The Nuclear Energy Density Functional Formalism

Abstract

The present document focuses on the theoretical foundations of the nuclear energy density functional (EDF) method. As such, it does not aim at reviewing the status of the field, at covering all possible ramifications of the approach or at presenting recent achievements and applications. The objective is to provide a modern account of the nuclear EDF formalism that is at variance with traditional presentations that rely, at one point or another, on a Hamiltonian-based picture. The latter is not general enough to encompass what the nuclear EDF method represents as of today. Specifically, the traditional Hamiltonian-based picture does not allow one to grasp the difficulties associated with the fact that currently available parametrizations of the energy kernel E[g′,g] at play in the method do not derive from a genuine Hamilton operator, would the latter be effective. The method is formulated from the outset through the most general multi-reference, i.e. beyond mean-field, implementation such that the single-reference, i.e. “mean-field”, derives as a particular case. As such, a key point of the presentation provided here is to demonstrate that the multi-reference EDF method can indeed be formulated in a mathematically meaningful fashion even if E[g′,g] does not derive from a genuine Hamilton operator. In particular, the restoration of symmetries can be entirely formulated without making any reference to a projected state, i.e. within a genuine EDF framework. However, and as is illustrated in the present document, a mathematically meaningful formulation does not guarantee that the formalism is sound from a physical standpoint. The price at which the latter can be enforced as well in the future is eventually alluded to.

Appendix: F-Functions

Kinetic densities are expressed in INM in terms of functions $F^{(0)}_{m}(I_{\tau},I_{\sigma},I_{\sigma\tau})$, $F^{(\tau)}_{m}(I_{\tau},I_{\sigma},I_{\sigma\tau})$, $F^{(\sigma)}_{m}(I_{\tau},I_{\sigma},I_{\sigma\tau})$ and $F^{(\sigma\tau)}_{m}(I_{\tau},I_{\sigma},I_{\sigma\tau})$ defined through [114]

$$\begin{aligned} F^{(0)}_{m} &\equiv\frac{1}{4} \bigl[ (1+I_\tau+I_\sigma+I_{\sigma\tau})^m + (1+I_\tau-I_\sigma-I_{\sigma\tau})^m \\ &\quad +(1-I_\tau+I_\sigma-I_{\sigma\tau})^m +(1-I_\tau-I_\sigma+I_{\sigma\tau})^m \bigr] , \end{aligned}$$

(7.90a)

$$\begin{aligned} F^{(\tau)}_{m}&\equiv \frac{1}{4} \bigl[ (1+I_\tau+I_\sigma+I_{\sigma\tau})^m + (1+I_\tau-I_\sigma-I_{\sigma\tau})^m \\ &\quad -(1-I_\tau+I_\sigma-I_{\sigma\tau})^m -(1-I_\tau-I_\sigma+I_{\sigma\tau})^m \bigr] , \end{aligned}$$

(7.90b)

$$\begin{aligned} F^{(\sigma)}_{m} &\equiv \frac{1}{4} \bigl[ (1+I_\tau+I_\sigma+I_{\sigma\tau})^m - (1+I_\tau-I_\sigma-I_{\sigma\tau})^m \\ &\quad +(1-I_\tau+I_\sigma-I_{\sigma\tau})^m -(1-I_\tau-I_\sigma+I_{\sigma\tau})^m \bigr] , \end{aligned}$$

(7.90c)

$$\begin{aligned} F^{(\sigma\tau)}_{m} &\equiv \frac{1}{4} \bigl[ (1+I_\tau+I_\sigma+I_{\sigma\tau})^m - (1+I_\tau-I_\sigma-I_{\sigma\tau})^m \\ &\quad -(1-I_\tau+I_\sigma-I_{\sigma\tau})^m +(1-I_\tau-I_\sigma+I_{\sigma\tau})^m \bigr] . \end{aligned}$$

(7.90d)

