This work studies in detail the possibility of defining a one-to-one mapping from charge densities as obtained by the time-dependent Schrödinger equation to external potentials. Such a mapping is provided by the Runge-Gross theorem and lies at the very core of time-dependent density functional theory. After introducing the necessary mathematical concepts, the usual mapping "there" - from potentials to wave functions as solutions to the Schrödinger equation - is revisited paying special attention to Sobolev regularity. This is scrutinised further when the question of functional differentiability of the solution with respect to the potential arises, a concept related to linear response theory. Finally, after a brief introduction to general density functional theory, the mapping "back again" - from densities to potentials thereby inverting the Schrödinger equation for a fixed initial state - is defined. Apart from utilising the original Runge-Gross proof this is achieved through a fixed-point procedure. Both approaches give rise to mathematical issues, previously unresolved, which however could be dealt with to some extent within the framework at hand.
A generalization of normal ordering and of Wick’s theorem with respect to an arbitrary reference function Φ as some generalized “physical vacuum” is formulated in a different (but essentially equivalent) way than that suggested previously by one of the present authors. Guiding principles are that normal order operators with respect to any reference state must be expressible as linear combinations of those with respect to the genuine vacuum, that the vacuum expectation value of a normal order operator must vanish (with respect to the vacuum to which it is in normal order), and that the well-known formalism for a single Slater determinant as physical vacuum must be contained as a special case. The derivation is largely based on the concepts of “Quantum Chemistry in Fock space,” which means that particle-number-conserving operators (excitation operators) play a central role. Nevertheless, the contraction rules in the frame of a generalized Wick theorem are derived, that hold for non-particle-number-conserving operators as well. The contraction rules are formulated and illustrated in terms of diagrams. The contractions involve the “residual n-particle density matrices” λ, which are the irreducible (non-factorizable) parts of the conventional n-particle density matrices γ, in the sense of a cumulant expansion for the density. A spinfree formulation is presented as well. The expression of the Hamiltonian in normal order with respect to a multiconfiguration reference function leads to a natural definition of a generalized Fock operator. MC-SCF-theory is easily worked out in this context. The paper concludes with a discussion of the excited configurations and the first-order interacting space, that underlies a perturbative coupled cluster type correction to the MCSCF function for an arbitrary reference function, and with general implications of the new formalism, that is related to “internally contracted multireference configuration interaction.” The present generalization of normal ordering is not only valid for arbitrary reference functions, but also if the reference state is an ensemble state.
The formula for differentiation of functional $A[\ensuremath{\rho}]$ with respect to $\ensuremath{\rho}(\mathbf{r})$ while keeping the normalization $\ensuremath{\int}\ensuremath{\rho}(\mathbf{r})d\mathbf{r}$ of $\ensuremath{\rho}(\mathbf{r})$ fixed is derived, and basic properties arising from it are concerned. The results are then generalized for time-dependent theories. One of the essential consequences of normalization conservation for functional differentiation, namely, that it ruins the symmetry in r of multiple derivatives, is shown to give the resolution of the causality paradox of response functions in time-dependent density-functional theory. The formula for the differentiation of functionals $A[\ensuremath{\rho}]$ that conserves the shape of $\ensuremath{\rho}(\mathbf{r})$ is also presented.
The study of the change of the Schroedinger wave functionals on space-like surfaces as the surface changes is formulated simply. In this formulation, we let
S be any space-like surface with the curvilinear coordinates
u, introduce
〈q(u)S∣〉 as the Schroedinger functional on it, with the nature of the functions
q(u) not specified for the moment, and write the Schroedinger equation as $\ensuremath{\hbar}i{〈q(u),{S}^{\ensuremath{'}}|〉\ensuremath{-}〈q(u),S|〉}=J〈q(u),S|〉$ where ${S}^{\ensuremath{'}}$ is a surface slightly different from
S, and
J is an operator operating on
q(u) and depending on
S, ${S}^{\ensuremath{'}}$, and the choice of their coordinates
u. Obviously, the only necessary conditions for
J is that
J is Hermitian and that (1) is integrable. To get a theory resembling the existing ones, we require further that if we construct the expectation value of a certain field quantity at a point
P from
〈q(u),S∣〉,
S being a surface passing through
P, the expectation value is independent of choice of
S and its coordinates
u, and satisfies certain differential equations.When these differential equations follow from a Langrangian principle, an expression for
J was effectively given by Weiss. His work is given here in a complete form and the proof of the satisfaction of all requirements completed.Finally, by a simple transformation on the wave functions and the observables, we deduce from the Weiss's formalism the Tomonaga's formalism. As a consequence, it is pointed out that the Tomonaga's formalism will encounter essentially the same difficulties as the orthodox Heisenberg-Pauli formalism.
