Time-dependent pair density from the principle of minimum Fisher information

Abstract

The Euler equation for the time-dependent pair density is derived from the principle of minimum Fisher information. The same Euler equation is also derived from the recently introduced time-dependent pair density functional theory. The concept of steric effect to electron pairs is proposed and the steric pair energy is defined as the Weizsäcker kinetic energy of electron pairs.

Appendix: derivation of the Euler equation of the time-dependent pair density

The EE can be derived from AE (8). The functions χ_I(x₁,x₂,t) can be written in the form

$$\begin{array}{@{}rcl@{}} \chi_{I}(\mathbf{x}_{1},\mathbf{x}_{2},t) &=& n(\mathbf{q},t)^{1/2} \phi_{I}(\mathbf{q},t) \gamma(\sigma_{1},\sigma_{2}) exp\\&&\times \left[ i \left( S(\mathbf{q},t)-S_{I}(t) \right) \right] , \end{array} $$

(29)

where S and S_I are real, while ϕ_I are generally complex functions. S_I are functions of t only. γ is the normalized, antisymmetric spin function of the electron pair with opposite spins. (Note that in the present version of the TDPDFT, the pairs contain electrons with opposite spins.) Substituting (29) into the pair density (7) we are led to

$$\begin{array}{@{}rcl@{}} 1 = (N-1) \sum\limits_{I = 1}^{M} |\phi_{I}(\mathbf{q},t)|^{2} .\end{array} $$

(30)

The derivative of Eq. 30 with respect to the time is

$$\begin{array}{@{}rcl@{}} 0 = \sum\limits_{I} \left[ \phi_{I}^{*}\frac{\partial \phi_{I}}{\partial t} + \phi_{I}\frac{\partial \phi_{I}^{*}}{\partial t} \right] , \end{array} $$

(31)

while the first and second spatial derivatives take the form

$$\begin{array}{@{}rcl@{}} 0 = \sum\limits_{I} \left[ \phi_{I}^{*} \nabla_{\mathbf{q}}\phi_{I} + \phi_{I}\nabla_{\mathbf{q}}\phi_{I}^{*} \right] , \end{array} $$

(32)

$$\begin{array}{@{}rcl@{}} 0 = \sum\limits_{I} \left[ \phi_{I}^{*}\nabla^{2}_{\mathbf{q}}\phi_{I} + \phi_{I}\nabla_{\mathbf{q}}^{2}\phi_{I}^{*} + 2 |\nabla_{\mathbf{q}}\phi_{I}|^{2} \right] . \end{array} $$

(33)

Substituting (29) into the AE (8), multiplying (8) with $\phi _{I}^{*}$, summing over all I and spins and then adding the complex conjugate we arrive at the equation:

$$\begin{array}{@{}rcl@{}} &&\frac18 \left( \frac{\nabla_{\mathbf{q}}n}{n}\right)^{2} - \frac14 \frac{\nabla^{2}_{\mathbf{q}}n}{n} + \frac12 (\nabla_{\mathbf{q}}S)^{2} + v_{eff} \\ &=&\frac12 i\left( \nabla^{2}_{\mathbf{q}}S + \nabla_{\mathbf{q}}S \frac{\nabla_{\mathbf{q}}n}{n} + \frac{1}{n} \frac{\partial n}{\partial t}\right) \\ &&+ \frac{N-1}{4} \sum\limits_{I} \left[ \phi_{I}^{*}\nabla^{2}_{\mathbf{q}}\phi_{I} + \phi_{I}\nabla^{2}_{\mathbf{q}}\phi_{I}^{*} \right] - \frac{\partial S}{\partial t} \\&&+ (N-1) \sum\limits_{I}\frac{\partial S_{I}}{\partial t} |\phi_{I}|^{2} . \end{array} $$

(34)

To improve transparency in Eq. 34, the arguments of the functions are not shown. In the derivation (7), (30), (31), (32) and (33) were utilized. Substituting functions χ_I (29) into the current density of AS (10) we are led to

$$\begin{array}{@{}rcl@{}} \mathbf{j}^{0} = n \nabla_{\mathbf{q}}S . \end{array} $$

(35)

Therefore CE (2) takes the form

$$\begin{array}{@{}rcl@{}} \frac{\partial n}{\partial t} + \nabla_{\mathbf{q}}n \nabla_{\mathbf{q}}S + n \nabla^{2}_{\mathbf{q}}S = 0 , \end{array} $$

(36)

that also can be written in the form of Eq. 22. We can observe that the imaginary part in the right-hand side of Eq. 34 disappears because of CE (36). Consequently, Eq. 34 takes the form

$$\begin{array}{@{}rcl@{}} \frac18 \left( \frac{\nabla_{\mathbf{q}}n}{n}\right)^{2} &-& \frac14 \frac{\nabla^{2}_{\mathbf{q}}n}{n} + {\tilde v}_{P} + \frac12 (\nabla_{\mathbf{q}}S)^{2} + v_{eff} \\&+& \frac{\partial S}{\partial t} = 0 , \end{array} $$

(37)

where

$$\begin{array}{@{}rcl@{}} {\tilde v}_{P} &=& - \frac{N-1}{4} \sum\limits_{I} \left[ \phi_{I}^{*}\nabla^{2}_{\mathbf{q}}\phi_{I} + \phi_{I}\nabla^{2}_{\mathbf{q}}\phi_{I}^{*} \right] \\&&- (N-1) \sum\limits_{I}\frac{\partial S_{I}}{\partial t} |\phi_{I}|^{2} \\ &=& \frac{N-1}{2} \sum\limits_{I} |\nabla_{\mathbf{q}}\phi_{I}|^{2} - (N-1) \sum\limits_{I}\frac{\partial S_{I}}{\partial t} |\phi_{I}|^{2} . \end{array} $$

(38)

Utilizing (9) and (20)

$$\begin{array}{@{}rcl@{}} {\tilde v}_{P} = v_{P} - v_{k} . \end{array} $$

(39)

Finally, we turn to the special case of time-independent external potential. We proved earlier [29,30,31] that in the ground state all auxiliary functions χ_I = χ₁ are the same and the pair density satisfies EE

$$\begin{array}{@{}rcl@{}} \frac18 \left( \frac{\nabla_{\mathbf{q}}n}{n}\right)^{2} - \frac14 \frac{\nabla^{2}_{\mathbf{q}}n}{n} + v_{eff} = \mu , \end{array} $$

(40)

where $\mu = {E_{0}^{N}} - E_{0}^{N-2}$ is the “pair ionization energy”, that is, the energy needed to remove an electron pair from the system. (${E_{0}^{N}}$ and $E_{0}^{N-2}$ are the ground-state energies of the N and N − 2 electron systems, respectively.)