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Regular article Equation for the direct determination of the density matrix: Time-dependent density equation and perturbation theory*

Hiroshi Nakatsuji

Published Jan 1, 1999• 0 citations
OpenAlex

Abstract

The density equation proposed previously for the direct determination of the density matrix, i.e. for the wave mechanics without wave, is extended to a time- dependent case. The time-dependent density equation has been shown to be equivalent to the time-dependent Schrodinger equation so long as the density matrix, included as a self-contained variable, is N-representable. Formally, it is obtainable from the previous time- independent equation by replacing the energy E with ih @ @t . The perturbation theory formulas for the density equation have also been given for both the time- dependent and time-independent cases.

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A nonvariational equation, called the density equation, is proposed for the direct determination of the density matrix without using a wave function. It is connected with the Schr\"odinger equation by a necessary and sufficient theorem. The equation for the lowest order depends explicitly only on the fourth-order density matrix ${\ensuremath{\Gamma}}^{(4)}$ (or ${\ensuremath{\Gamma}}^{(3)}$ in a special case) and not on the higher-order density matrices. The equation always gives the density matrix and the associated energy which coincide with those obtained indirectly from the Schr\"odinger equation. This is true even if we solve the equation only with the known and tractable $N$-representability conditions, although in such a case some unphysical solution may also occur in the non-$N$-representable space. The equation is applicable to both fermion and boson systems, and to both ground and excited states. In contrast to the Schr\"odinger equation, the labor of solving the equation does not increase when the number of particles of the system is increased. When we have the Hartree-Fock solution, the equation is transformed such that the correlated density matrix and the correlation energy are the direct solution. The correlated density equation thus obtained is suitable for the study of electron correlations.

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The density equation proposed previously for the direct determination of the density matrix, i.e. for the wave mechanics without wave, is extended to a time- dependent case. The time-dependent density equation has been shown to be equivalent to the time-dependent Schrodinger equation so long as the density matrix, included as a self-contained variable, is N-representable. Formally, it is obtainable from the previous time- independent equation by replacing the energy E with ih @ @t . The perturbation theory formulas for the density equation have also been given for both the time- dependent and time-independent cases.
A nonvariational equation, called the density equation, is proposed for the direct determination of the density matrix without using a wave function. It is connected with the Schr\"odinger equation by a necessary and sufficient theorem. The equation for the lowest order depends explicitly only on the fourth-order density matrix ${\ensuremath{\Gamma}}^{(4)}$ (or ${\ensuremath{\Gamma}}^{(3)}$ in a special case) and not on the higher-order density matrices. The equation always gives the density matrix and the associated energy which coincide with those obtained indirectly from the Schr\"odinger equation. This is true even if we solve the equation only with the known and tractable NNN-representability conditions, although in such a case some unphysical solution may also occur in the non-NNN-representable space. The equation is applicable to both fermion and boson systems, and to both ground and excited states. In contrast to the Schr\"odinger equation, the labor of solving the equation does not increase when the number of particles of the system is increased. When we have the Hartree-Fock solution, the equation is transformed such that the correlated density matrix and the correlation energy are the direct solution. The correlated density equation thus obtained is suitable for the study of electron correlations.
For pt.I see ibid. vol.16, p.3987-4000, (1983). The determinantal formalism for solving the time-dependent Schrodinger equation of the density matrix has been developed, in I, assuming a constant external perturbation. The important case of an harmonic perturbation is presented here. Detailed expressions are derived for diagonal and nondiagonal elements of the response, and the trace conservation theorem is shown to be satisfied at all times and any order. The results are again applied to the simple illustrative quantum system considered in I.
In all the applications of the formalism of quantum theory, we have considered mainly time-independent potential functions, V ( r , t ) = V ( r ). Such potentials made it easier to solve exactly the Schrodinger's equation or by solving the Schrodinger's equation approximately by using perturbation theory. The solution of the Schrodinger equation for time-independent potentials is obtained by using method of separable variables. The solution of time-independent Schrodinger gives stationary states, and the time dependence of the wave function is obtained by adding a phase factor to the stationary state. The schematic that follows describes such an approach ( Figure 7.1 ).
Basic concepts of quantum mechanics: Schroedinger equation; Dirac notation; the energy representation; expectation value; Hermite operators; coherent superposition of states and motion in the quantum world; perturbation Hamiltonian. Time-dependent perturbation theory: harmonic perturbation. Transition rate: Fermi’s golden rule. The density matrix; pure and mixed states. Temporal dependence of the density operator: von Neuman equation. Randomizing Hamiltonian. Longitudinal and transverse relaxation times. Density matrix and entropy.
The previous time-dependent variation-perturbation theory, which is derived from the conventional time-dependent perturbation equations, contains two inconvenient features. When the perturbation becomes static, the wave function so obtained does not satisfy the time-dependent Schr\"odinger equation, and the perturbation equations cannot be reduced to the ordinary time-independent perturbation equations. To remove these difficulties, we introduce a perturbed time-dependent phase factor in deriving a new set of perturbation equations. We then construct the variation-perturbation functionals as in the previous theory; the perturbed wave functions are solved by minimizing these functionals. The same method is also applied to many-electron systems, using the coupled and uncoupled Hartree-Fock scheme.