Within the framework of zero-temperature Fock-space density-functional theory (DFT), we prove that the G\^ateaux functional derivative of the universal density functional, ${\ensuremath{\delta}{F}^{\ensuremath{\lambda}}[\ensuremath{\rho}]∕\ensuremath{\delta}\ensuremath{\rho}(\mathbit{r})\ensuremath{\mid}}_{\ensuremath{\rho}={\ensuremath{\rho}}_{0}}$, at ground-state densities with arbitrary normalizations $(⟨{\ensuremath{\rho}}_{0}(\mathbit{r})⟩=n∊{\mathcal{R}}_{+})$ and an electron-electron interaction strength $\ensuremath{\lambda}$, is uniquely defined, but is discontinuous when the number of electrons
n becomes an integer, thus providing a mathematically rigorous confirmation for the ``derivative discontinuity'' initially discovered by Perdew et al. [Phys. Rev. Lett. 49, 1691 (1982)]. However, the functional derivative of the exchange-correlation functional is continuous with respect to the number of electrons in Fock space; i.e., there is no ``derivative discontinuity'' for the exchange-correlation functional at an integer electron number. For a ground-state density ${\ensuremath{\rho}}_{0,n}^{v,\ensuremath{\lambda}}(\mathbit{r})$ of an external potential $v(\mathbit{r})$, we show that ${\ensuremath{\delta}{F}^{\ensuremath{\lambda}}[\ensuremath{\rho}]∕\ensuremath{\delta}\ensuremath{\rho}(\mathbit{r})\ensuremath{\mid}}_{\ensuremath{\rho}={\ensuremath{\rho}}_{0,n}^{v,\ensuremath{\lambda}}}={\ensuremath{\mu}}_{\mathrm{SM}}^{n}\ensuremath{-}v(\mathbit{r})$, where the constant ${\ensuremath{\mu}}_{\mathrm{SM}}^{n}$ is given by the following chain of dependences: ${\ensuremath{\rho}}_{0,n}^{v,\ensuremath{\lambda}}(\mathbit{r})\ensuremath{\mapsto}[v]\ensuremath{\mapsto}{E}_{0}^{v,\ensuremath{\lambda}}(n)\ensuremath{\mapsto}{\ensuremath{\mu}}_{\mathrm{SM}}^{n}={\ensuremath{\partial}{E}_{0}^{v,\ensuremath{\lambda}}(k)∕\ensuremath{\partial}k\ensuremath{\mid}}_{k=n}$. Here
[v] is the class of the external potential $v(\mathbit{r})$ up to a real constant, and ${\ensuremath{\mu}}_{\mathrm{SM}}^{n}$ is the chemical potential defined according to statistical mechanics. At an integer electron number
N, we find that there is no freedom of adding an arbitrary constant to the value of the chemical potential ${\ensuremath{\mu}}_{\mathrm{SM}}^{N}$, whose exact value is generally not the popular preference of the negative of Mulliken's electronegativity, $\ensuremath{-}\frac{1}{2}(I+A)$, where
I and
A are the first ionization potential and the first electron affinity, respectively. In addition, for any external potential converging to the same constant at infinity in all directions, we resolve that ${\ensuremath{\mu}}_{\mathrm{SM}}^{N}=\ensuremath{-}I$. Finally, the equality ${\ensuremath{\mu}}_{\mathrm{DFT}}={\ensuremath{\mu}}_{\mathrm{SM}}^{n}$ is rigorously derived via an alternative route, where ${\ensuremath{\mu}}_{\mathrm{DFT}}$ is the Lagrangian multiplier used to constrain the normalization of the density in the traditional DFT approach.
The problem of expanding a density operator $\ensuremath{\rho}$ in forms that simplify the evaluation of important classes of quantum-mechanical expectation values is studied. The weight function $P(\ensuremath{\alpha})$ of the
P representation, the Wigner distribution $W(\ensuremath{\alpha})$, and the function $〈\ensuremath{\alpha}|\ensuremath{\rho}|\ensuremath{\alpha}〉$, where $|\ensuremath{\alpha}〉$ is a coherent state, are discussed from a unified point of view. Each of these quasiprobability distributions is examined as the expectation value of a Hermitian operator, as the weight function of an integral representation for the density operator and as the function associated with the density operator by one of the operator-function correspondences defined in the preceding paper. The weight function $P(\ensuremath{\alpha})$ of the
P representation is shown to be the expectation value of a Hermitian operator all of whose eigenvalues are infinite. The existence of the function $P(\ensuremath{\alpha})$ as an infinitely differentiable function is found to be equivalent to the existence of a well-defined antinormally ordered series expansion for the density operator in powers of the annihilation and creation operators
a and ${a}^{\ifmmode\dagger\else\textdagger\fi{}}$. The Wigner distribution $W(\ensuremath{\alpha})$ is shown to be a continuous, uniformly bounded, square-integrable weight function for an integral expansion of the density operator and to be the function associated with the symmetrically ordered power-series expansion of the density operator. The function $〈\ensuremath{\alpha}|\ensuremath{\rho}|\ensuremath{\alpha}〉$, which is infinitely differentiable, corresponds to the normally ordered form of the density operator. Its use as a weight function in an integral expansion of the density operator is shown to involve singularities that are closely related to those which occur in the
P representation. A parametrized integral expansion of the density operator is introduced in which the weight function $W(\ensuremath{\alpha},s)$ may be identified with the weight function $P(\ensuremath{\alpha})$ of the
P representation, with the Wigner distribution $W(\ensuremath{\alpha})$, and with the function $〈\ensuremath{\alpha}|\ensuremath{\rho}|\ensuremath{\alpha}〉$ when the order parameter
s assumes the values $s=+1, 0, \ensuremath{-}1$, respectively. The function $W(\ensuremath{\alpha},s)$ is shown to be the expectation value of the ordered operator analog of the $\ensuremath{\delta}$ function defined in the preceding paper. This operator is in the trace class for
Res<0, has bounded eigenvalues for
Res=0, and has infinite eigenvalues for
s=1. Marked changes in the properties of the quasiprobability distribution $W(\ensuremath{\alpha},s)$ are exhibited as the order parameter
s is varied continuously from $s=\ensuremath{-}1$, corresponding to the function $〈\ensuremath{\alpha}|\ensuremath{\rho}|\ensuremath{\alpha}〉$, to
s=+1, corresponding to the function $P(\ensuremath{\alpha})$. Methods for constructing these functions and for using them to compute expectation values are presented and illustrated with several examples. One of these examples leads to a physical characterization of the density operators for which the
P representation is appropriate.