. Rev."
    },
    {
      "key": "10.1016/S0081-1947(08)60678-5_bib88",
      "unstructured": "The values for the theoretical pressure were calculated by the author using data for the constants in Eq. (11.1) kindly supplied by Brooks. The values of the lattice constants and cohesive energies of Li, K, and Rb were given by Brooks,5 and those for Na and Cs were calculated by Brooks in an unpublished revision of that work. In these revised calculations, account was taken of errors discovered7,84 in Kuhn's tables.30"
    },
    {
      "key": "10.1016/S0081-1947(08)60678-5_bib89",
      "unstructured": "The revised calculations using the form of QDM presented in this paper and the recent values of correlation and exchange energy calculated by Pines24 undoubtedly will change slightly the theoretical values of Table V. The values of the theoretical pressure will be altered the most, for these values are particularly sensitive to small changes in the constants in Eq. (11.1). Hence in Table V the theoretical pressure values are probably significant only to the extent that they are correct in order of magnitude. A more exacting comparison of theory and experiment must await the completion of the revised calculations. These will include the contribution of the k4 term in the energy and will make use of a more accurate expression for the pressure than that derived from (11.1), which is not very satisfactory from either a theoretical or experimental80 point of view."
    },
    {
      "key": "10.1016/S0081-1947(08)60678-5_bib90",
      "first-page": "1411",
      "volume": "94",
      "author": "Brooks",
      "year": "1954",
      "journal-title": "Phys. Rev."
    },
    {
      "key": "10.1016/S0081-1947(08)60678-5_bib91",
      "unstructured": "This section is based upon Kuhn's analysis of the Coulomb wave functions30 and upon a recent paper by the author33 which corrects an oversight in Kuhn's work. The second paper develops both convergent and asymptotic series in (1/n2) for the irregular functions. The reader is referred to the papers for proofs of statements made above. Tables of the coefficients Uk(L)(z) in (A.2) have been prepared by the author for the series expansions of (z/2)J21+1n and (z/2) N2L+1n(z). These are valid for L = 0 and for values of z of interest in band calculations. The tables, together with recurrence relations connecting functions of different L, make possible accurate and relatively simple computations of the Coulomb functions needed in such work.34 The tables are more detailed than those published by Kuhn. They may be obtained from the author or from Cruft Laboratory, Harvard University."
    }
  ],
  "container-title": "Solid State Physics",
  "original-title": [],
  "deposited": {
    "date-parts": [
      [
        2021,
        9,
        11
      ]
    ],
    "date-time": "2021-09-11T09:26:46Z",
    "timestamp": 1631352406000
  },
  "score": 1,
  "resource": {
    "primary": {
      "URL": "https://linkinghub.elsevier.com/retrieve/pii/S0081194708606785"
    }
  },
  "subtitle": [],
  "short-title": [],
  "issued": {
    "date-parts": [
      [
        1955
      ]
    ]
  },
  "ISBN": [
    "9780126077018"
  ],
  "references-count": 93,
  "URL": "http://dx.doi.org/10.1016/S0081-1947(08)60678-5",
  "relation": {},
  "ISSN": [
    "0081-1947"
  ],
  "subject": [],
  "published": {
    "date-parts": [
      [
        1955
      ]
    ]
  }
}