Their first derivatives with respect to spin, isospin and spin-isospin excesses are

$$\begin{aligned} \frac{\partial F^{(\tau)}_{m}}{\partial I_\tau} &= \frac{\partial F^{(\sigma)}_{m}}{\partial I_\sigma} = \frac{\partial F^{(\sigma \tau)}_{m}}{\partial I_{\sigma\tau}} = m F^{(0)}_{m-1} , \end{aligned}$$

(7.91a)

$$\begin{aligned} \frac{\partial F^{(0)}_{m}}{\partial I_\tau} &= \frac{\partial F^{(\sigma)}_{m}}{\partial I_{\sigma\tau}} = \frac{\partial F^{(\sigma\tau)}_{m}}{\partial I_\sigma} = m F^{(\tau)}_{m-1} , \end{aligned}$$

(7.91b)

$$\begin{aligned} \frac{\partial F^{(0)}_{m}}{\partial I_\sigma} &= \frac{\partial F^{(\tau)}_{m}}{\partial I_{\sigma\tau}} = \frac{\partial F^{(\sigma\tau)}_{m}}{\partial I_\tau} = m F^{(\sigma)}_{m-1} , \end{aligned}$$

(7.91c)

$$\begin{aligned} \frac{\partial F^{(0)}_{m}}{\partial I_{\sigma\tau}}& = \frac {\partial F^{(\tau)}_{m}}{\partial I_\sigma} = \frac{\partial F^{(\sigma)}_{m}}{\partial I_\tau} = m F^{(\sigma\tau)}_{m-1} , \end{aligned}$$

(7.91d)

while their second derivatives are

$$\begin{aligned} \frac{\partial^2 F^{(j)}_{m}}{\partial I_i^2} = m (m-1) F^{(j)}_{m-2} , \end{aligned}$$

(7.92)

for any i,j∈{0,τ,σ,στ}. Remarkable values are

$$\begin{aligned} F^{(0)}_{0}(I_\tau,I_\sigma,I_{\sigma\tau}) &=1 , \qquad F^{(i)}_{0}(I_\tau,I_\sigma,I_{\sigma\tau}) =0 , \end{aligned}$$

(7.93a)

$$\begin{aligned} F^{(0)}_{1}(I_\tau,I_\sigma,I_{\sigma\tau}) &=1 , \qquad F^{(i)}_{1}(I_\tau,I_\sigma,I_{\sigma\tau}) =I_i , \end{aligned}$$

(7.93b)

and

$$\begin{aligned} F^{(0)}_{m} (0,0,0) &= 1 , \end{aligned}$$

(7.94a)

$$\begin{aligned} F^{(i)}_{m} (0,0,0) &= 0 , \end{aligned}$$

(7.94b)

$$\begin{aligned} F^{(\tau)}_{m} (0,1,0) &= F^{(\tau)}_{m} (0,0,1) = 0 , \end{aligned}$$

(7.94c)

$$\begin{aligned} F^{(\sigma)}_{m} (1,0,0)& = F^{(\sigma)}_{m} (0,0,1) = 0 , \end{aligned}$$

(7.94d)

$$\begin{aligned} F^{(\sigma\tau)}_{m} (1,0,0) &= F^{(\sigma\tau)}_{m} (0,1,0) = 0 , \end{aligned}$$

(7.94e)

$$\begin{aligned} F^{(0)}_{m} (1,0,0) &= F^{(0)}_{m} (0,1,0) = F^{(0)}_{m} (0,0,1) = 2^{m-1} , \end{aligned}$$

(7.94f)

$$\begin{aligned} F^{(\tau)}_{m} (1,0,0) &= F^{(\sigma)}_{m} (0,1,0) = F^{(\sigma\tau )}_{m} (0,0,1) = 2^{m-1} , \end{aligned}$$

(7.94g)

$$\begin{aligned} F^{(0)}_{m} (1,1,1) &= F^{(i)}_{m} (1,1,1) = 4^{m-1} , \end{aligned}$$

(7.94h)

where i∈{τ,σ,στ}.