Although every stationary-state density
ni(r\ensuremath{\rightarrow}) of a many-particle system is not an extremum of the ground-state density functional
Ev[n], every extremum of
Ev[n] [i.e., every solution of the Euler equation \ensuremath{\delta}
Ev/\ensuremath{\delta}n(r\ensuremath{\rightarrow})=\ensuremath{\lambda}] is a stationary-state density
ni(r\ensuremath{\rightarrow}). Always,
Ev[
ni]\ensuremath{\le}
Ei, where
Ei is the lowest stationary-state energy for density
ni(r\ensuremath{\rightarrow}); the equality holds if and only if
ni(r\ensuremath{\rightarrow}) is an extremum of
Ev[n]. The extrema lying above the absolute minimum are excited-state densities which fail to be pure-state v-representable. Surprisingly, infinitesimal number-conserving density variations \ensuremath{\delta}n(r\ensuremath{\rightarrow}) about an extremum n(r\ensuremath{\rightarrow}) do not lead to energy variations \ensuremath{\delta}
Ev of order (\ensuremath{\delta}n
)2 when \ensuremath{\delta}n(r\ensuremath{\rightarrow})/
n1/2(r\ensuremath{\rightarrow}) fails to be square-integrable; in fact, variations \ensuremath{\delta}
Ev of order \ensuremath{\Vert}\ensuremath{\delta}n\ensuremath{\Vert} about the ground state are exemplified by the recently discovered ``derivative discontinuities of the energy.'' This unconventional behavior of
Ev[n] may be traced in part to an asymptotic divergence of ${\ensuremath{\delta}}^{2}$
Ev/\ensuremath{\delta}n(r\ensuremath{\rightarrow})\ensuremath{\delta}n(r\ensuremath{\rightarrow}'). Conditions are presented under which a self-consistent solution of the Kohn-Sham single-particle problem represents an extremum of
Ev[n]. The multiplets of the ground-state orbital configuration of the carbon atom are examined. The local-density and Langreth-Mehl approximations are found to yield a remarkably accurate account of the degeneracy of the various ground-state densities for this system, but no estimate of the multiplet splitting is obtained. Finally, aspects of v-representability are discussed, with emphasis on the iron atom.
It is proven that the exact excited-state wave function and energy may be obtained by minimizing the energy expectation value of trial wave functions that are constrained only to have the correct nodes of the state of interest. This excited-state nodal minimum principle has the advantage that it requires neither minimization with the constraint of wave-function orthogonality to all lower eigenstates nor the antisymmetry of the trial wave functions. It is also found that the minimization over the entire space can be partitioned into several interconnected minimizations within the individual nodal regions, and the exact excited-state energy may be obtained by a minimization in just one or several of these nodal regions. For the proofs of the theorem, it is observed that the many-electron eigenfunction (excited state as well as ground state), restricted to a nodal region, is equivalent to a ground-state wave function of one electron in a higher-dimensional space; and, alternatively, an explicit excited-state energy variational expression is utilized by generalizing the Jacobi method of multiplicative variation. In corollaries, error functions are constructed for cases for which the nodes are not necessarily exact. The exact nodes minimize the energy error functions with respect to nodal variations. Simple numerical illustrations of the error functions are presented.
Recent work [Phys. Rev. A 9, 108 (1974)] on variational principles for diagonal bound state matrix elements of arbitrary Hermitian operators is extended. In particular, it is shown that the previously derived minimum principle for the trial auxiliary function appearing in such variational principles can be constructed using a modified Hamiltonian possessing not heretofore recognized positive definite properties. Thus there is at least one alternative to the particular modified Hamiltonian on which the results of Phys. Rev. A 9, 108 (1974) originally were based.
a minimum ; the value of the integral is the corresponding eigenvalue El of the Hamiltonian H. It is also well known that this property leads to a powerful method of approximating, at least qualitatively, 4,1 and Er, by minimizing (1) among a restricted class of functions. No similarly simple minimum property exists for the higher eigenvalues . One can obtain the nth eigenvalue by minimizing (1) with the subsidiary condition that >G be orthogonal to ¢l, 42, y n-r• However, this procedure becomes progressively more cumbersome as n increases and besides it is of no use unless 4,1, 02, • • • , 4'n-1 are very exactly known. For problems of thermal equilibrium one is often concerned with the free energy, rather than with the individual eigenvalues . It is the purpose of this note to draw attention to a simple minimum property of the free energy which may be considered as a generalization of the variation principle for the lowest eigenvalue. The free energy has the following property: If v1, W2 , • • • , pn, • • • are an arbitrary set of orthogonal and normalized functions, and
a minimum ; the value of the integral is the corresponding eigenvalue El of the Hamiltonian H. It is also well known that this property leads to a powerful method of approximating, at least qualitatively, 4,1 and Er, by minimizing (1) among a restricted class of functions. No similarly simple minimum property exists for the higher eigenvalues . One can obtain the nth eigenvalue by minimizing (1) with the subsidiary condition that >G be orthogonal to ¢l, 42, y n-r• However, this procedure becomes progressively more cumbersome as n increases and besides it is of no use unless 4,1, 02, • • • , 4'n-1 are very exactly known. For problems of thermal equilibrium one is often concerned with the free energy, rather than with the individual eigenvalues . It is the purpose of this note to draw attention to a simple minimum property of the free energy which may be considered as a generalization of the variation principle for the lowest eigenvalue. The free energy has the following property: If v1, W2 , • • • , pn, • • • are an arbitrary set of orthogonal and normalized functions, and