Density-potential mapping in time-dependent density-functional theory

Physical Review A - Atomic, Molecular, and Optical Physics

Volumn 81, Issue 4, 2010, Pages

  Maitra, N T ^a   Todorov, T N ^b   Woodward, C ^c   Burke, K ^d  

^a CITY UNIVERSITY OF NEW YORK   (United States)

^b QUEEN'S UNIVERSITY BELFAST   (United Kingdom)

^c RUTGERS UNIVERSITY   (United States)

^d UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

DINGER EQUATION; POTENTIAL MAPPING; QUANTUM MECHANICS; TIME DEPENDENT DENSITY FUNCTIONAL THEORY;

NONLINEAR EQUATIONS; TIME VARYING SYSTEMS; WAVE FUNCTIONS;

DENSITY FUNCTIONAL THEORY;

EID: 77951741059     PISSN: 10502947     EISSN: 10941622     Source Type: Journal    
DOI: 10.1103/PhysRevA.81.042525     Document Type: Article

References (24)

1. P. Hohenberg and W. Kohn, Phys. Rev. PHRVAO 0031-899X 10.1103/PhysRev.136.B864 136, B864 (1964).
2. W. Kohn and L. J. Sham, Phys. Rev. PHRVAO 0031-899X 10.1103/PhysRev.140. A1133 140, A1133 (1965).
3. P. Elliott, F. Furche, and K. Burke, in Reviews in Computational Chemistry, edited by, K. B. Lipkowitz, and, T. R. Cundari, (Wiley, Hoboken, NJ, 2009), p. 91.
4. M. A. L. Marques, F. Nogueira, A. Rubio, K. Burke, C. A. Ullrich, and, E. K. U. Gross, eds., Time-Dependent Density Functional Theory (Springer, Berlin, 2006).
5. E. Runge and E. K. U. Gross, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.52.997 52, 997 (1984).
6. R. van Leeuwen, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.82. 3863 82, 3863 (1999).
7. G. Vignale, Phys. Rev. B PRLTAO 1098-0121 10.1103/PhysRevB.70.201102 70, 201102 (R) (2004).
8. T. K. Ng and K. S. Singwi, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.59.2627 59, 2627 (1987).
9. R. van Leeuwen, Int. J. Mod. Phys. B IJPBEV 0217-9792 10.1142/S021797920100499X 15, 1969 (2001).
10. T. Cazenave, Semilinear Schrödinger Equations (Courant Institute of Mathematical Sciences, New York, 2003).
11. See, e.g., Theorem 17.2.7 in L. Hörmander, Analysis of Linear Partial Differential Operators, III: Pseudo Differential Operators (Springer-Verlag, Berlin, 2007).
12. We note that, although not stated there, the same restrictions apply also to the proof in Ref. [6] for the t -analytic case. Thorough discussion of the latter is given by M. Ruggenthaler, M. Penz, and D. Bauer, J. Phys. A PRLTAO 1751-8113 10.1088/1751-8113/42/42/425207 42, 425207 (2009).
13. J. Rauch, in Partial Differential Equations (Springer-Verlag, New York, 1991), Theorem 2, p. 125.
14. An alternative formulation is to replace the left-hand side of Eq. (5) with (1/m)•[nΨ(r,t) v(r,t)]. Subtracting Eqs. (3) and (5) then gives Δn (r,t)=0, where Δn(r,t)=nΨ(r,t)-naim(r,t). The given initial conditions Δn(r,0)=0, Δn (r,0)=0 then guarantee Δn(r,t)=0 for all t≥0. Existence and uniqueness of the solution, for [v(r,t),|Ψ(t)], to this modified formulation is then a necessary and sufficient condition for unique v-representability of the chosen target density naim(r,t). However, this formulation makes v(r,t), as defined by Eq. (5), a more complicated functional of |Ψ(t). The distinction between these two formulations is left for future work.
15. See, e.g., E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations (McGraw-Hill, New York, 1955), for the Picard-Lindelöf theorem (equivalently, the Cauchy-Lipschitz theorem).
16. J. Ginibre and G. Velo, Math. Z. MAZEAX 0025-5874 10.1007/BF01214768 170, 109 (1980).
17. N. Burq, P. Gerard, and N. Tzvetkov, Am. J. Math. AJMAAN 0002-9327 10.1353/ajm.2004.0016 126, 569 (Theorem 3.1) (2004).
18. C. E. Kenig, G. Ponce, and L. Vega, Invent. Math. INVMBH 0020-9910 10.1007/s002220050272 134, 489 (1998).
19. +.
20. B. R. Holstein and A. R. Swift, Am. J. Phys. AJPIAS 0002-9505 10.1119/1.1986678 40, 829 (1972).
21. Z. Yang and K. Burke (unpublished).
22. I. V. Tokatly, Phys. Rev. B AJPIAS 1098-0121 10.1103/PhysRevB.75.125105 75, 125105 (2007);
23. I. V. Tokatly, Phys. Chem. Chem. Phys. PPCPFQ 1463-9076 10.1039/b903666k 11, 4621 (2009).
24. Since Eq. (A2) is of second order, the complementary function in the general solution for w (r, t) contains a second term, in addition to φ 0 (r, t). It, too, has been eliminated in the solution in Eq. (A6).