Project: s41467-024-45674-5 | Function: calc_F_el (cpp)

system: You are an expert C++ programmer implementing algorithms from scientific papers.
Complete the masked function based on the algorithm description.
Follow C++ best practices and maintain consistent code style.
Return **only** the complete C++ function code.

user: Complete this C++ function:

**DESCRIPTION:** Calculates the elastic free energy of liquid crystal using Landau-de Gennes theory, implementing the Frank elastic energy functional

**KEY TERMS:** {
  "landau_t": "reduced temperature parameter",
  "qijk": "Q-tensor components in real space",
  "qnlm": "Q-tensor components in spectral space",
  "trQ2": "trace of Q-tensor squared",
  "s0": "equilibrium scalar order parameter"
}

**CONTEXT CODE:** 
/**
 * @file   calculate_Fel.h
 * @author onedimension <onedimension@onedimension-PC>
 * @date   Mon Nov  3 21:28:52 2014
 * 
 * @brief  
 * 
 * 
 */
#ifndef _CALCULATE_FEL_H
#define _CALCULATE_FEL_H

using namespace std;

#include <iostream>
#include "global.h"
#include "Basis.h"

<think>We have given a C++ code snippet from the file calculate_Fel.h which says:

double calc_F_el(double *qnlm, double *qijk)
{
    std::cerr << "Function calc_F_el has been masked" << std::endl;
    throw std::runtime_error("Function implementation has been masked");
    return -1.0;
}
And we receive a description: "Complete this CPP function: Calculates the elastic free energy of liquid crystal using Landau-de Gennes theory, implementing the Frank elastic energy functional", with some key terms provided as follows:
{
  "landau_t": "reduced temperature parameter",
  "qijk": "Q-tensor components in real space",
  "qnlm": "Q-tensor components in spectral space",
  "trQ2": "trace of Q-tensor squared",
  "s0": "equilibrium scalar order parameter"
}

We are told to complete function "calc_F_el" code. We must complete the function that calculates the elastic free energy of liquid crystal using Landau-de Gennes theory.

We need to know the Frank elastic energy functional. However Landau-de Gennes is typically expressed in terms of the Q-tensor. Let us see: The theory is for nematic liquid crystals (LLC). The free energy has components: bulk free energy and elastic free energy. The elastic free energy in Landau-de Gennes theory is given by an elastic term:

E_elastic = (1/2) * K1 * (div Q)^2 + (1/2)*K2*(Q dot grad Q + grad Q.T dot Q)^2 + (1/2)*K3 (grad Q …)

However in many Landau-de Gennes formulations, the elastic part is usually E_elastic = 1/2 * (\sum_{ijkl} L_{ijkl} \partial_{m} Q_{ij}\partial_{m} Q_{kl}) which is often discretized or approximated. 

What is in our `calc_F_el`? The parameters include qnlm (spectral Q-tensor components and qijk (real space Q-tensor components). Possibly the algorithm uses quadrature in spectral space to compute gradients via derivatives as provided in the basis functions for spatial discretization. But this may be very complicated.

The code is incomplete. However, given the "KEY TERMS", we have mention of:

landau_t: reduced temperature parameter
qijk: Q-tensor components in real space
qnlm: Q-tensor components in spectral space
trQ2: trace of Q-tensor squared
s0: equilibrium scalar order parameter

Thus possibly the free energy expression (elastic) to be computed is derived from the Landau-de Gennes theory free energy functional:

F = F_bulk + F_elastic
F_bulk = a1 tr(Q^2) + a2 tr(Q^2)^2 + a3 Q^3, where a1, a2, a3 are constants with a1 being related to temperature and s0. But we want elastic free energy functional specifically, the Frank elastic free energy. It might be related to the derivative of the order parameter tensor Q. Landau-deGennes elastic free energy is usually written as: F_elastic = 1/2 * (L1 (grad n)^2 + L2 Q: grad n: grad n) etc. But in our context, what do we want?

Alternatively, one well-known approach is:

F_elastic = 1/(2) * K * sum_{i,j} (partial_i Q_{jk} partial_i Q_{kl}). Actually then the sum over i (spatial indices) and j,k. The constants might be L^4 and gradient squared depending on how many terms. In Frank elastic theory, there are three terms: splay energy (1/2)*K_11*(div n)^2, twist energy (1/2)*K_22*(n.x curl n)**2, bend energy (1/2)*K_33*(curl n - (div n) n)**2.

But the "calc_F_el" function signature uses two separate arrays: qnlm: Q-tensor components in spectral space, and qijk: Q-tensor components in real space. Possibly these represent: qijk is a discretization of "real space" values and qnlm is the Fourier transform? Possibly the code uses qnlm for derivatives in spectral domain.

Given some project structure. Although we have a lot of missing details.

Alternatively, maybe we can assume a simplified integration over the domain areas. The code likely calculates the elastic energy functional as the sum over contributions from each element in the mesh. For instance, energy = sum over i { [some expression] }.

