# Many Body Perturbation Theory for Non-Uniform Electronic Systems

• D. F. Scofield
Part of the NATO Advanced Study Institutes Series book series (NSSB, volume 2)

## Abstract

In the now famous papers by P. Hohenberg and W. Kohn [1] and in a sequel by W. Kohn and L.J. Sham [2] use was made of the fact that the total energy of an interacting electron system is a functional of the electronic charge density:
$$E[n(r)]=\int{\upsilon (r)n(r)dr+F[n(r)]}$$
(1)
where υ(r) is the external potential and F is a universal functional of the density. For the exact n(r), E[n(r)] is a minimum relative to variations of n(r) consistent with a variation of υ(r). Unfortunately, for an approximate F = F a the approximate energy functional E a obtained from equation (1) by replacing F[n] with F a[n] does not have to have any relation in particular to E. That is, E a may be greater or smaller or equal to E. An example is the Slater X α scheme [3] in which one can vary α to obtain any E a that one wants. Another fundamental objection is that given an F a there may or may not exist a solution to the Kohn and Sham equations in a region of physical interest. Even when the approximate energy functional has the appropriate properties leading to a n a(r) (there exists a point ñ a(r) such that E a[n] = min), the constraint used by Kohn and Sham that
$$\tilde{n}(r)=\sum\limits_{i}{{{\left| {{{\tilde{\phi }}}_{i}}(r) \right|}^{2}}}N=\int{\tilde{n}(r)}dr$$
(2)
(using the Lagrange multiplier metyhod this constraint leads to the K-S equations upon minimizing the functional [E a[n(r)] − λxn(r)dr] with respect to variations in n compatible with being produced by a variation in the external potential, supposing, of course, that the functional derivative exists) effectively raises the calculated energy even though it does not necessarily ensure N-representability. In principle other N-conserving constraints than (2) could be used.

## Keywords

Banach Space Normed Linear Space Contraction Mapping Principle Nonlinear Operator Equation Skeleton Graph
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. 1.
Hohenberg, P. and Kohn, W. (1964). Phys. Rev., B136, 864.
2. 2.
Kohn, W. and Sham, L.J. (1965). Phys. Rev., A140, 1133.
3. 3.
Slater, J.C. (1951). Phys. Rev., 81, 385;
4. Gaspar, R. (1954). Acta Phys. Sci. Hung., 3, 263.
5. 4.
Scofield, D.F. (1971). Int. J. Quantum Chem., 5, 489.
6. 5.
Herman, F., Van Dyke, J.P. and Ortenburger, I.B. (1969). Phys. Rev. Lett., 22, 807.
7. 6.
Ma, S. and Brueckner, K.A. (1968). Phys. Rev., 165, 18.
8. 7.
Some excellent references: Fetter, A.L. and Walecka, J.D. (1971). Quantum Theory of Many Particle Systems, (McGraw-Hill Book Co., New York);Google Scholar
9. March, N.H., Young, W.H. and Sampanthar, S. (1967). The Many Body Problem in Quantum Mechanics, ( Cambridge University Press, London);Google Scholar
10. Hedin, L. Lundquist S. (1969). In Solid State Physics, Vol 23, (eds. Seitz, F. and Turnbull, D), ( Academic Press, New York);Google Scholar
11. Kumar, K. (1962). Perturbation Theory and The Nuclear Many Body Problem, ( North Holland, Amsterdam);Google Scholar
12. Pines, D. (1962). The Many Body Problem, (W.A. Benjamin, New York);
13. Nozieres, P. (1964). Theory of Interacting Fermi Systems, (W.A. Benjamin, New York);
14. Mattuck, R.D. (1967). A Guide to Feynman Diagrams in The Many Body Problem, ( McGraw-Hill Book Co., New York).Google Scholar
15. 8.
Martin, P.C. and Schwinger, J.S. (1959). Phys. Rev., 115, 1342.
16. 9.
Dyson, F.J. (1949). Phys. Rev., 75, 486, 1736.
17. 10.
Brueckner, K.A. (1955). Phys. Rev., 100, 36.
18. 11.
Goldstone, J. (1957). Proc. Roy. Soc., A239, 267.
19. 12.
20. 13.
Kelly, H.P. (1964). Phys. Rev., 8136, 896.
21. 14.
Huzinaga, S. and Arnau, C. (1970). Phys. Rev., A1, 1285.
22. 15.
Scofield, D.F., Dutta, N.C. and Dutta, C.M. (1972). Int. J. Quantum Chem., 6, 9.
23. 16.
Miller, J.H. and Kelly, H.P. (1971). Phys. Rev, A3, 578.
24. 17.
Hartree, D.R. (1957). The Calculation of Atomic Structures, ( John Wiley and Sons, New York).
25. 18.
Froese, C. (1963). Can. J. Phys., 41, 1895.
26. 19.
Harris, F.E., Kumar, L. and Monkhorst, H.J. (1973). Phys. Rev., B7, 2850.
27. 20.
Hazelgrove, C.B. (1961). Math. Comp., 15, 323;
28. Davis, P.J. and Rabinowitz, P. (1967). Numerical Integration, (Blaisdell Publishing Co., Waltham, Massachusetts).
29. 21.
Brillouin, L. (1933). Actual, Sci. Ind., No. 71; (1934). Actual. Sci. Ind., No. 159.Google Scholar
30. 22.
Slater, J.C. (1951). Phys. Rev., 81, 385;
31. Gaspar, R. (1954). Acta Phys. Sci. Hung., 3, 263.
32. 23.
Graves-Morris, P.R. (ed.) (1973). Padé Approximants, ( Lectures delivered at a Summer School Held at the University of Kent, July 1972, (The Institute of Physics, London ).
33. 24.
Scofield, D.F. (1972). Phys. Rev. Lett., 29, 811.
34. 25.
The material presented here is standard. It may be found in any of the following: Kato, T. (1966). Perturbation Theory for Linear Operators, (Springer-Verlag, New York);
35. Reisz, F. and Nagy, B. Sz. (1955). Functional Analysis, ( Frederick Ungar, New York).Google Scholar
36. 26.
A very good source for what follows is: Ball, L.B. (1969). Computational Solutions of Nonlinear Operator Equations, (John Wiley and Sons, New York). Rall has also given a very nice survey of Vainberg’s book in: Anselone, P.M. (1964). Nonlinear Integral Equations, (University of Wisconsin Press, Madison).Google Scholar
37. 27.
Vainberg, M.M. (1964). Variational Methods for The Study of Nonlinear Operators, ( Holden-Day, San Francisco). This book also has a chapter on Newton’s method in Banach spaces by L.V. Kantorovich and G.P. Akilov.
38. 28.
Zmuidzinas, J.S. (1970). Phys. Rev., B2, 4445.
39. 29.
Kadanoff, L.P. and Baym, G. (1962). Quantum Statistical Mechanics, (W.A. Benjamin, New York).
40. 30.
Schneider, B., Taylor, H.S. and Yaris, R. (1970). Phys. Rev., A1, 855.
41. 31.
Scofield, D.F. Self-Consistent Green’s Function Method for Energy Bands, (to be published).Google Scholar