I should literally complete the function by calculating the elastic free energy by summing over terms. But which variables do we have? I only have pointers to doubles representing arrays qnlm and qijk. Likely their lengths and arrangement are known. Possibly qijk holds 9 values representing the Q-tensor (which is 3x3 symmetric, so there are 6 degrees of freedom), but potentially allowing for degeneracy. Similarly qnlm is a representation on a spectral basis with corresponding coefficients multiplied by basis functions giving q tensor in the lattice geometry. But we don't have information on their dimensions. We have key terms: "trQ2" = "trace of Q-tensor squared" possibly will be computed from real qijk. "s0: equilibrium scalar order parameter" probably requires computing scalar value Q = tr(Q^T Q)? but Q is usually not symmetric positive but sometimes can be diagonalized.

Given connection with elasticity, maybe we should differentiate the Q tensor. But the function should "calculate the elastic free energy of liquid crystal using Landau-de Gennes theory, implementing the Frank elastic energy functional". The theory has a bulk part involving temperature and Q. But the description says "elastic free energy", so we should exclude the bulk free energy and only the elastic part.

I found many equations online: "The Landau-de Gennes elastic free energy density is given by F_el = (1/2) L1 (nabla Q)^2 + (1/2) L2 tr((nabla Q)^2) + ...", but specifics vary with elasticity. Alternatively, a common model is:
F_elastic = K/2 * [ (div n)^2 + (curl n)^2 + ...]. But wait, this is the Frank elastic free energy, where Q = QI - (3/2)(s) n_i n_j, for uniaxial nematic, n is the unit vector representing the director and s is scalar order parameter. But then the invariants include gradients of n. But then the Landau-deGennes approach described earlier uses Q for order parameter. Frank elastic energy is sometimes given by:

For a given coordinate direction, F_elastic = (1/2)*K1 (div n)^2 + (1/2)*K2 (n·curl n)^2 + (1/2)*K3(curl n - div n n)^2. 

But no information about derivatives of n is explicitly provided.

Alternatively, in Landau-de Gennes formulation, the elastic free energy is represented as:

F_elastic = 1/2 * L (sum_{i,j} (∂_i Q_{jk} ∂_i Q_{kl})). 
Maybe using some form of summation over index i.

It might be that in the code, the elastic energy is computed by differentiating the spectral representation of q. Because qnlm contains spectral coefficients for the basis functions. And then derivatives in spectral space are just multiplication by factors.

For example, if we assume Fourier space and periodic boundaries, then derivative of a Fourier mode includes multiplication by i * k. And then the elastic free energy becomes:

F_elastic = (1/2) ∑_{lm} k^2 |q_{ij}(k)|^2 <Φ_{ij}(r) Φ_{ij}(r')> coefficient factor. Usually, the elastic free energy might include summation terms like ∑_{q} K |q|² |Φ_{ij}(k)|².

But without more detail, the function might be just a stub that sums over certain indices.

Maybe the function could act as follows (pseudocode):

Start by computing the squared norm of q via tr(Q^2-) and subtract out the equilibrium part scaled by s0. Then add an elastic term that involves derivatives of q using qnlm in spectral space.

I recall that Frank elastic free energy energy density for a nematic liquid crystal is: 
F_el ~ ∫ dr { [ (K1/2)(∇.n)^2 + K2/2 (n· (∇ × n) - ω )^2 + K3/2 (∇×n - (∇·n) n - ω n)^2 }.

But in Landau-de Gennes formulation, there is a similar expression: F_elastic = (1/2) L1 (Q: ∇)^2 + (1/2) L2 (∇ Q)^2 + (1/2) L3 (∇^2 Q)^2.

I need to decide on a formula. Perhaps the simplest is to use a gradient squared term. Alternatively, maybe we can assume:

F_el = Σ_{(i, j)} [ (1/2) * (ε^(i) + constant) ] where ε^(i) is something like the real space energy density where the derivatives of Q are computed in spectral space coefficients qnlm.

Given the function signature, which is "double calc_F_el(double* qnlm, double* qijk)", probably do something like: 
compute F_el = F_bulk + F_elastic, but because description says "elastic free energy", so just the F_elastic term.

The code might approach a discrete domain integration:
- The free energy is computed by a sum over indices i for each spatial mode. Possibly using discrete finite difference derivative on qijk.
- The derivative of the tensor might be computed from the spectral representation qnlm (which might contain wavevector modes) by multiplying with squared wave vector magnitude.

Because we have qnlm and qijk arrays, maybe they are the same but in different representations.

Maybe the function intended to sum up differences between qijk's at different positions, for instance: 
energy += (qijk[n] - qijk[m])^2 weighted by corresponding grid factors. But we are given only pointers without grid sizes.

Maybe we assume some grid size N_spacepoints. But not provided in the code. Also, qnlm is spectral space array, typically the Fourier transform of qijk. But then the derivatives in spectral space are easier to compute for an operator that gives multiplication by squared magnitude of wave vector times.

I search for "calc_F_el" in original Landau-de Gennes code. Possibly we can compute the elastic free energy by: sum over q indices: (landau_t * trace(Q^2)?) However, the given key term "trQ2: trace of Q-tensor squared" is used. Possibly: F_bulk = s0*trQ2 maybe uses equilibrium scalar order parameter s0.


Complete the function implementation replacing "throw std::runtime_error("Not implemented");".